CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Mathematical optimization - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"dd299ca0-789f-4225-8ac7-e60c58c1a30a","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Mathematical_optimization","wgTitle":"Mathematical optimization","wgCurRevisionId":1257438818,"wgRevisionId":1257438818,"wgArticleId":52033,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description matches Wikidata","Commons category link from Wikidata","Mathematical optimization","Operations research","Mathematical and quantitative methods (economics)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Mathematical_optimization","wgRelevantArticleId":52033,"wgIsProbablyEditable":true, "wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q141495","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics": true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar", "ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/1200px-Max_paraboloid.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="960"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/800px-Max_paraboloid.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="640"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/640px-Max_paraboloid.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="512"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Mathematical optimization - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Mathematical_optimization"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Mathematical_optimization&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Mathematical_optimization"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Mathematical_optimization rootpage-Mathematical_optimization skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Mathematical+optimization" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Mathematical+optimization" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Mathematical+optimization" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Mathematical+optimization" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Optimization_problems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Optimization_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Optimization problems</span> </div> </a> <ul id="toc-Optimization_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Notation</span> </div> </a> <button aria-controls="toc-Notation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Notation subsection</span> </button> <ul id="toc-Notation-sublist" class="vector-toc-list"> <li id="toc-Minimum_and_maximum_value_of_a_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimum_and_maximum_value_of_a_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Minimum and maximum value of a function</span> </div> </a> <ul id="toc-Minimum_and_maximum_value_of_a_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Optimal_input_arguments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimal_input_arguments"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Optimal input arguments</span> </div> </a> <ul id="toc-Optimal_input_arguments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Major_subfields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Major_subfields"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Major subfields</span> </div> </a> <button aria-controls="toc-Major_subfields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Major subfields subsection</span> </button> <ul id="toc-Major_subfields-sublist" class="vector-toc-list"> <li id="toc-Multi-objective_optimization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multi-objective_optimization"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Multi-objective optimization</span> </div> </a> <ul id="toc-Multi-objective_optimization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multi-modal_or_global_optimization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multi-modal_or_global_optimization"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Multi-modal or global optimization</span> </div> </a> <ul id="toc-Multi-modal_or_global_optimization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Classification_of_critical_points_and_extrema" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Classification_of_critical_points_and_extrema"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Classification of critical points and extrema</span> </div> </a> <button aria-controls="toc-Classification_of_critical_points_and_extrema-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Classification of critical points and extrema subsection</span> </button> <ul id="toc-Classification_of_critical_points_and_extrema-sublist" class="vector-toc-list"> <li id="toc-Feasibility_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feasibility_problem"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Feasibility problem</span> </div> </a> <ul id="toc-Feasibility_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Existence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Existence"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Existence</span> </div> </a> <ul id="toc-Existence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Necessary_conditions_for_optimality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Necessary_conditions_for_optimality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Necessary conditions for optimality</span> </div> </a> <ul id="toc-Necessary_conditions_for_optimality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sufficient_conditions_for_optimality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sufficient_conditions_for_optimality"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Sufficient conditions for optimality</span> </div> </a> <ul id="toc-Sufficient_conditions_for_optimality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sensitivity_and_continuity_of_optima" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sensitivity_and_continuity_of_optima"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Sensitivity and continuity of optima</span> </div> </a> <ul id="toc-Sensitivity_and_continuity_of_optima-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculus_of_optimization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculus_of_optimization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Calculus of optimization</span> </div> </a> <ul id="toc-Calculus_of_optimization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Global_convergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Global_convergence"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Global convergence</span> </div> </a> <ul id="toc-Global_convergence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Computational_optimization_techniques" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Computational_optimization_techniques"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Computational optimization techniques</span> </div> </a> <button aria-controls="toc-Computational_optimization_techniques-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computational optimization techniques subsection</span> </button> <ul id="toc-Computational_optimization_techniques-sublist" class="vector-toc-list"> <li id="toc-Optimization_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optimization_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Optimization algorithms</span> </div> </a> <ul id="toc-Optimization_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iterative_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iterative_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Iterative methods</span> </div> </a> <ul id="toc-Iterative_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Heuristics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heuristics"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Heuristics</span> </div> </a> <ul id="toc-Heuristics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Mechanics</span> </div> </a> <ul id="toc-Mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Economics_and_finance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Economics_and_finance"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Economics and finance</span> </div> </a> <ul id="toc-Economics_and_finance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrical_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrical_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Electrical engineering</span> </div> </a> <ul id="toc-Electrical_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Civil_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Civil_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.4</span> <span>Civil engineering</span> </div> </a> <ul id="toc-Civil_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operations_research" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operations_research"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.5</span> <span>Operations research</span> </div> </a> <ul id="toc-Operations_research-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Control_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Control_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.6</span> <span>Control engineering</span> </div> </a> <ul id="toc-Control_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geophysics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geophysics"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.7</span> <span>Geophysics</span> </div> </a> <ul id="toc-Geophysics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molecular_modeling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Molecular_modeling"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.8</span> <span>Molecular modeling</span> </div> </a> <ul id="toc-Molecular_modeling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_systems_biology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computational_systems_biology"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.9</span> <span>Computational systems biology</span> </div> </a> <ul id="toc-Computational_systems_biology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Machine_learning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Machine_learning"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.10</span> <span>Machine learning</span> </div> </a> <ul id="toc-Machine_learning-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Solvers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Solvers"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Solvers</span> </div> </a> <ul id="toc-Solvers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mathematical optimization</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 55 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-55" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">55 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D9%85%D8%AB%D8%A7%D9%84_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="استمثال (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="استمثال (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-hyw mw-list-item"><a href="https://hyw.wikipedia.org/wiki/%D5%88%D6%82%D5%BD%D5%B8%D5%B2%D5%A1%D5%AF%D5%A1%D5%B6_%D4%BE%D6%80%D5%A1%D5%A3%D6%80%D5%A1%D6%82%D5%B8%D6%80%D5%B8%D6%82%D5%B4" title="Ուսողական Ծրագրաւորում – Western Armenian" lang="hyw" hreflang="hyw" data-title="Ուսողական Ծրագրաւորում" data-language-autonym="Արեւմտահայերէն" data-language-local-name="Western Armenian" class="interlanguage-link-target"><span>Արեւմտահայերէն</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Optimalla%C5%9Fd%C4%B1rma" title="Optimallaşdırma – Azerbaijani" lang="az" hreflang="az" data-title="Optimallaşdırma" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%95%E0%A6%BE%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%A4%E0%A6%AE%E0%A6%95%E0%A6%B0%E0%A6%A3" title="গাণিতিক কাম্যতমকরণ – Bangla" lang="bn" hreflang="bn" data-title="গাণিতিক কাম্যতমকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9E%D0%BF%D1%82%D0%B8%D0%BC%D0%B0%D0%BB%D0%BB%D3%99%D1%88%D1%82%D0%B5%D1%80%D0%B5%D2%AF_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Оптималләштереү (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Оптималләштереү (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BE%D0%BF%D1%82%D0%B8%D0%BC%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F" title="Математическа оптимизация – Bulgarian" lang="bg" hreflang="bg" data-title="Математическа оптимизация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Optimitzaci%C3%B3_matem%C3%A0tica" title="Optimització matemàtica – Catalan" lang="ca" hreflang="ca" data-title="Optimització matemàtica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Optimalizace_(matematika)" title="Optimalizace (matematika) – Czech" lang="cs" hreflang="cs" data-title="Optimalizace (matematika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Optimering_(matematik)" title="Optimering (matematik) – Danish" lang="da" hreflang="da" data-title="Optimering (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mathematische_Optimierung" title="Mathematische Optimierung – German" lang="de" hreflang="de" data-title="Mathematische Optimierung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%92%CE%B5%CE%BB%CF%84%CE%B9%CF%83%CF%84%CE%BF%CF%80%CE%BF%CE%AF%CE%B7%CF%83%CE%B7" title="Βελτιστοποίηση – Greek" lang="el" hreflang="el" data-title="Βελτιστοποίηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Optimizaci%C3%B3n_(matem%C3%A1tica)" title="Optimización (matemática) – Spanish" lang="es" hreflang="es" data-title="Optimización (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Optimumigo_(matematiko)" title="Optimumigo (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Optimumigo (matematiko)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Optimizazio_(matematika)" title="Optimizazio (matematika) – Basque" lang="eu" hreflang="eu" data-title="Optimizazio (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D9%87%DB%8C%D9%86%D9%87%E2%80%8C%D8%B3%D8%A7%D8%B2%DB%8C" title="بهینه‌سازی – Persian" lang="fa" hreflang="fa" data-title="بهینه‌سازی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Optimisation_(math%C3%A9matiques)" title="Optimisation (mathématiques) – French" lang="fr" hreflang="fr" data-title="Optimisation (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Optimizaci%C3%B3n_matem%C3%A1tica" title="Optimización matemática – Galician" lang="gl" hreflang="gl" data-title="Optimización matemática" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99%EC%A0%81_%EC%B5%9C%EC%A0%81%ED%99%94" title="수학적 최적화 – Korean" lang="ko" hreflang="ko" data-title="수학적 최적화" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D6%85%D5%BA%D5%BF%D5%AB%D5%B4%D5%AB%D5%A6%D5%A1%D6%81%D5%AB%D5%A1" title="Մաթեմատիկական օպտիմիզացիա – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական օպտիմիզացիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%87%E0%A4%B7%E0%A5%8D%E0%A4%9F%E0%A4%A4%E0%A4%AE%E0%A4%95%E0%A4%B0%E0%A4%A3" title="इष्टतमकरण – Hindi" lang="hi" hreflang="hi" data-title="इष्टतमकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Optimizacija_(matematika)" title="Optimizacija (matematika) – Croatian" lang="hr" hreflang="hr" data-title="Optimizacija (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Optimisasi" title="Optimisasi – Indonesian" lang="id" hreflang="id" data-title="Optimisasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ottimizzazione_(matematica)" title="Ottimizzazione (matematica) – Italian" lang="it" hreflang="it" data-title="Ottimizzazione (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A4%D7%98%D7%99%D7%9E%D7%99%D7%96%D7%A6%D7%99%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="אופטימיזציה (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="אופטימיזציה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BE%D0%BF%D1%82%D0%B8%D0%BC%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F" title="Математикалық оптимизация – Kazakh" lang="kk" hreflang="kk" data-title="Математикалық оптимизация" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Optimizasyon_(mat%C3%A9matik)" title="Optimizasyon (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Optimizasyon (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%81%E0%BA%B2%E0%BA%99%E0%BA%8A%E0%BA%AD%E0%BA%81%E0%BA%84%E0%BB%88%E0%BA%B2%E0%BB%80%E0%BB%9D%E0%BA%B2%E0%BA%B0%E0%BA%AA%E0%BA%BB%E0%BA%A1_(%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94)" title="ການຊອກຄ່າເໝາະສົມ (ຄະນິດສາດ) – Lao" lang="lo" hreflang="lo" data-title="ການຊອກຄ່າເໝາະສົມ (ຄະນິດສາດ)" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Optimizavimas_(matematika)" title="Optimizavimas (matematika) – Lithuanian" lang="lt" hreflang="lt" data-title="Optimizavimas (matematika)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_optimaliz%C3%A1l%C3%A1s" title="Matematikai optimalizálás – Hungarian" lang="hu" hreflang="hu" data-title="Matematikai optimalizálás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BE%D0%BF%D1%82%D0%B8%D0%BC%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Математичка оптимизација – Macedonian" lang="mk" hreflang="mk" data-title="Математичка оптимизација" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pengoptimuman" title="Pengoptimuman – Malay" lang="ms" hreflang="ms" data-title="Pengoptimuman" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskundige_optimalisatie" title="Wiskundige optimalisatie – Dutch" lang="nl" hreflang="nl" data-title="Wiskundige optimalisatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E6%9C%80%E9%81%A9%E5%8C%96" title="数理最適化 – Japanese" lang="ja" hreflang="ja" data-title="数理最適化" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Optimering" title="Optimering – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Optimering" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_programmering" title="Matematisk programmering – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk programmering" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Optymalizacja_(matematyka)" title="Optymalizacja (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Optymalizacja (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Otimiza%C3%A7%C3%A3o" title="Otimização – Portuguese" lang="pt" hreflang="pt" data-title="Otimização" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Optimizare" title="Optimizare – Romanian" lang="ro" hreflang="ro" data-title="Optimizare" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BF%D1%82%D0%B8%D0%BC%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Оптимизация (математика) – Russian" lang="ru" hreflang="ru" data-title="Оптимизация (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Optimizimi_matematikor" title="Optimizimi matematikor – Albanian" lang="sq" hreflang="sq" data-title="Optimizimi matematikor" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_optimization" title="Mathematical optimization – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical optimization" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Optimaliz%C3%A1cia_(matematika)" title="Optimalizácia (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Optimalizácia (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Optimizacija_(matematika)" title="Optimizacija (matematika) – Slovenian" lang="sl" hreflang="sl" data-title="Optimizacija (matematika)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D0%BF%D1%82%D0%B8%D0%BC%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%98%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Оптимизација (математика) – Serbian" lang="sr" hreflang="sr" data-title="Оптимизација (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Optimisasi_(matematik)" title="Optimisasi (matematik) – Sundanese" lang="su" hreflang="su" data-title="Optimisasi (matematik)" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matemaattinen_optimointi" title="Matemaattinen optimointi – Finnish" lang="fi" hreflang="fi" data-title="Matemaattinen optimointi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Optimeringsl%C3%A4ra" title="Optimeringslära – Swedish" lang="sv" hreflang="sv" data-title="Optimeringslära" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Optimisasyong_matematikal" title="Optimisasyong matematikal – Tagalog" lang="tl" hreflang="tl" data-title="Optimisasyong matematikal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%AB%E0%B8%B2%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B9%80%E0%B8%AB%E0%B8%A1%E0%B8%B2%E0%B8%B0%E0%B8%97%E0%B8%B5%E0%B9%88%E0%B8%AA%E0%B8%B8%E0%B8%94" title="การหาค่าเหมาะที่สุด – Thai" lang="th" hreflang="th" data-title="การหาค่าเหมาะที่สุด" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Optimizasyon" title="Optimizasyon – Turkish" lang="tr" hreflang="tr" data-title="Optimizasyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%BF%D1%82%D0%B8%D0%BC%D1%96%D0%B7%D0%B0%D1%86%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Оптимізація (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Оптимізація (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%A7%D9%85%D9%84%DB%8C%D8%AA_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="کاملیت (ریاضیات) – Urdu" lang="ur" hreflang="ur" data-title="کاملیت (ریاضیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%91i_%C6%B0u_h%C3%B3a_(to%C3%A1n_h%E1%BB%8Dc)" title="Tối ưu hóa (toán học) – Vietnamese" lang="vi" hreflang="vi" data-title="Tối ưu hóa (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9C%80%E4%BD%B3%E5%8C%96" title="最佳化 – Cantonese" lang="yue" hreflang="yue" data-title="最佳化" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9C%80%E4%BC%98%E5%8C%96" title="最优化 – Chinese" lang="zh" hreflang="zh" data-title="最优化" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q141495#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Mathematical_optimization" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Mathematical_optimization" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Mathematical_optimization"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Mathematical_optimization"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Mathematical_optimization" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Mathematical_optimization" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&oldid=1257438818" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Mathematical_optimization&id=1257438818&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMathematical_optimization"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMathematical_optimization"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Mathematical_optimization&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Mathematical_optimization&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_optimization" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q141495" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Study of mathematical algorithms for optimization problems</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Optimization" and "Optimum" redirect here. For other uses, see <a href="/wiki/Optimization_(disambiguation)" class="mw-disambig" title="Optimization (disambiguation)">Optimization (disambiguation)</a> and <a href="/wiki/Optimum_(disambiguation)" class="mw-disambig" title="Optimum (disambiguation)">Optimum (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Mathematical programming" redirects here. For the peer-reviewed journal, see <a href="/wiki/Mathematical_Programming" title="Mathematical Programming">Mathematical Programming</a>.</div><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Max_paraboloid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/220px-Max_paraboloid.svg.png" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/330px-Max_paraboloid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/440px-Max_paraboloid.svg.png 2x" data-file-width="700" data-file-height="560" /></a><figcaption>Graph of a surface given by <i>z</i> = f(<i>x</i>, <i>y</i>) = −(<i>x</i>² + <i>y</i>²) + 4. The global <a href="/wiki/Maximum_(mathematics)" class="mw-redirect" title="Maximum (mathematics)">maximum</a> at (<i>x, y, z</i>) = (0, 0, 4) is indicated by a blue dot.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Nelder-Mead_Simionescu.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Nelder-Mead_Simionescu.gif/220px-Nelder-Mead_Simionescu.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Nelder-Mead_Simionescu.gif/330px-Nelder-Mead_Simionescu.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Nelder-Mead_Simionescu.gif/440px-Nelder-Mead_Simionescu.gif 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>Nelder-Mead minimum search of <a href="/wiki/Test_functions_for_optimization" title="Test functions for optimization">Simionescu's function</a>. Simplex vertices are ordered by their values, with 1 having the lowest (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> best) value.</figcaption></figure> <p><b>Mathematical optimization</b> (alternatively spelled <i>optimisation</i>) or <b>mathematical programming</b> is the selection of a best element, with regard to some criteria, from some set of available alternatives.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is generally divided into two subfields: <a href="/wiki/Discrete_optimization" title="Discrete optimization">discrete optimization</a> and <a href="/wiki/Continuous_optimization" title="Continuous optimization">continuous optimization</a>. Optimization problems arise in all quantitative disciplines from <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a><sup id="cite_ref-edo2021_3-0" class="reference"><a href="#cite_note-edo2021-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> to <a href="/wiki/Operations_research" title="Operations research">operations research</a> and <a href="/wiki/Economics" title="Economics">economics</a>, and the development of solution methods has been of interest in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> for centuries.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In the more general approach, an <a href="/wiki/Optimization_problem" title="Optimization problem">optimization problem</a> consists of <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">maximizing or minimizing</a> a <a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">real function</a> by systematically choosing <a href="/wiki/Argument_of_a_function" title="Argument of a function">input</a> values from within an allowed set and computing the <a href="/wiki/Value_(mathematics)" title="Value (mathematics)">value</a> of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Optimization_problems">Optimization problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=1" title="Edit section: Optimization problems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Optimization_problem" title="Optimization problem">Optimization problem</a></div> <p>Optimization problems can be divided into two categories, depending on whether the <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> are <a href="/wiki/Continuous_variable" class="mw-redirect" title="Continuous variable">continuous</a> or <a href="/wiki/Discrete_variable" class="mw-redirect" title="Discrete variable">discrete</a>: </p> <ul><li>An optimization problem with discrete variables is known as a <i><a href="/wiki/Discrete_optimization" title="Discrete optimization">discrete optimization</a></i>, in which an <a href="/wiki/Mathematical_object" title="Mathematical object">object</a> such as an <a href="/wiki/Integer" title="Integer">integer</a>, <a href="/wiki/Permutation" title="Permutation">permutation</a> or <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a> must be found from a <a href="/wiki/Countable_set" title="Countable set">countable set</a>.</li> <li>A problem with continuous variables is known as a <i><a href="/wiki/Continuous_optimization" title="Continuous optimization">continuous optimization</a></i>, in which optimal arguments from a continuous set must be found. They can include <a href="/wiki/Constrained_optimization" title="Constrained optimization">constrained problems</a> and multimodal problems.</li></ul> <p>An optimization problem can be represented in the following way: </p> <dl><dd><i>Given:</i> a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml"><i>f</i> : <i>A</i> → <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span></span> from some <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <span class="texhtml mvar" style="font-style:italic;">A</span> to the <a href="/wiki/Real_number" title="Real number">real numbers</a></dd> <dd><i>Sought:</i> an element <span class="texhtml"><b>x</b><sub>0</sub> ∈ <i>A</i></span> such that <span class="texhtml"><i>f</i>(<b>x</b><sub>0</sub>) ≤ <i>f</i>(<b>x</b>)</span> for all <span class="texhtml"><b>x</b> ∈ <i>A</i></span> ("minimization") or such that <span class="texhtml"><i>f</i>(<b>x</b><sub>0</sub>) ≥ <i>f</i>(<b>x</b>)</span> for all <span class="texhtml"><b>x</b> ∈ <i>A</i></span> ("maximization").</dd></dl> <p>Such a formulation is called an <b><a href="/wiki/Optimization_problem" title="Optimization problem">optimization problem</a></b> or a <b>mathematical programming problem</b> (a term not directly related to <a href="/wiki/Computer_programming" title="Computer programming">computer programming</a>, but still in use for example in <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a> – see <a href="#History">History</a> below). Many real-world and theoretical problems may be modeled in this general framework. </p><p>Since the following is valid: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b16621e3e03e9543826d663fced8f3f167561a0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.178ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} _{0})\geq f(\mathbf {x} )\Leftrightarrow -f(\mathbf {x} _{0})\leq -f(\mathbf {x} ),}"></span></dd></dl> <p>it suffices to solve only minimization problems. However, the opposite perspective of considering only maximization problems would be valid, too. </p><p>Problems formulated using this technique in the fields of <a href="/wiki/Physics" title="Physics">physics</a> may refer to the technique as <i><a href="/wiki/Energy" title="Energy">energy</a> minimization</i>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> speaking of the value of the function <span class="texhtml mvar" style="font-style:italic;">f</span> as representing the energy of the <a href="/wiki/System" title="System">system</a> being <a href="/wiki/Mathematical_model" title="Mathematical model">modeled</a>. In <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, it is always necessary to continuously evaluate the quality of a data model by using a <a href="/wiki/Loss_function" title="Loss function">cost function</a> where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error. </p><p>Typically, <span class="texhtml mvar" style="font-style:italic;">A</span> is some <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>, often specified by a set of <i><a href="/wiki/Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a></i>, equalities or inequalities that the members of <span class="texhtml mvar" style="font-style:italic;">A</span> have to satisfy. The <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> <span class="texhtml mvar" style="font-style:italic;">A</span> of <span class="texhtml mvar" style="font-style:italic;">f</span> is called the <i>search space</i> or the <i>choice set</i>, while the elements of <span class="texhtml mvar" style="font-style:italic;">A</span> are called <i><a href="/wiki/Candidate_solution" class="mw-redirect" title="Candidate solution">candidate solutions</a></i> or <i>feasible solutions</i>. </p><p>The function <span class="texhtml mvar" style="font-style:italic;">f</span> is variously called an <i>objective function</i>, <i>criterion function</i>, <i><a href="/wiki/Loss_function" title="Loss function">loss function</a></i>, <i>cost function</i> (minimization),<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <i>utility function</i> or <i>fitness function</i> (maximization), or, in certain fields, an <i>energy function</i> or <i>energy <a href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</a></i>. A feasible solution that minimizes (or maximizes) the objective function is called an <i>optimal solution</i>. </p><p>In mathematics, conventional optimization problems are usually stated in terms of minimization. </p><p>A <i>local minimum</i> <span class="texhtml"><b>x</b>*</span> is defined as an element for which there exists some <span class="texhtml"><i>δ</i> > 0</span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>∈</mo> <mi>A</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> <mspace width="thickmathspace" /> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo symmetric="true">‖</mo> </mrow> <mo>≤</mo> <mi>δ</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c45ce12a77abe61366d96dcffae4378f357976" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.134ex; height:2.843ex;" alt="{\displaystyle \forall \mathbf {x} \in A\;{\text{where}}\;\left\Vert \mathbf {x} -\mathbf {x} ^{\ast }\right\Vert \leq \delta ,\,}"></span></dd></dl> <p>the expression <span class="texhtml"><i>f</i>(<b>x</b>*) ≤ <i>f</i>(<b>x</b>)</span> holds; </p><p>that is to say, on some region around <span class="texhtml"><b>x</b>*</span> all of the function values are greater than or equal to the value at that element. Local maxima are defined similarly. </p><p>While a local minimum is at least as good as any nearby elements, a <a href="/wiki/Global_minimum" class="mw-redirect" title="Global minimum">global minimum</a> is at least as good as every feasible element. Generally, unless the objective function is <a href="/wiki/Convex_function" title="Convex function">convex</a> in a minimization problem, there may be several local minima. In a <a href="/wiki/Convex_optimization" title="Convex optimization">convex problem</a>, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima. </p><p>A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. <a href="/wiki/Global_optimization" title="Global optimization">Global optimization</a> is the branch of <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> and <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem. </p> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=2" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Optimization problems are often expressed with special notation. Here are some examples: </p> <div class="mw-heading mw-heading3"><h3 id="Minimum_and_maximum_value_of_a_function">Minimum and maximum value of a function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=3" title="Edit section: Minimum and maximum value of a function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the following notation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63529e0c09654712ebdefef0445a12017794b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.424ex; height:4.343ex;" alt="{\displaystyle \min _{x\in \mathbb {R} }\;\left(x^{2}+1\right)}"></span></dd></dl> <p>This denotes the minimum <a href="/wiki/Value_(mathematics)" title="Value (mathematics)">value</a> of the objective function <span class="texhtml"><i>x</i><sup>2</sup> + 1</span>, when choosing <span class="texhtml mvar" style="font-style:italic;">x</span> from the set of <a href="/wiki/Real_number" title="Real number">real numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. The minimum value in this case is 1, occurring at <span class="texhtml"><i>x</i> = 0</span>. </p><p>Similarly, the notation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{x\in \mathbb {R} }\;2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> <mspace width="thickmathspace" /> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max _{x\in \mathbb {R} }\;2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a128d2c279975694f4c98533554a03f500780804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.85ex; height:4.009ex;" alt="{\displaystyle \max _{x\in \mathbb {R} }\;2x}"></span></dd></dl> <p>asks for the maximum value of the objective function <span class="texhtml">2<i>x</i></span>, where <span class="texhtml mvar" style="font-style:italic;">x</span> may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "<a href="/wiki/Infinity" title="Infinity">infinity</a>" or "<a href="/wiki/Undefined_(mathematics)" title="Undefined (mathematics)">undefined</a>". </p> <div class="mw-heading mw-heading3"><h3 id="Optimal_input_arguments">Optimal input arguments</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=4" title="Edit section: Optimal input arguments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Arg_max" title="Arg max">Arg max</a></div> <p>Consider the following notation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </munder> </mrow> <mspace width="thickmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c071258fcecb43aaf25920b9833589bc35036c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.292ex; height:5.343ex;" alt="{\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}"></span></dd></dl> <p>or equivalently </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <mi>x</mi> </munder> </mrow> <mspace width="thickmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to:</mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ecc7504e271936684860c5d738740d0841edac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.837ex; height:4.676ex;" alt="{\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}"></span></dd></dl> <p>This represents the value (or values) of the <a href="/wiki/Argument_of_a_function" title="Argument of a function">argument</a> <span class="texhtml mvar" style="font-style:italic;">x</span> in the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> <span class="texhtml">(−∞,−1]</span> that minimizes (or minimize) the objective function <span class="texhtml"><i>x</i><sup>2</sup> + 1</span> (the actual minimum value of that function is not what the problem asks for). In this case, the answer is <span class="texhtml"><i>x</i> = −1</span>, since <span class="texhtml"><i>x</i> = 0</span> is infeasible, that is, it does not belong to the <a href="/wiki/Feasible_set" class="mw-redirect" title="Feasible set">feasible set</a>. </p><p>Similarly, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> <mrow> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </munder> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3de833a31bfb33700e63b8d5df04565e899915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.196ex; height:4.343ex;" alt="{\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos y,}"></span></dd></dl> <p>or equivalently </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> <mrow> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> </mrow> </munder> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mi>cos</mi> <mo>⁡</mo> <mi>y</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>subject to:</mtext> </mrow> <mspace width="thickmathspace" /> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="thickmathspace" /> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29cd276141e8e134eb156b2fa537ced38c14a5b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.398ex; height:4.676ex;" alt="{\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos y,\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}"></span></dd></dl> <p>represents the <span class="texhtml">{<i>x</i>, <i>y</i>}</span> pair (or pairs) that maximizes (or maximize) the value of the objective function <span class="texhtml"><i>x</i> cos <i>y</i></span>, with the added constraint that <span class="texhtml mvar" style="font-style:italic;">x</span> lie in the interval <span class="texhtml">[−5,5]</span> (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form <span class="texhtml">{5, 2<i>k</i><span class="texhtml mvar" style="font-style:italic;">π</span>}</span> and <span class="texhtml">{−5, (2<i>k</i> + 1)<span class="texhtml mvar" style="font-style:italic;">π</span>}</span>, where <span class="texhtml mvar" style="font-style:italic;">k</span> ranges over all <a href="/wiki/Integer" title="Integer">integers</a>. </p><p>Operators <span class="texhtml">arg min</span> and <span class="texhtml">arg max</span> are sometimes also written as <span class="texhtml">argmin</span> and <span class="texhtml">argmax</span>, and stand for <i>argument of the minimum</i> and <i>argument of the maximum</i>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=5" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat</a> and <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a> found calculus-based formulae for identifying optima, while <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> proposed iterative methods for moving towards an optimum. </p><p>The term "<a href="/wiki/Linear_programming" title="Linear programming">linear programming</a>" for certain optimization cases was due to <a href="/wiki/George_Dantzig" title="George Dantzig">George B. Dantzig</a>, although much of the theory had been introduced by <a href="/wiki/Leonid_Kantorovich" title="Leonid Kantorovich">Leonid Kantorovich</a> in 1939. (<i>Programming</i> in this context does not refer to <a href="/wiki/Computer_programming" title="Computer programming">computer programming</a>, but comes from the use of <i>program</i> by the <a href="/wiki/United_States" title="United States">United States</a> military to refer to proposed training and <a href="/wiki/Logistics" title="Logistics">logistics</a> schedules, which were the problems Dantzig studied at that time.) Dantzig published the <a href="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm</a> in 1947, and also <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a> and other researchers worked on the theoretical aspects of linear programming (like the theory of <a href="/wiki/Linear_programming#Duality" title="Linear programming">duality</a>) around the same time.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Other notable researchers in mathematical optimization include the following: </p> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Richard_Bellman" class="mw-redirect" title="Richard Bellman">Richard Bellman</a></li> <li><a href="/wiki/Dimitri_Bertsekas" title="Dimitri Bertsekas">Dimitri Bertsekas</a></li> <li><a href="/wiki/Michel_Bierlaire" title="Michel Bierlaire">Michel Bierlaire</a></li> <li><a href="/wiki/Stephen_P._Boyd" title="Stephen P. Boyd">Stephen P. Boyd</a></li> <li><a href="/wiki/Roger_Fletcher_(mathematician)" title="Roger Fletcher (mathematician)">Roger Fletcher</a></li> <li><a href="/wiki/Martin_Gr%C3%B6tschel" title="Martin Grötschel">Martin Grötschel</a></li> <li><a href="/wiki/Ronald_A._Howard" title="Ronald A. Howard">Ronald A. Howard</a></li> <li><a href="/wiki/Fritz_John" title="Fritz John">Fritz John</a></li> <li><a href="/wiki/Narendra_Karmarkar" title="Narendra Karmarkar">Narendra Karmarkar</a></li> <li><a href="/wiki/William_Karush" title="William Karush">William Karush</a></li> <li><a href="/wiki/Leonid_Khachiyan" title="Leonid Khachiyan">Leonid Khachiyan</a></li> <li><a href="/wiki/Bernard_Koopman" title="Bernard Koopman">Bernard Koopman</a></li> <li><a href="/wiki/Harold_Kuhn" class="mw-redirect" title="Harold Kuhn">Harold Kuhn</a></li> <li><a href="/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a></li> <li><a href="/wiki/David_Luenberger" title="David Luenberger">David Luenberger</a></li> <li><a href="/wiki/Arkadi_Nemirovski" title="Arkadi Nemirovski">Arkadi Nemirovski</a></li> <li><a href="/wiki/Yurii_Nesterov" title="Yurii Nesterov">Yurii Nesterov</a></li> <li><a href="/wiki/Lev_Pontryagin" title="Lev Pontryagin">Lev Pontryagin</a></li> <li><a href="/wiki/R._Tyrrell_Rockafellar" title="R. Tyrrell Rockafellar">R. Tyrrell Rockafellar</a></li> <li><a href="/wiki/Naum_Z._Shor" title="Naum Z. Shor">Naum Z. Shor</a></li> <li><a href="/wiki/Albert_W._Tucker" title="Albert W. Tucker">Albert Tucker</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Major_subfields">Major subfields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=6" title="Edit section: Major subfields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Convex_programming" class="mw-redirect" title="Convex programming">Convex programming</a> studies the case when the objective function is <a href="/wiki/Convex_function" title="Convex function">convex</a> (minimization) or <a href="/wiki/Concave_function" title="Concave function">concave</a> (maximization) and the constraint set is <a href="/wiki/Convex_set" title="Convex set">convex</a>. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming. <ul><li><a href="/wiki/Linear_programming" title="Linear programming">Linear programming</a> (LP), a type of convex programming, studies the case in which the objective function <i>f</i> is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> or a <a href="/wiki/Polytope" title="Polytope">polytope</a> if it is <a href="/wiki/Bounded_set" title="Bounded set">bounded</a>.</li> <li><a href="/wiki/Second-order_cone_programming" title="Second-order cone programming">Second-order cone programming</a> (SOCP) is a convex program, and includes certain types of quadratic programs.</li> <li><a href="/wiki/Semidefinite_programming" title="Semidefinite programming">Semidefinite programming</a> (SDP) is a subfield of convex optimization where the underlying variables are <a href="/wiki/Semidefinite" class="mw-redirect" title="Semidefinite">semidefinite</a> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>. It is a generalization of linear and convex quadratic programming.</li> <li><a href="/wiki/Conic_programming" class="mw-redirect" title="Conic programming">Conic programming</a> is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.</li> <li><a href="/wiki/Geometric_programming" title="Geometric programming">Geometric programming</a> is a technique whereby objective and inequality constraints expressed as <a href="/wiki/Posynomials" class="mw-redirect" title="Posynomials">posynomials</a> and equality constraints as <a href="/wiki/Monomials" class="mw-redirect" title="Monomials">monomials</a> can be transformed into a convex program.</li></ul></li> <li><a href="/wiki/Integer_programming" title="Integer programming">Integer programming</a> studies linear programs in which some or all variables are constrained to take on <a href="/wiki/Integer" title="Integer">integer</a> values. This is not convex, and in general much more difficult than regular linear programming.</li> <li><a href="/wiki/Quadratic_programming" title="Quadratic programming">Quadratic programming</a> allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.</li> <li><a href="/wiki/Fractional_programming" title="Fractional programming">Fractional programming</a> studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.</li> <li><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a> studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.</li> <li><a href="/wiki/Stochastic_programming" title="Stochastic programming">Stochastic programming</a> studies the case in which some of the constraints or parameters depend on <a href="/wiki/Random_variable" title="Random variable">random variables</a>.</li> <li><a href="/wiki/Robust_optimization" title="Robust optimization">Robust optimization</a> is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty set.</li> <li><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Combinatorial optimization</a> is concerned with problems where the set of feasible solutions is discrete or can be reduced to a <a href="/wiki/Discrete_mathematics" title="Discrete mathematics">discrete</a> one.</li> <li><a href="/wiki/Stochastic_optimization" title="Stochastic optimization">Stochastic optimization</a> is used with random (noisy) function measurements or random inputs in the search process.</li> <li><a href="/wiki/Infinite-dimensional_optimization" title="Infinite-dimensional optimization">Infinite-dimensional optimization</a> studies the case when the set of feasible solutions is a subset of an infinite-<a href="/wiki/Dimension" title="Dimension">dimensional</a> space, such as a space of functions.</li> <li><a href="/wiki/Heuristic_(computer_science)" title="Heuristic (computer science)">Heuristics</a> and <a href="/wiki/Metaheuristic" title="Metaheuristic">metaheuristics</a> make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.</li> <li><a href="/wiki/Constraint_satisfaction" title="Constraint satisfaction">Constraint satisfaction</a> studies the case in which the objective function <i>f</i> is constant (this is used in <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, particularly in <a href="/wiki/Automated_reasoning" title="Automated reasoning">automated reasoning</a>). <ul><li><a href="/wiki/Constraint_programming" title="Constraint programming">Constraint programming</a> is a programming paradigm wherein relations between variables are stated in the form of constraints.</li></ul></li> <li>Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.</li> <li><a href="/wiki/Space_mapping" title="Space mapping">Space mapping</a> is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or <a href="/wiki/Surrogate_model" title="Surrogate model">surrogate model</a>.</li></ul> <p>In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time): </p> <ul><li><a href="/wiki/Calculus_of_variations" title="Calculus of variations">Calculus of variations</a> is concerned with finding the best way to achieve some goal, such as finding a surface whose boundary is a specific curve, but with the least possible area.</li> <li><a href="/wiki/Optimal_control" title="Optimal control">Optimal control</a> theory is a generalization of the calculus of variations which introduces control policies.</li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a> is the approach to solve the <a href="/wiki/Stochastic_optimization" title="Stochastic optimization">stochastic optimization</a> problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the <a href="/wiki/Bellman_equation" title="Bellman equation">Bellman equation</a>.</li> <li><a href="/wiki/Mathematical_programming_with_equilibrium_constraints" title="Mathematical programming with equilibrium constraints">Mathematical programming with equilibrium constraints</a> is where the constraints include <a href="/wiki/Variational_inequalities" class="mw-redirect" title="Variational inequalities">variational inequalities</a> or <a href="/wiki/Complementarity_theory" title="Complementarity theory">complementarities</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Multi-objective_optimization">Multi-objective optimization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=7" title="Edit section: Multi-objective optimization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Multi-objective_optimization" title="Multi-objective optimization">Multi-objective optimization</a></div> <p>Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the <a href="/wiki/Pareto_set" class="mw-redirect" title="Pareto set">Pareto set</a>. The curve created plotting weight against stiffness of the best designs is known as the <a href="/wiki/Pareto_frontier" class="mw-redirect" title="Pareto frontier">Pareto frontier</a>. </p><p>A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal. </p><p>The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker. </p><p>Multi-objective optimization problems have been generalized further into <a href="/wiki/Vector_optimization" title="Vector optimization">vector optimization</a> problems where the (partial) ordering is no longer given by the Pareto ordering. </p> <div class="mw-heading mw-heading3"><h3 id="Multi-modal_or_global_optimization">Multi-modal or global optimization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=8" title="Edit section: Multi-modal or global optimization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer. </p><p>Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm. </p><p>Common approaches to <a href="/wiki/Global_optimization" title="Global optimization">global optimization</a> problems, where multiple local extrema may be present include <a href="/wiki/Evolutionary_algorithm" title="Evolutionary algorithm">evolutionary algorithms</a>, <a href="/wiki/Bayesian_optimization" title="Bayesian optimization">Bayesian optimization</a> and <a href="/wiki/Simulated_annealing" title="Simulated annealing">simulated annealing</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_of_critical_points_and_extrema">Classification of critical points and extrema</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=9" title="Edit section: Classification of critical points and extrema"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Feasibility_problem">Feasibility problem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=10" title="Edit section: Feasibility problem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i><a href="/wiki/Satisfiability_problem" class="mw-redirect" title="Satisfiability problem">satisfiability problem</a></i>, also called the <i>feasibility problem</i>, is just the problem of finding any <a href="/wiki/Feasible_solution" class="mw-redirect" title="Feasible solution">feasible solution</a> at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal. </p><p>Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to <a href="/wiki/Relaxation_(approximation)" title="Relaxation (approximation)">relax</a> the feasibility conditions using a <a href="/wiki/Slack_variable" title="Slack variable">slack variable</a>; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative. </p> <div class="mw-heading mw-heading3"><h3 id="Existence">Existence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=11" title="Edit section: Existence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a> of <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view. </p> <div class="mw-heading mw-heading3"><h3 id="Necessary_conditions_for_optimality">Necessary conditions for optimality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=12" title="Edit section: Necessary conditions for optimality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Fermat%27s_theorem_(stationary_points)" title="Fermat's theorem (stationary points)">One of Fermat's theorems</a> states that optima of unconstrained problems are found at <a href="/wiki/Stationary_point" title="Stationary point">stationary points</a>, where the first derivative or the gradient of the objective function is zero (see <a href="/wiki/First_derivative_test" class="mw-redirect" title="First derivative test">first derivative test</a>). More generally, they may be found at <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a>, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions. </p><p>Optima of equality-constrained problems can be found by the <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> method. The optima of problems with equality and/or inequality constraints can be found using the '<a href="/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a>'. </p> <div class="mw-heading mw-heading3"><h3 id="Sufficient_conditions_for_optimality">Sufficient conditions for optimality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=13" title="Edit section: Sufficient conditions for optimality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a>) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the <a href="/wiki/Hessian_matrix#Bordered_Hessian" title="Hessian matrix">bordered Hessian</a> in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see '<a href="/wiki/Second_derivative_test" class="mw-redirect" title="Second derivative test">Second derivative test</a>'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality. </p> <div class="mw-heading mw-heading3"><h3 id="Sensitivity_and_continuity_of_optima">Sensitivity and continuity of optima</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=14" title="Edit section: Sensitivity and continuity of optima"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Envelope_theorem" title="Envelope theorem">envelope theorem</a> describes how the value of an optimal solution changes when an underlying <a href="/wiki/Parameter" title="Parameter">parameter</a> changes. The process of computing this change is called <a href="/wiki/Comparative_statics" title="Comparative statics">comparative statics</a>. </p><p>The <a href="/wiki/Maximum_theorem" title="Maximum theorem">maximum theorem</a> of <a href="/wiki/Claude_Berge" title="Claude Berge">Claude Berge</a> (1963) describes the continuity of an optimal solution as a function of underlying parameters. </p> <div class="mw-heading mw-heading3"><h3 id="Calculus_of_optimization">Calculus of optimization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=15" title="Edit section: Calculus of optimization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" title="Karush–Kuhn–Tucker conditions">Karush–Kuhn–Tucker conditions</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">Critical point (mathematics)</a>, <a href="/wiki/Differential_calculus" title="Differential calculus">Differential calculus</a>, <a href="/wiki/Gradient" title="Gradient">Gradient</a>, <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a>, <a href="/wiki/Definite_matrix" title="Definite matrix">Definite matrix</a>, <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a>, <a href="/wiki/Rademacher%27s_theorem" title="Rademacher's theorem">Rademacher's theorem</a>, <a href="/wiki/Convex_function" title="Convex function">Convex function</a>, and <a href="/wiki/Convex_analysis" title="Convex analysis">Convex analysis</a></div> <p>For unconstrained problems with twice-differentiable functions, some <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> can be found by finding the points where the <a href="/wiki/Gradient" title="Gradient">gradient</a> of the objective function is zero (that is, the stationary points). More generally, a zero <a href="/wiki/Subgradient" class="mw-redirect" title="Subgradient">subgradient</a> certifies that a local minimum has been found for <a href="/wiki/Convex_optimization" title="Convex optimization">minimization problems with convex</a> <a href="/wiki/Convex_function" title="Convex function">functions</a> and other <a href="/wiki/Rademacher%27s_theorem" title="Rademacher's theorem">locally</a> <a href="/wiki/Lipschitz_function" class="mw-redirect" title="Lipschitz function">Lipschitz functions</a>, which meet in loss function minimization of the neural network. The positive-negative momentum estimation lets to avoid the local minimum and converges at the objective function global minimum.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Further, critical points can be classified using the <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">definiteness</a> of the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a>: If the Hessian is <i>positive</i> definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of <a href="/wiki/Saddle_point" title="Saddle point">saddle point</a>. </p><p>Constrained problems can often be transformed into unconstrained problems with the help of <a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multipliers</a>. <a href="/wiki/Lagrangian_relaxation" title="Lagrangian relaxation">Lagrangian relaxation</a> can also provide approximate solutions to difficult constrained problems. </p><p>When the objective function is a <a href="/wiki/Convex_function" title="Convex function">convex function</a>, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as <a href="/wiki/Interior-point_method" title="Interior-point method">interior-point methods</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Global_convergence">Global convergence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=16" title="Edit section: Global convergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on <a href="/wiki/Line_search" title="Line search">line searches</a>, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses <a href="/wiki/Trust_region" title="Trust region">trust regions</a>. Both line searches and trust regions are used in modern methods of <a href="/wiki/Subgradient_method" title="Subgradient method">non-differentiable optimization</a>. Usually, a global optimizer is much slower than advanced local optimizers (such as <a href="/wiki/BFGS_method" class="mw-redirect" title="BFGS method">BFGS</a>), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points. </p> <div class="mw-heading mw-heading2"><h2 id="Computational_optimization_techniques">Computational optimization techniques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=17" title="Edit section: Computational optimization techniques"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To solve problems, researchers may use <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> that terminate in a finite number of steps, or <a href="/wiki/Iterative_method" title="Iterative method">iterative methods</a> that converge to a solution (on some specified class of problems), or <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a> that may provide approximate solutions to some problems (although their iterates need not converge). </p> <div class="mw-heading mw-heading3"><h3 id="Optimization_algorithms">Optimization algorithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=18" title="Edit section: Optimization algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For a more comprehensive list, see <a href="/wiki/List_of_optimization_algorithms" class="mw-redirect" title="List of optimization algorithms">List of optimization algorithms</a>.</div> <ul><li><a href="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm</a> of <a href="/wiki/George_Dantzig" title="George Dantzig">George Dantzig</a>, designed for <a href="/wiki/Linear_programming" title="Linear programming">linear programming</a></li> <li>Extensions of the simplex algorithm, designed for <a href="/wiki/Quadratic_programming" title="Quadratic programming">quadratic programming</a> and for <a href="/wiki/Linear-fractional_programming" title="Linear-fractional programming">linear-fractional programming</a></li> <li>Variants of the simplex algorithm that are especially suited for <a href="/wiki/Flow_network" title="Flow network">network optimization</a></li> <li><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Combinatorial algorithms</a></li> <li><a href="/wiki/Quantum_optimization_algorithms" title="Quantum optimization algorithms">Quantum optimization algorithms</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Iterative_methods">Iterative methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=19" title="Edit section: Iterative methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Iterative_method" title="Iterative method">Iterative method</a></div> <p>The <a href="/wiki/Iterative_methods" class="mw-redirect" title="Iterative methods">iterative methods</a> used to solve problems of <a href="/wiki/Nonlinear_programming" title="Nonlinear programming">nonlinear programming</a> differ according to whether they <a href="/wiki/Subroutine" class="mw-redirect" title="Subroutine">evaluate</a> <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessians</a>, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity</a> (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high. </p><p>One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself. </p> <ul><li>Methods that evaluate Hessians (or approximate Hessians, using <a href="/wiki/Finite_difference" title="Finite difference">finite differences</a>): <ul><li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li> <li><a href="/wiki/Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a>: A Newton-based method for small-medium scale <i>constrained</i> problems. Some versions can handle large-dimensional problems.</li> <li><a href="/wiki/Interior_point_methods" class="mw-redirect" title="Interior point methods">Interior point methods</a>: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.</li></ul></li> <li>Methods that evaluate gradients, or approximate gradients in some way (or even subgradients): <ul><li><a href="/wiki/Coordinate_descent" title="Coordinate descent">Coordinate descent</a> methods: Algorithms which update a single coordinate in each iteration</li> <li><a href="/wiki/Conjugate_gradient_method" title="Conjugate gradient method">Conjugate gradient methods</a>: <a href="/wiki/Iterative_method" title="Iterative method">Iterative methods</a> for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)</li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient descent</a> (alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.</li> <li><a href="/wiki/Subgradient_method" title="Subgradient method">Subgradient methods</a>: An iterative method for large <a href="/wiki/Rademacher%27s_theorem" title="Rademacher's theorem">locally</a> <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz functions</a> using <a href="/wiki/Subgradient" class="mw-redirect" title="Subgradient">generalized gradients</a>. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.</li> <li>Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for <a href="/wiki/Convex_optimization" title="Convex optimization">convex minimization</a> problems (similar to conjugate gradient methods).</li> <li><a href="/wiki/Ellipsoid_method" title="Ellipsoid method">Ellipsoid method</a>: An iterative method for small problems with <a href="/wiki/Quasiconvex_function" title="Quasiconvex function">quasiconvex</a> objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.</li> <li><a href="/wiki/Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Conditional gradient method (Frank–Wolfe)</a> for approximate minimization of specially structured problems with <a href="/wiki/Linear_constraints" class="mw-redirect" title="Linear constraints">linear constraints</a>, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).</li> <li><a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton methods</a>: Iterative methods for medium-large problems (e.g. N<1000).</li> <li><a href="/wiki/Simultaneous_perturbation_stochastic_approximation" title="Simultaneous perturbation stochastic approximation">Simultaneous perturbation stochastic approximation</a> (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.</li></ul></li> <li>Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used. <ul><li><a href="/wiki/Interpolation" title="Interpolation">Interpolation</a> methods</li> <li><a href="/wiki/Pattern_search_(optimization)" title="Pattern search (optimization)">Pattern search</a> methods, which have better convergence properties than the <a href="/wiki/Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead heuristic (with simplices)</a>, which is listed below.</li> <li><a href="/wiki/Mirror_descent" title="Mirror descent">Mirror descent</a></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Heuristics">Heuristics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=20" title="Edit section: Heuristics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">Heuristic algorithm</a></div> <p>Besides (finitely terminating) <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> and (convergent) <a href="/wiki/Iterative_method" title="Iterative method">iterative methods</a>, there are <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a>. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li><a href="/wiki/Differential_evolution" title="Differential evolution">Differential evolution</a></li> <li><a href="/wiki/Dynamic_relaxation" title="Dynamic relaxation">Dynamic relaxation</a></li> <li><a href="/wiki/Evolutionary_algorithms" class="mw-redirect" title="Evolutionary algorithms">Evolutionary algorithms</a></li> <li><a href="/wiki/Genetic_algorithms" class="mw-redirect" title="Genetic algorithms">Genetic algorithms</a></li> <li><a href="/wiki/Hill_climbing" title="Hill climbing">Hill climbing</a> with random restart</li> <li><a href="/wiki/Memetic_algorithm" title="Memetic algorithm">Memetic algorithm</a></li> <li><a href="/wiki/Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead simplicial heuristic</a>: A popular heuristic for approximate minimization (without calling gradients)</li> <li><a href="/wiki/Particle_swarm_optimization" title="Particle swarm optimization">Particle swarm optimization</a></li> <li><a href="/wiki/Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li> <li><a href="/wiki/Stochastic_tunneling" title="Stochastic tunneling">Stochastic tunneling</a></li> <li><a href="/wiki/Tabu_search" title="Tabu search">Tabu search</a></li></ul></div> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=21" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Mechanics">Mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=22" title="Edit section: Mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Problems in <a href="/wiki/Rigid_body_dynamics" title="Rigid body dynamics">rigid body dynamics</a> (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an <a href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> on a constraint manifold;<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a <a href="/wiki/Linear_complementarity_problem" title="Linear complementarity problem">linear complementarity problem</a>, which can also be viewed as a QP (quadratic programming) problem. </p><p>Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the <a href="/wiki/Engineering_optimization" title="Engineering optimization">engineering optimization</a>, and another recent and growing subset of this field is <a href="/wiki/Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">multidisciplinary design optimization</a>, which, while useful in many problems, has in particular been applied to <a href="/wiki/Aerospace_engineering" title="Aerospace engineering">aerospace engineering</a> problems. </p><p>This approach may be applied in cosmology and astrophysics.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Economics_and_finance">Economics and finance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=23" title="Edit section: Economics and finance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Economics" title="Economics">Economics</a> is closely enough linked to optimization of <a href="/wiki/Agent_(economics)" title="Agent (economics)">agents</a> that an influential definition relatedly describes economics <i>qua</i> science as the "study of human behavior as a relationship between ends and <a href="/wiki/Scarce" class="mw-redirect" title="Scarce">scarce</a> means" with alternative uses.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Modern optimization theory includes traditional optimization theory but also overlaps with <a href="/wiki/Game_theory" title="Game theory">game theory</a> and the study of economic <a href="/wiki/Equilibrium_(economics)" class="mw-redirect" title="Equilibrium (economics)">equilibria</a>. The <i><a href="/wiki/Journal_of_Economic_Literature" title="Journal of Economic Literature">Journal of Economic Literature</a></i> <a href="/wiki/JEL_classification_codes" title="JEL classification codes">codes</a> classify mathematical programming, optimization techniques, and related topics under <a href="/wiki/JEL_classification_codes#Mathematical_and_quantitative_methods_JEL:_C_Subcategories" title="JEL classification codes">JEL:C61-C63</a>. </p><p>In microeconomics, the <a href="/wiki/Utility_maximization_problem" title="Utility maximization problem">utility maximization problem</a> and its <a href="/wiki/Dual_problem" class="mw-redirect" title="Dual problem">dual problem</a>, the <a href="/wiki/Expenditure_minimization_problem" title="Expenditure minimization problem">expenditure minimization problem</a>, are economic optimization problems. Insofar as they behave consistently, <a href="/wiki/Consumer" title="Consumer">consumers</a> are assumed to maximize their <a href="/wiki/Utility" title="Utility">utility</a>, while <a href="/wiki/Firm" class="mw-redirect" title="Firm">firms</a> are usually assumed to maximize their <a href="/wiki/Profit_(economics)" title="Profit (economics)">profit</a>. Also, agents are often modeled as being <a href="/wiki/Risk_aversion" title="Risk aversion">risk-averse</a>, thereby preferring to avoid risk. <a href="/wiki/Asset_pricing" title="Asset pricing">Asset prices</a> are also modeled using optimization theory, though the underlying mathematics relies on optimizing <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a> rather than on static optimization. <a href="/wiki/International_trade_theory" title="International trade theory">International trade theory</a> also uses optimization to explain trade patterns between nations. The optimization of <a href="/wiki/Portfolio_(finance)" title="Portfolio (finance)">portfolios</a> is an example of multi-objective optimization in economics. </p><p>Since the 1970s, economists have modeled dynamic decisions over time using <a href="/wiki/Control_theory" title="Control theory">control theory</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> For example, dynamic <a href="/wiki/Search_theory" title="Search theory">search models</a> are used to study <a href="/wiki/Labor_economics" class="mw-redirect" title="Labor economics">labor-market behavior</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> A crucial distinction is between deterministic and stochastic models.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Macroeconomics" title="Macroeconomics">Macroeconomists</a> build <a href="/wiki/Dynamic_stochastic_general_equilibrium" title="Dynamic stochastic general equilibrium">dynamic stochastic general equilibrium (DSGE)</a> models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electrical_engineering">Electrical engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=24" title="Edit section: Electrical engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some common applications of optimization techniques in <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a> include <a href="/wiki/Active_filter" title="Active filter">active filter</a> design,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> stray field reduction in superconducting magnetic energy storage systems, <a href="/wiki/Space_mapping" title="Space mapping">space mapping</a> design of <a href="/wiki/Microwave" title="Microwave">microwave</a> structures,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> handset antennas,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical <a href="/wiki/Surrogate_model" title="Surrogate model">surrogate model</a> and <a href="/wiki/Space_mapping" title="Space mapping">space mapping</a> methodologies since the discovery of <a href="/wiki/Space_mapping" title="Space mapping">space mapping</a> in 1993.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Optimization techniques are also used in <a href="/wiki/Power-flow_analysis" class="mw-redirect" title="Power-flow analysis">power-flow analysis</a>.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Civil_engineering">Civil engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=25" title="Edit section: Civil engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Optimization has been widely used in civil engineering. <a href="/wiki/Construction_management" title="Construction management">Construction management</a> and <a href="/wiki/Transportation_engineering" title="Transportation engineering">transportation engineering</a> are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Resource_leveling" title="Resource leveling">resource leveling</a>,<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_30-0" class="reference"><a href="#cite_note-:0-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Hydrological_optimization" title="Hydrological optimization">water resource allocation</a>, <a href="/wiki/Traffic" title="Traffic">traffic</a> management<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> and schedule optimization. </p> <div class="mw-heading mw-heading3"><h3 id="Operations_research">Operations research</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=26" title="Edit section: Operations research"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another field that uses optimization techniques extensively is <a href="/wiki/Operations_research" title="Operations research">operations research</a>.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses <a href="/wiki/Stochastic_programming" title="Stochastic programming">stochastic programming</a> to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and <a href="/wiki/Stochastic_optimization" title="Stochastic optimization">stochastic optimization</a> methods. </p> <div class="mw-heading mw-heading3"><h3 id="Control_engineering">Control engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=27" title="Edit section: Control engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical optimization is used in much modern controller design. High-level controllers such as <a href="/wiki/Model_predictive_control" title="Model predictive control">model predictive control</a> (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled. </p> <div class="mw-heading mw-heading3"><h3 id="Geophysics">Geophysics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=28" title="Edit section: Geophysics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Optimization techniques are regularly used in <a href="/wiki/Geophysics" title="Geophysics">geophysical</a> parameter estimation problems. Given a set of geophysical measurements, e.g. <a href="/wiki/Seismology" title="Seismology">seismic recordings</a>, it is common to solve for the <a href="/wiki/Mineral_physics" title="Mineral physics">physical properties</a> and <a href="/wiki/Structure_of_the_earth" class="mw-redirect" title="Structure of the earth">geometrical shapes</a> of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used. </p> <div class="mw-heading mw-heading3"><h3 id="Molecular_modeling">Molecular modeling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=29" title="Edit section: Molecular modeling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Molecular_modeling" class="mw-redirect" title="Molecular modeling">Molecular modeling</a></div> <p>Nonlinear optimization methods are widely used in <a href="/wiki/Conformational_analysis" class="mw-redirect" title="Conformational analysis">conformational analysis</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Computational_systems_biology">Computational systems biology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=30" title="Edit section: Computational systems biology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Computational_systems_biology" class="mw-redirect" title="Computational systems biology">Computational systems biology</a></div> <p>Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.<sup id="cite_ref-Papoutsakis_1984_33-0" class="reference"><a href="#cite_note-Papoutsakis_1984-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Linear_programming" title="Linear programming">Linear programming</a> has been applied to calculate the maximal possible yields of fermentation products,<sup id="cite_ref-Papoutsakis_1984_33-1" class="reference"><a href="#cite_note-Papoutsakis_1984-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> and to infer gene regulatory networks from multiple microarray datasets<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> as well as transcriptional regulatory networks from high-throughput data.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a> has been used to analyze energy metabolism<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> and has been applied to metabolic engineering and parameter estimation in biochemical pathways.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Machine_learning">Machine learning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=31" title="Edit section: Machine learning"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Machine_learning#Optimization" title="Machine learning">Machine learning § Optimization</a></div> <div class="mw-heading mw-heading2"><h2 id="Solvers">Solvers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=32" title="Edit section: Solvers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_optimization_software" title="List of optimization software">List of optimization software</a></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=33" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Brachistochrone_curve" title="Brachistochrone curve">Brachistochrone curve</a></li> <li><a href="/wiki/Curve_fitting" title="Curve fitting">Curve fitting</a></li> <li><a href="/wiki/Deterministic_global_optimization" title="Deterministic global optimization">Deterministic global optimization</a></li> <li><a href="/wiki/Goal_programming" title="Goal programming">Goal programming</a></li> <li><a href="/wiki/List_of_publications_in_mathematics#Optimization" class="mw-redirect" title="List of publications in mathematics">Important publications in optimization</a></li> <li><a href="/wiki/Least_squares" title="Least squares">Least squares</a></li> <li><a href="/wiki/Mathematical_Optimization_Society" title="Mathematical Optimization Society">Mathematical Optimization Society</a> (formerly Mathematical Programming Society)</li> <li><a href="/wiki/Category:Optimization_algorithms_and_methods" title="Category:Optimization algorithms and methods">Mathematical optimization algorithms</a></li> <li><a href="/wiki/Category:Mathematical_optimization_software" title="Category:Mathematical optimization software">Mathematical optimization software</a></li> <li><a href="/wiki/Process_optimization" title="Process optimization">Process optimization</a></li> <li><a href="/wiki/Simulation-based_optimization" title="Simulation-based optimization">Simulation-based optimization</a></li> <li><a href="/wiki/Test_functions_for_optimization" title="Test functions for optimization">Test functions for optimization</a></li> <li><a href="/wiki/Vehicle_routing_problem" title="Vehicle routing problem">Vehicle routing problem</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=34" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">"<a rel="nofollow" class="external text" href="http://glossary.computing.society.informs.org/index.php?page=nature.html">The Nature of Mathematical Programming</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140305080324/http://glossary.computing.society.informs.org/index.php?page=nature.html">Archived</a> 2014-03-05 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>," <i>Mathematical Programming Glossary</i>, INFORMS Computing Society.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.mit.edu/15.053/www/AppliedMathematicalProgramming.pdf">"Mathematical Programming: An Overview"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">26 April</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematical+Programming%3A+An+Overview&rft_id=https%3A%2F%2Fweb.mit.edu%2F15.053%2Fwww%2FAppliedMathematicalProgramming.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-edo2021-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-edo2021_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartinsNing2021" class="citation book cs1">Martins, Joaquim R. R. A.; Ning, Andrew (2021-10-01). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/352413464"><i>Engineering Design Optimization</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1108833417" title="Special:BookSources/978-1108833417"><bdi>978-1108833417</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Design+Optimization&rft.pub=Cambridge+University+Press&rft.date=2021-10-01&rft.isbn=978-1108833417&rft.aulast=Martins&rft.aufirst=Joaquim+R.+R.+A.&rft.au=Ning%2C+Andrew&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F352413464&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuPardalosWu2008" class="citation book cs1">Du, D. Z.; Pardalos, P. M.; Wu, W. (2008). "History of Optimization". In <a href="/wiki/Christodoulos_Floudas" title="Christodoulos Floudas">Floudas, C.</a>; Pardalos, P. (eds.). <i>Encyclopedia of Optimization</i>. Boston: Springer. pp. 1538–1542.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=History+of+Optimization&rft.btitle=Encyclopedia+of+Optimization&rft.place=Boston&rft.pages=1538-1542&rft.pub=Springer&rft.date=2008&rft.aulast=Du&rft.aufirst=D.+Z.&rft.au=Pardalos%2C+P.+M.&rft.au=Wu%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.engati.com/glossary/mathematical-optimization">"Mathematical optimization"</a>. <i>Engati</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Engati&rft.atitle=Mathematical+optimization&rft_id=https%3A%2F%2Fwww.engati.com%2Fglossary%2Fmathematical-optimization&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ojmo.centre-mersenne.org">"Open Journal of Mathematical Optimization"</a>. <i>ojmo.centre-mersenne.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ojmo.centre-mersenne.org&rft.atitle=Open+Journal+of+Mathematical+Optimization&rft_id=https%3A%2F%2Fojmo.centre-mersenne.org&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartmannRieger2002" class="citation book cs1">Hartmann, Alexander K; Rieger, Heiko (2002). <i>Optimization algorithms in physics</i>. Citeseer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optimization+algorithms+in+physics&rft.pub=Citeseer&rft.date=2002&rft.aulast=Hartmann&rft.aufirst=Alexander+K&rft.au=Rieger%2C+Heiko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErwin_Diewert2017" class="citation cs2">Erwin Diewert, W. (2017), <a rel="nofollow" class="external text" href="https://link.springer.com/referenceworkentry/10.1057/978-1-349-95121-5_659-2">"Cost Functions"</a>, <i>The New Palgrave Dictionary of Economics</i>, London: Palgrave Macmillan UK, pp. 1–12, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1057%2F978-1-349-95121-5_659-2">10.1057/978-1-349-95121-5_659-2</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-349-95121-5" title="Special:BookSources/978-1-349-95121-5"><bdi>978-1-349-95121-5</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2024-08-18</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+Palgrave+Dictionary+of+Economics&rft.atitle=Cost+Functions&rft.pages=1-12&rft.date=2017&rft_id=info%3Adoi%2F10.1057%2F978-1-349-95121-5_659-2&rft.isbn=978-1-349-95121-5&rft.aulast=Erwin+Diewert&rft.aufirst=W.&rft_id=https%3A%2F%2Flink.springer.com%2Freferenceworkentry%2F10.1057%2F978-1-349-95121-5_659-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBixby2012" class="citation journal cs1">Bixby, Robert E (2012). <a rel="nofollow" class="external text" href="https://www.math.uni-bielefeld.de/documenta/vol-ismp/25_bixby-robert.pdf">"A brief history of linear and mixed-integer programming computation"</a> <span class="cs1-format">(PDF)</span>. <i>Documenta Mathematica</i>. Documenta Mathematica Series. <b>2012</b>: 107–121. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4171%2Fdms%2F6%2F16">10.4171/dms/6/16</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-936609-58-5" title="Special:BookSources/978-3-936609-58-5"><bdi>978-3-936609-58-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Documenta+Mathematica&rft.atitle=A+brief+history+of+linear+and+mixed-integer+programming+computation&rft.volume=2012&rft.pages=107-121&rft.date=2012&rft_id=info%3Adoi%2F10.4171%2Fdms%2F6%2F16&rft.isbn=978-3-936609-58-5&rft.aulast=Bixby&rft.aufirst=Robert+E&rft_id=https%3A%2F%2Fwww.math.uni-bielefeld.de%2Fdocumenta%2Fvol-ismp%2F25_bixby-robert.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbdulkadirovLyakhovBergermanReznikov2024" class="citation journal cs1">Abdulkadirov, R.; Lyakhov, P.; Bergerman, M.; Reznikov, D. (February 2024). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S0960077923013346">"Satellite image recognition using ensemble neural networks and difference gradient positive-negative momentum"</a>. <i>Chaos, Solitons & Fractals</i>. <b>179</b>: 114432. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2024CSF...17914432A">2024CSF...17914432A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.chaos.2023.114432">10.1016/j.chaos.2023.114432</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chaos%2C+Solitons+%26+Fractals&rft.atitle=Satellite+image+recognition+using+ensemble+neural+networks+and+difference+gradient+positive-negative+momentum&rft.volume=179&rft.pages=114432&rft.date=2024-02&rft_id=info%3Adoi%2F10.1016%2Fj.chaos.2023.114432&rft_id=info%3Abibcode%2F2024CSF...17914432A&rft.aulast=Abdulkadirov&rft.aufirst=R.&rft.au=Lyakhov%2C+P.&rft.au=Bergerman%2C+M.&rft.au=Reznikov%2C+D.&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2FS0960077923013346&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVereshchagin1989" class="citation journal cs1">Vereshchagin, A.F. (1989). "Modelling and control of motion of manipulation robots". <i>Soviet Journal of Computer and Systems Sciences</i>. <b>27</b> (5): 29–38.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Soviet+Journal+of+Computer+and+Systems+Sciences&rft.atitle=Modelling+and+control+of+motion+of+manipulation+robots&rft.volume=27&rft.issue=5&rft.pages=29-38&rft.date=1989&rft.aulast=Vereshchagin&rft.aufirst=A.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaggagDesokeyRamadan2017" class="citation journal cs1">Haggag, S.; Desokey, F.; Ramadan, M. (2017). "A cosmological inflationary model using optimal control". <i>Gravitation and Cosmology</i>. <b>23</b> (3): 236–239. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017GrCo...23..236H">2017GrCo...23..236H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1134%2FS0202289317030069">10.1134/S0202289317030069</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1995-0721">1995-0721</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125980981">125980981</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Gravitation+and+Cosmology&rft.atitle=A+cosmological+inflationary+model+using+optimal+control&rft.volume=23&rft.issue=3&rft.pages=236-239&rft.date=2017&rft_id=info%3Adoi%2F10.1134%2FS0202289317030069&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125980981%23id-name%3DS2CID&rft.issn=1995-0721&rft_id=info%3Abibcode%2F2017GrCo...23..236H&rft.aulast=Haggag&rft.aufirst=S.&rft.au=Desokey%2C+F.&rft.au=Ramadan%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="/wiki/Lionel_Robbins" title="Lionel Robbins">Lionel Robbins</a> (1935, 2nd ed.) <i><a href="/wiki/An_Essay_on_the_Nature_and_Significance_of_Economic_Science#Major_propositions" title="An Essay on the Nature and Significance of Economic Science">An Essay on the Nature and Significance of Economic Science</a></i>, Macmillan, p. 16.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDorfman1969" class="citation journal cs1"><a href="/wiki/Robert_Dorfman" title="Robert Dorfman">Dorfman, Robert</a> (1969). "An Economic Interpretation of Optimal Control Theory". <i><a href="/wiki/American_Economic_Review" title="American Economic Review">American Economic Review</a></i>. <b>59</b> (5): 817–831. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1810679">1810679</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Economic+Review&rft.atitle=An+Economic+Interpretation+of+Optimal+Control+Theory&rft.volume=59&rft.issue=5&rft.pages=817-831&rft.date=1969&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1810679%23id-name%3DJSTOR&rft.aulast=Dorfman&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSargent1987" class="citation book cs1"><a href="/wiki/Thomas_J._Sargent" title="Thomas J. Sargent">Sargent, Thomas J.</a> (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nVuyXF8ibeIC&pg=PA57">"Search"</a>. <i>Dynamic Macroeconomic Theory</i>. Harvard University Press. pp. 57–91. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674043084" title="Special:BookSources/9780674043084"><bdi>9780674043084</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Search&rft.btitle=Dynamic+Macroeconomic+Theory&rft.pages=57-91&rft.pub=Harvard+University+Press&rft.date=1987&rft.isbn=9780674043084&rft.aulast=Sargent&rft.aufirst=Thomas+J.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnVuyXF8ibeIC%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">A.G. Malliaris (2008). "stochastic optimal control," <i>The New Palgrave Dictionary of Economics</i>, 2nd Edition. <a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2008_S000269&edition=&field=keyword&q=Taylor's%20th&topicid=&result_number=1">Abstract</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171018182459/http://www.dictionaryofeconomics.com/article?id=pde2008_S000269&edition=&field=keyword&q=Taylor's%20th&topicid=&result_number=1">Archived</a> 2017-10-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaves_MazaFedrianiOrdaz_Sanz2018" class="citation journal cs1">Chaves Maza, Manuel; Fedriani, Eugenio M.; Ordaz Sanz, José Antonio (2018-07-01). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.15446/innovar.v28n69.71693">"Factores relevantes para optimizar los servicios públicos de apoyo a los emprendedores y la tasa de supervivencia de las empresas"</a>. <i>Innovar</i>. <b>28</b> (69): 9–24. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.15446%2Finnovar.v28n69.71693">10.15446/innovar.v28n69.71693</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2248-6968">2248-6968</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Innovar&rft.atitle=Factores+relevantes+para+optimizar+los+servicios+p%C3%BAblicos+de+apoyo+a+los+emprendedores+y+la+tasa+de+supervivencia+de+las+empresas&rft.volume=28&rft.issue=69&rft.pages=9-24&rft.date=2018-07-01&rft_id=info%3Adoi%2F10.15446%2Finnovar.v28n69.71693&rft.issn=2248-6968&rft.aulast=Chaves+Maza&rft.aufirst=Manuel&rft.au=Fedriani%2C+Eugenio+M.&rft.au=Ordaz+Sanz%2C+Jos%C3%A9+Antonio&rft_id=http%3A%2F%2Fdx.doi.org%2F10.15446%2Finnovar.v28n69.71693&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotembergWoodford1997" class="citation journal cs1"><a href="/wiki/Julio_Rotemberg" title="Julio Rotemberg">Rotemberg, Julio</a>; <a href="/wiki/Michael_Woodford_(economist)" class="mw-redirect" title="Michael Woodford (economist)">Woodford, Michael</a> (1997). <a rel="nofollow" class="external text" href="http://www.nber.org/chapters/c11041.pdf">"An Optimization-based Econometric Framework for the Evaluation of Monetary Policy"</a> <span class="cs1-format">(PDF)</span>. <i>NBER Macroeconomics Annual</i>. <b>12</b>: 297–346. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3585236">10.2307/3585236</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3585236">3585236</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NBER+Macroeconomics+Annual&rft.atitle=An+Optimization-based+Econometric+Framework+for+the+Evaluation+of+Monetary+Policy&rft.volume=12&rft.pages=297-346&rft.date=1997&rft_id=info%3Adoi%2F10.2307%2F3585236&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3585236%23id-name%3DJSTOR&rft.aulast=Rotemberg&rft.aufirst=Julio&rft.au=Woodford%2C+Michael&rft_id=http%3A%2F%2Fwww.nber.org%2Fchapters%2Fc11041.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">From <i><a href="/wiki/The_New_Palgrave_Dictionary_of_Economics" title="The New Palgrave Dictionary of Economics">The New Palgrave Dictionary of Economics</a></i> (2008), 2nd Edition with Abstract links:<br />• "<a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2008_N000148&edition=current&q=optimization&topicid=&result_number=1">numerical optimization methods in economics</a>" by Karl Schmedders<br />• "<a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2008_C000348&edition=current&q=optimization&topicid=&result_number=4">convex programming</a>" by <a href="/wiki/Lawrence_E._Blume" title="Lawrence E. Blume">Lawrence E. Blume</a><br />• "<a rel="nofollow" class="external text" href="http://www.dictionaryofeconomics.com/article?id=pde2008_A000133&edition=current&q=optimization&topicid=&result_number=20">Arrow–Debreu model of general equilibrium</a>" by <a href="/wiki/John_Geanakoplos" title="John Geanakoplos">John Geanakoplos</a>.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeKarMandalGhoshal2014" class="citation journal cs1">De, Bishnu Prasad; Kar, R.; Mandal, D.; Ghoshal, S.P. (2014-09-27). "Optimal selection of components value for analog active filter design using simplex particle swarm optimization". <i>International Journal of Machine Learning and Cybernetics</i>. <b>6</b> (4): 621–636. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs13042-014-0299-0">10.1007/s13042-014-0299-0</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1868-8071">1868-8071</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13071135">13071135</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Machine+Learning+and+Cybernetics&rft.atitle=Optimal+selection+of+components+value+for+analog+active+filter+design+using+simplex+particle+swarm+optimization&rft.volume=6&rft.issue=4&rft.pages=621-636&rft.date=2014-09-27&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13071135%23id-name%3DS2CID&rft.issn=1868-8071&rft_id=info%3Adoi%2F10.1007%2Fs13042-014-0299-0&rft.aulast=De&rft.aufirst=Bishnu+Prasad&rft.au=Kar%2C+R.&rft.au=Mandal%2C+D.&rft.au=Ghoshal%2C+S.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKozielBandler2008" class="citation journal cs1">Koziel, Slawomir; Bandler, John W. (January 2008). "Space Mapping With Multiple Coarse Models for Optimization of Microwave Components". <i>IEEE Microwave and Wireless Components Letters</i>. <b>18</b> (1): 1–3. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.147.5407">10.1.1.147.5407</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLMWC.2007.911969">10.1109/LMWC.2007.911969</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11086218">11086218</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Microwave+and+Wireless+Components+Letters&rft.atitle=Space+Mapping+With+Multiple+Coarse+Models+for+Optimization+of+Microwave+Components&rft.volume=18&rft.issue=1&rft.pages=1-3&rft.date=2008-01&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.147.5407%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11086218%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FLMWC.2007.911969&rft.aulast=Koziel&rft.aufirst=Slawomir&rft.au=Bandler%2C+John+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuChengZhangBandler2013" class="citation journal cs1">Tu, Sheng; Cheng, Qingsha S.; Zhang, Yifan; Bandler, John W.; Nikolova, Natalia K. (July 2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAP.2013.2254695">"Space Mapping Optimization of Handset Antennas Exploiting Thin-Wire Models"</a>. <i>IEEE Transactions on Antennas and Propagation</i>. <b>61</b> (7): 3797–3807. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013ITAP...61.3797T">2013ITAP...61.3797T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAP.2013.2254695">10.1109/TAP.2013.2254695</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Antennas+and+Propagation&rft.atitle=Space+Mapping+Optimization+of+Handset+Antennas+Exploiting+Thin-Wire+Models&rft.volume=61&rft.issue=7&rft.pages=3797-3807&rft.date=2013-07&rft_id=info%3Adoi%2F10.1109%2FTAP.2013.2254695&rft_id=info%3Abibcode%2F2013ITAP...61.3797T&rft.aulast=Tu&rft.aufirst=Sheng&rft.au=Cheng%2C+Qingsha+S.&rft.au=Zhang%2C+Yifan&rft.au=Bandler%2C+John+W.&rft.au=Nikolova%2C+Natalia+K.&rft_id=https%3A%2F%2Fdoi.org%2F10.1109%252FTAP.2013.2254695&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">N. Friedrich, <a rel="nofollow" class="external text" href="http://mwrf.com/software/space-mapping-outpaces-em-optimization-handset-antenna-design">“Space mapping outpaces EM optimization in handset-antenna design,”</a> microwaves&rf, August 30, 2013.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCervantes-GonzálezRayas-SánchezLópezCamacho-Pérez2016" class="citation journal cs1">Cervantes-González, Juan C.; Rayas-Sánchez, José E.; López, Carlos A.; Camacho-Pérez, José R.; Brito-Brito, Zabdiel; Chávez-Hurtado, José L. (February 2016). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmmce.20945">"Space mapping optimization of handset antennas considering EM effects of mobile phone components and human body"</a>. <i><a href="/wiki/International_Journal_of_RF_and_Microwave_Computer-Aided_Engineering" title="International Journal of RF and Microwave Computer-Aided Engineering">International Journal of RF and Microwave Computer-Aided Engineering</a></i>. <b>26</b> (2): 121–128. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmmce.20945">10.1002/mmce.20945</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:110195165">110195165</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+RF+and+Microwave+Computer-Aided+Engineering&rft.atitle=Space+mapping+optimization+of+handset+antennas+considering+EM+effects+of+mobile+phone+components+and+human+body&rft.volume=26&rft.issue=2&rft.pages=121-128&rft.date=2016-02&rft_id=info%3Adoi%2F10.1002%2Fmmce.20945&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A110195165%23id-name%3DS2CID&rft.aulast=Cervantes-Gonz%C3%A1lez&rft.aufirst=Juan+C.&rft.au=Rayas-S%C3%A1nchez%2C+Jos%C3%A9+E.&rft.au=L%C3%B3pez%2C+Carlos+A.&rft.au=Camacho-P%C3%A9rez%2C+Jos%C3%A9+R.&rft.au=Brito-Brito%2C+Zabdiel&rft.au=Ch%C3%A1vez-Hurtado%2C+Jos%C3%A9+L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Fmmce.20945&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBandlerBiernackiChenGrobelny1994" class="citation journal cs1">Bandler, J.W.; Biernacki, R.M.; Chen, Shao Hua; Grobelny, P.A.; Hemmers, R.H. (1994). "Space mapping technique for electromagnetic optimization". <i>IEEE Transactions on Microwave Theory and Techniques</i>. <b>42</b> (12): 2536–2544. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994ITMTT..42.2536B">1994ITMTT..42.2536B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F22.339794">10.1109/22.339794</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Microwave+Theory+and+Techniques&rft.atitle=Space+mapping+technique+for+electromagnetic+optimization&rft.volume=42&rft.issue=12&rft.pages=2536-2544&rft.date=1994&rft_id=info%3Adoi%2F10.1109%2F22.339794&rft_id=info%3Abibcode%2F1994ITMTT..42.2536B&rft.aulast=Bandler&rft.aufirst=J.W.&rft.au=Biernacki%2C+R.M.&rft.au=Chen%2C+Shao+Hua&rft.au=Grobelny%2C+P.A.&rft.au=Hemmers%2C+R.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBandlerBiernackiShao_Hua_ChenHemmers1995" class="citation journal cs1">Bandler, J.W.; Biernacki, R.M.; Shao Hua Chen; Hemmers, R.H.; Madsen, K. (1995). "Electromagnetic optimization exploiting aggressive space mapping". <i>IEEE Transactions on Microwave Theory and Techniques</i>. <b>43</b> (12): 2874–2882. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995ITMTT..43.2874B">1995ITMTT..43.2874B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F22.475649">10.1109/22.475649</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Microwave+Theory+and+Techniques&rft.atitle=Electromagnetic+optimization+exploiting+aggressive+space+mapping&rft.volume=43&rft.issue=12&rft.pages=2874-2882&rft.date=1995&rft_id=info%3Adoi%2F10.1109%2F22.475649&rft_id=info%3Abibcode%2F1995ITMTT..43.2874B&rft.aulast=Bandler&rft.aufirst=J.W.&rft.au=Biernacki%2C+R.M.&rft.au=Shao+Hua+Chen&rft.au=Hemmers%2C+R.H.&rft.au=Madsen%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation conference cs1"><a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/6629391"><i>Convex relaxation of optimal power flow: A tutorial</i></a>. 2013 iREP Symposium on Bulk Power System Dynamics and Control. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIREP.2013.6629391">10.1109/IREP.2013.6629391</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Convex+relaxation+of+optimal+power+flow%3A+A+tutorial&rft_id=info%3Adoi%2F10.1109%2FIREP.2013.6629391&rft_id=https%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F6629391&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPiryonesiTavakolan2017" class="citation journal cs1">Piryonesi, Sayed Madeh; Tavakolan, Mehdi (9 January 2017). "A mathematical programming model for solving cost-safety optimization (CSO) problems in the maintenance of structures". <i>KSCE Journal of Civil Engineering</i>. <b>21</b> (6): 2226–2234. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017KSJCE..21.2226P">2017KSJCE..21.2226P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12205-017-0531-z">10.1007/s12205-017-0531-z</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:113616284">113616284</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=KSCE+Journal+of+Civil+Engineering&rft.atitle=A+mathematical+programming+model+for+solving+cost-safety+optimization+%28CSO%29+problems+in+the+maintenance+of+structures&rft.volume=21&rft.issue=6&rft.pages=2226-2234&rft.date=2017-01-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A113616284%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs12205-017-0531-z&rft_id=info%3Abibcode%2F2017KSJCE..21.2226P&rft.aulast=Piryonesi&rft.aufirst=Sayed+Madeh&rft.au=Tavakolan%2C+Mehdi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHegazy1999" class="citation journal cs1">Hegazy, Tarek (June 1999). "Optimization of Resource Allocation and Leveling Using Genetic Algorithms". <i>Journal of Construction Engineering and Management</i>. <b>125</b> (3): 167–175. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2F%28ASCE%290733-9364%281999%29125%3A3%28167%29">10.1061/(ASCE)0733-9364(1999)125:3(167)</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Construction+Engineering+and+Management&rft.atitle=Optimization+of+Resource+Allocation+and+Leveling+Using+Genetic+Algorithms&rft.volume=125&rft.issue=3&rft.pages=167-175&rft.date=1999-06&rft_id=info%3Adoi%2F10.1061%2F%28ASCE%290733-9364%281999%29125%3A3%28167%29&rft.aulast=Hegazy&rft.aufirst=Tarek&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-:0-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_30-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPiryonesiNasseriRamezani2018" class="citation journal cs1">Piryonesi, S. Madeh; Nasseri, Mehran; Ramezani, Abdollah (9 July 2018). "Piryonesi, S. M., Nasseri, M., & Ramezani, A. (2018). Resource leveling in construction projects with activity splitting and resource constraints: a simulated annealing optimization". <i>Canadian Journal of Civil Engineering</i>. <b>46</b>: 81–86. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1139%2Fcjce-2017-0670">10.1139/cjce-2017-0670</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1807%2F93364">1807/93364</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:116480238">116480238</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Civil+Engineering&rft.atitle=Piryonesi%2C+S.+M.%2C+Nasseri%2C+M.%2C+%26+Ramezani%2C+A.+%282018%29.+Resource+leveling+in+construction+projects+with+activity+splitting+and+resource+constraints%3A+a+simulated+annealing+optimization.&rft.volume=46&rft.pages=81-86&rft.date=2018-07-09&rft_id=info%3Ahdl%2F1807%2F93364&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A116480238%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1139%2Fcjce-2017-0670&rft.aulast=Piryonesi&rft.aufirst=S.+Madeh&rft.au=Nasseri%2C+Mehran&rft.au=Ramezani%2C+Abdollah&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHertyKlar2003" class="citation journal cs1">Herty, M.; Klar, A. (2003-01-01). <a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/10.1137/S106482750241459X">"Modeling, Simulation, and Optimization of Traffic Flow Networks"</a>. <i>SIAM Journal on Scientific Computing</i>. <b>25</b> (3): 1066–1087. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003SJSC...25.1066H">2003SJSC...25.1066H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS106482750241459X">10.1137/S106482750241459X</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1064-8275">1064-8275</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIAM+Journal+on+Scientific+Computing&rft.atitle=Modeling%2C+Simulation%2C+and+Optimization+of+Traffic+Flow+Networks&rft.volume=25&rft.issue=3&rft.pages=1066-1087&rft.date=2003-01-01&rft.issn=1064-8275&rft_id=info%3Adoi%2F10.1137%2FS106482750241459X&rft_id=info%3Abibcode%2F2003SJSC...25.1066H&rft.aulast=Herty&rft.aufirst=M.&rft.au=Klar%2C+A.&rft_id=https%3A%2F%2Fepubs.siam.org%2Fdoi%2F10.1137%2FS106482750241459X&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141218090504/http://www.seophonist-wahl.de/">"New force on the political scene: the Seophonisten"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.seophonist-wahl.de/">the original</a> on 18 December 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">14 September</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=New+force+on+the+political+scene%3A+the+Seophonisten&rft_id=http%3A%2F%2Fwww.seophonist-wahl.de%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-Papoutsakis_1984-33"><span class="mw-cite-backlink">^ <a href="#cite_ref-Papoutsakis_1984_33-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Papoutsakis_1984_33-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPapoutsakis1984" class="citation journal cs1">Papoutsakis, Eleftherios Terry (February 1984). "Equations and calculations for fermentations of butyric acid bacteria". <i>Biotechnology and Bioengineering</i>. <b>26</b> (2): 174–187. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fbit.260260210">10.1002/bit.260260210</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0006-3592">0006-3592</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18551704">18551704</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:25023799">25023799</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Biotechnology+and+Bioengineering&rft.atitle=Equations+and+calculations+for+fermentations+of+butyric+acid+bacteria&rft.volume=26&rft.issue=2&rft.pages=174-187&rft.date=1984-02&rft.issn=0006-3592&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A25023799%23id-name%3DS2CID&rft_id=info%3Apmid%2F18551704&rft_id=info%3Adoi%2F10.1002%2Fbit.260260210&rft.aulast=Papoutsakis&rft.aufirst=Eleftherios+Terry&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangJoshiZhangXu2006" class="citation journal cs1">Wang, Yong; Joshi, Trupti; Zhang, Xiang-Sun; Xu, Dong; Chen, Luonan (2006-07-24). "Inferring gene regulatory networks from multiple microarray datasets". <i>Bioinformatics</i>. <b>22</b> (19): 2413–2420. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtl396">10.1093/bioinformatics/btl396</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1460-2059">1460-2059</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16864593">16864593</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bioinformatics&rft.atitle=Inferring+gene+regulatory+networks+from+multiple+microarray+datasets&rft.volume=22&rft.issue=19&rft.pages=2413-2420&rft.date=2006-07-24&rft.issn=1460-2059&rft_id=info%3Apmid%2F16864593&rft_id=info%3Adoi%2F10.1093%2Fbioinformatics%2Fbtl396&rft.aulast=Wang&rft.aufirst=Yong&rft.au=Joshi%2C+Trupti&rft.au=Zhang%2C+Xiang-Sun&rft.au=Xu%2C+Dong&rft.au=Chen%2C+Luonan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangWangZhangChen2007" class="citation journal cs1">Wang, Rui-Sheng; Wang, Yong; Zhang, Xiang-Sun; Chen, Luonan (2007-09-22). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtm465">"Inferring transcriptional regulatory networks from high-throughput data"</a>. <i>Bioinformatics</i>. <b>23</b> (22): 3056–3064. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2Fbtm465">10.1093/bioinformatics/btm465</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1460-2059">1460-2059</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17890736">17890736</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bioinformatics&rft.atitle=Inferring+transcriptional+regulatory+networks+from+high-throughput+data&rft.volume=23&rft.issue=22&rft.pages=3056-3064&rft.date=2007-09-22&rft.issn=1460-2059&rft_id=info%3Apmid%2F17890736&rft_id=info%3Adoi%2F10.1093%2Fbioinformatics%2Fbtm465&rft.aulast=Wang&rft.aufirst=Rui-Sheng&rft.au=Wang%2C+Yong&rft.au=Zhang%2C+Xiang-Sun&rft.au=Chen%2C+Luonan&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fbioinformatics%252Fbtm465&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVoPaul_LeePalsson2007" class="citation journal cs1">Vo, Thuy D.; Paul Lee, W.N.; Palsson, Bernhard O. (May 2007). "Systems analysis of energy metabolism elucidates the affected respiratory chain complex in Leigh's syndrome". <i>Molecular Genetics and Metabolism</i>. <b>91</b> (1): 15–22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ymgme.2007.01.012">10.1016/j.ymgme.2007.01.012</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1096-7192">1096-7192</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17336115">17336115</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Molecular+Genetics+and+Metabolism&rft.atitle=Systems+analysis+of+energy+metabolism+elucidates+the+affected+respiratory+chain+complex+in+Leigh%27s+syndrome&rft.volume=91&rft.issue=1&rft.pages=15-22&rft.date=2007-05&rft.issn=1096-7192&rft_id=info%3Apmid%2F17336115&rft_id=info%3Adoi%2F10.1016%2Fj.ymgme.2007.01.012&rft.aulast=Vo&rft.aufirst=Thuy+D.&rft.au=Paul+Lee%2C+W.N.&rft.au=Palsson%2C+Bernhard+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendesKell1998" class="citation journal cs1"><a href="/wiki/Pedro_Pedrosa_Mendes" class="mw-redirect" title="Pedro Pedrosa Mendes">Mendes, P.</a>; Kell, D. (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2F14.10.869">"Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation"</a>. <i>Bioinformatics</i>. <b>14</b> (10): 869–883. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fbioinformatics%2F14.10.869">10.1093/bioinformatics/14.10.869</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1367-4803">1367-4803</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9927716">9927716</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bioinformatics&rft.atitle=Non-linear+optimization+of+biochemical+pathways%3A+applications+to+metabolic+engineering+and+parameter+estimation&rft.volume=14&rft.issue=10&rft.pages=869-883&rft.date=1998&rft.issn=1367-4803&rft_id=info%3Apmid%2F9927716&rft_id=info%3Adoi%2F10.1093%2Fbioinformatics%2F14.10.869&rft.aulast=Mendes&rft.aufirst=P.&rft.au=Kell%2C+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fbioinformatics%252F14.10.869&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=35" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Stephen_P._Boyd" title="Stephen P. Boyd">Boyd, Stephen P.</a>; Vandenberghe, Lieven (2004). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~boyd/cvxbook/"><i>Convex Optimization</i></a>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83378-7" title="Special:BookSources/0-521-83378-7"><bdi>0-521-83378-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+Optimization&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=0-521-83378-7&rft.aulast=Boyd&rft.aufirst=Stephen+P.&rft.au=Vandenberghe%2C+Lieven&rft_id=https%3A%2F%2Fweb.stanford.edu%2F~boyd%2Fcvxbook%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Gill, P. E.; Murray, W.; <a href="/wiki/Margaret_H._Wright" title="Margaret H. Wright">Wright, M. H.</a> (1982). <i>Practical Optimization</i>. London: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-283952-8" title="Special:BookSources/0-12-283952-8"><bdi>0-12-283952-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Practical+Optimization&rft.place=London&rft.pub=Academic+Press&rft.date=1982&rft.isbn=0-12-283952-8&rft.aulast=Gill&rft.aufirst=P.+E.&rft.au=Murray%2C+W.&rft.au=Wright%2C+M.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Jon_Lee_(mathematician)" title="Jon Lee (mathematician)">Lee, Jon</a> (2004). <i>A First Course in Combinatorial Optimization</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-01012-8" title="Special:BookSources/0-521-01012-8"><bdi>0-521-01012-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Combinatorial+Optimization&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=0-521-01012-8&rft.aulast=Lee&rft.aufirst=Jon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Jorge_Nocedal" title="Jorge Nocedal">Nocedal, Jorge</a>; Wright, Stephen J. (2006). <a rel="nofollow" class="external text" href="http://www.ece.northwestern.edu/~nocedal/book/num-opt.html"><i>Numerical Optimization</i></a> (2nd ed.). Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-30303-0" title="Special:BookSources/0-387-30303-0"><bdi>0-387-30303-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Optimization&rft.place=Berlin&rft.edition=2nd&rft.pub=Springer&rft.date=2006&rft.isbn=0-387-30303-0&rft.aulast=Nocedal&rft.aufirst=Jorge&rft.au=Wright%2C+Stephen+J.&rft_id=http%3A%2F%2Fwww.ece.northwestern.edu%2F~nocedal%2Fbook%2Fnum-opt.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_optimization&action=edit&section=36" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_optimization" class="extiw" title="commons:Category:Mathematical optimization">Mathematical optimization</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://plato.asu.edu/guide.html">"Decision Tree for Optimization Software"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Decision+Tree+for+Optimization+Software&rft_id=http%3A%2F%2Fplato.asu.edu%2Fguide.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span> Links to optimization source codes</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mat.univie.ac.at/~neum/glopt.html">"Global optimization"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Global+optimization&rft_id=https%3A%2F%2Fwww.mat.univie.ac.at%2F~neum%2Fglopt.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://see.stanford.edu/Course/EE364A">"EE364a: Convex Optimization I"</a>. <i>Course from Stanford University</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Course+from+Stanford+University&rft.atitle=EE364a%3A+Convex+Optimization+I&rft_id=https%3A%2F%2Fsee.stanford.edu%2FCourse%2FEE364A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaroquaux" class="citation web cs1">Varoquaux, Gaël. <a rel="nofollow" class="external text" href="https://scipy-lectures.org/advanced/mathematical_optimization/index.html">"Mathematical Optimization: Finding Minima of Functions"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mathematical+Optimization%3A+Finding+Minima+of+Functions&rft.aulast=Varoquaux&rft.aufirst=Ga%C3%ABl&rft_id=https%3A%2F%2Fscipy-lectures.org%2Fadvanced%2Fmathematical_optimization%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+optimization" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Optimization:_Algorithms,_methods,_and_heuristics" style="padding:3px"><table class="nowraplinks hlist mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Optimization_algorithms" title="Template:Optimization algorithms"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Optimization_algorithms" title="Template talk:Optimization algorithms"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Optimization_algorithms" title="Special:EditPage/Template:Optimization algorithms"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Optimization:_Algorithms,_methods,_and_heuristics" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Optimization</a>: <a href="/wiki/Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="/wiki/Iterative_method" title="Iterative method">methods</a>, and <a href="/wiki/Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Golden-section_search" title="Golden-section search">Golden-section search</a></li> <li><a href="/wiki/Powell%27s_method" title="Powell's method">Powell's method</a></li> <li><a href="/wiki/Line_search" title="Line search">Line search</a></li> <li><a href="/wiki/Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li> <li><a href="/wiki/Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Trust_region" title="Trust region">Trust region</a></li> <li><a href="/wiki/Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li> <li><a href="/wiki/Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="/wiki/Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li> <li><a href="/wiki/Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li> <li><a href="/wiki/Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li> <li><a href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li> <li><a href="/wiki/Gradient_descent" title="Gradient descent">Gradient</a></li> <li><a href="/wiki/Mirror_descent" title="Mirror descent">Mirror</a></li> <li><a href="/wiki/Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li> <li><a href="/wiki/Powell%27s_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li> <li><a href="/wiki/Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:Max_paraboloid.svg" class="mw-file-description" title="Optimization computes maxima and minima."><img alt="Graph of a strictly concave quadratic function with unique maximum." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/150px-Max_paraboloid.svg.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/225px-Max_paraboloid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Max_paraboloid.svg/300px-Max_paraboloid.svg.png 2x" data-file-width="700" data-file-height="560" /></a><figcaption>Optimization computes maxima and minima.</figcaption></figure></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear" style="font-size:114%;margin:0 4em"><a href="/wiki/Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barrier_function" title="Barrier function">Barrier methods</a></li> <li><a href="/wiki/Penalty_method" title="Penalty method">Penalty methods</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li> <li><a href="/wiki/Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li> <li><a href="/wiki/Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization" style="font-size:114%;margin:0 4em"><a href="/wiki/Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Convex_minimization" class="mw-redirect" title="Convex minimization">Convex<br /> minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li> <li><a href="/wiki/Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li> <li><a href="/wiki/Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming" title="Linear programming">Linear</a> and<br /><a href="/wiki/Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Affine_scaling" title="Affine scaling">Affine scaling</a></li> <li><a href="/wiki/Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li> <li><a href="/wiki/Karmarkar%27s_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matroid" title="Matroid">Basis-</a><a href="/wiki/Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li> <li><a href="/wiki/Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li> <li><a href="/wiki/Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li> <li><a href="/wiki/Lemke%27s_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial" style="font-size:114%;margin:0 4em"><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li> <li><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li> <li><a href="/wiki/Integer_programming" title="Integer programming">Integer programming</a> <ul><li><a href="/wiki/Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="/wiki/Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph<br /> algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Minimum_spanning_tree" title="Minimum spanning tree">Minimum<br /> spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bor%C5%AFvka%27s_algorithm" title="Borůvka's algorithm">Borůvka</a></li> <li><a href="/wiki/Prim%27s_algorithm" title="Prim's algorithm">Prim</a></li> <li><a href="/wiki/Kruskal%27s_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul> </div></td></tr></tbody></table><div> </div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a> <ul><li><a href="/wiki/Shortest_Path_Faster_Algorithm" class="mw-redirect" title="Shortest Path Faster Algorithm">SPFA</a></li></ul></li> <li><a href="/wiki/Dijkstra%27s_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li> <li><a href="/wiki/Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dinic%27s_algorithm" title="Dinic's algorithm">Dinic</a></li> <li><a href="/wiki/Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li> <li><a href="/wiki/Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li> <li><a href="/wiki/Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Metaheuristics" style="font-size:114%;margin:0 4em"><a href="/wiki/Metaheuristic" title="Metaheuristic">Metaheuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li> <li><a href="/wiki/Hill_climbing" title="Hill climbing">Hill climbing</a></li> <li><a href="/wiki/Local_search_(optimization)" title="Local search (optimization)">Local search</a></li> <li><a href="/wiki/Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li> <li><a href="/wiki/Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li> <li><a href="/wiki/Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li> <li><a href="/wiki/Tabu_search" title="Tabu search">Tabu search</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calculus" title="Calculus">Calculus</a></li> <li><a href="/wiki/Real_analysis" title="Real analysis">Real analysis</a></li> <li><a href="/wiki/Complex_analysis" title="Complex analysis">Complex analysis</a></li> <li><a href="/wiki/Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li> <li><a href="/wiki/Differential_equation" title="Differential equation">Differential equations</a></li> <li><a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a></li> <li><a href="/wiki/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a href="/wiki/Control_theory" title="Control theory">Control theory</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Computational_mathematics" title="Computational mathematics">Computational</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Computer_science" title="Computer science">Computer science</a></li> <li><a href="/wiki/Theory_of_computation" title="Theory of computation">Theory of computation</a></li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li> <li><a href="/wiki/Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li> <li><a class="mw-selflink selflink">Optimization</a></li> <li><a href="/wiki/Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a> <ul><li><a href="/wiki/List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li> <li><a href="/wiki/Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li> <li><a href="/wiki/List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li> <li><a href="/wiki/Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <b><a href="/wiki/Category:Fields_of_mathematics" title="Category:Fields of mathematics">Category</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw" title="commons:Category:Mathematics">Commons</a></b></li> <li><span class="noviewer" typeof="mw:File"><span title="WikiProject"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/16px-People_icon.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/24px-People_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/People_icon.svg/32px-People_icon.svg.png 2x" data-file-width="100" data-file-height="100" /></span></span> <b><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">WikiProject</a></b></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Systems_engineering" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Systems_engineering" title="Template:Systems engineering"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Systems_engineering" title="Template talk:Systems engineering"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Systems_engineering" title="Special:EditPage/Template:Systems engineering"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Systems_engineering" style="font-size:114%;margin:0 4em"><a href="/wiki/Systems_engineering" title="Systems engineering">Systems engineering</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Subfields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aerospace_engineering" title="Aerospace engineering">Aerospace engineering</a></li> <li><a href="/wiki/Biological_systems_engineering" title="Biological systems engineering">Biological systems engineering</a></li> <li><a href="/wiki/Cognitive_systems_engineering" title="Cognitive systems engineering">Cognitive systems engineering</a></li> <li><a href="/wiki/Configuration_management" title="Configuration management">Configuration management</a></li> <li><a href="/wiki/Earth_systems_engineering_and_management" title="Earth systems engineering and management">Earth systems engineering and management</a></li> <li><a href="/wiki/Electrical_engineering" title="Electrical engineering">Electrical engineering</a></li> <li><a href="/wiki/Enterprise_systems_engineering" title="Enterprise systems engineering">Enterprise systems engineering</a></li> <li><a href="/wiki/Health_systems_engineering" title="Health systems engineering">Health systems engineering</a></li> <li><a href="/wiki/Performance_engineering" title="Performance engineering">Performance engineering</a></li> <li><a href="/wiki/Reliability_engineering" title="Reliability engineering">Reliability engineering</a></li> <li><a href="/wiki/Safety_engineering" title="Safety engineering">Safety engineering</a></li> <li><a href="/wiki/Stanford_torus" title="Stanford torus">Sociocultural Systems Engineering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Processes</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Requirements_engineering" title="Requirements engineering">Requirements engineering</a></li> <li><a href="/wiki/Functional_specification" title="Functional specification">Functional specification</a></li> <li><a href="/wiki/System_integration" title="System integration">System integration</a></li> <li><a href="/wiki/Verification_and_validation" title="Verification and validation">Verification and validation</a></li> <li><a href="/wiki/Design_review" title="Design review">Design review</a></li> <li><a href="/wiki/System_of_systems_engineering" title="System of systems engineering">System of systems engineering</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Business_process" title="Business process">Business process</a></li> <li><a href="/wiki/Fault_tolerance" title="Fault tolerance">Fault tolerance</a></li> <li><a href="/wiki/System" title="System">System</a></li> <li><a href="/wiki/System_lifecycle" class="mw-redirect" title="System lifecycle">System lifecycle</a></li> <li><a href="/wiki/V-model" title="V-model">V-Model</a></li> <li><a href="/wiki/Systems_development_life_cycle" title="Systems development life cycle">Systems development life cycle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Decision-making" title="Decision-making">Decision-making</a></li> <li><a href="/wiki/Function_model" title="Function model">Function modelling</a></li> <li><a href="/wiki/IDEF" title="IDEF">IDEF</a></li> <li><a class="mw-selflink selflink">Optimization</a></li> <li><a href="/wiki/Quality_function_deployment" title="Quality function deployment">Quality function deployment</a></li> <li><a href="/wiki/System_dynamics" title="System dynamics">System dynamics</a></li> <li><a href="/wiki/Systems_Modeling_Language" class="mw-redirect" title="Systems Modeling Language">Systems Modeling Language</a></li> <li><a href="/wiki/Systems_analysis" title="Systems analysis">Systems analysis</a></li> <li><a href="/wiki/Systems_modeling" title="Systems modeling">Systems modeling</a></li> <li><a href="/wiki/Work_breakdown_structure" title="Work breakdown structure">Work breakdown structure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/James_S._Albus" title="James S. Albus">James S. Albus</a></li> <li><a href="/wiki/Ruzena_Bajcsy" title="Ruzena Bajcsy">Ruzena Bajcsy</a></li> <li><a href="/wiki/Benjamin_S._Blanchard" title="Benjamin S. Blanchard">Benjamin S. Blanchard</a></li> <li><a href="/wiki/Wernher_von_Braun" title="Wernher von Braun">Wernher von Braun</a></li> <li><a href="/wiki/Kathleen_Carley" title="Kathleen Carley">Kathleen Carley</a></li> <li><a href="/wiki/Harold_Chestnut" title="Harold Chestnut">Harold Chestnut</a></li> <li><a href="/wiki/Wolt_Fabrycky" title="Wolt Fabrycky">Wolt Fabrycky</a></li> <li><a href="/wiki/Barbara_Grosz" class="mw-redirect" title="Barbara Grosz">Barbara Grosz</a></li> <li><a href="/wiki/Arthur_David_Hall_III" title="Arthur David Hall III">Arthur David Hall III</a></li> <li><a href="/wiki/Derek_Hitchins" title="Derek Hitchins">Derek Hitchins</a></li> <li><a href="/wiki/Robert_E._Machol" title="Robert E. Machol">Robert E. Machol</a></li> <li><a href="/wiki/Radhika_Nagpal" title="Radhika Nagpal">Radhika Nagpal</a></li> <li><a href="/wiki/Simon_Ramo" title="Simon Ramo">Simon Ramo</a></li> <li><a href="/wiki/Joseph_Francis_Shea" title="Joseph Francis Shea">Joseph Francis Shea</a></li> <li><a href="/wiki/Katia_Sycara" title="Katia Sycara">Katia Sycara</a></li> <li><a href="/wiki/Manuela_M._Veloso" title="Manuela M. Veloso">Manuela M. Veloso</a></li> <li><a href="/wiki/John_N._Warfield" title="John N. Warfield">John N. Warfield</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related fields</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Control_engineering" title="Control engineering">Control engineering</a></li> <li><a href="/wiki/Computer_engineering" title="Computer engineering">Computer engineering</a></li> <li><a href="/wiki/Industrial_engineering" title="Industrial engineering">Industrial engineering</a></li> <li><a href="/wiki/Operations_research" title="Operations research">Operations research</a></li> <li><a href="/wiki/Project_management" title="Project management">Project management</a></li> <li><a href="/wiki/Quality_management" title="Quality management">Quality management</a></li> <li><a href="/wiki/Risk_management" title="Risk management">Risk management</a></li> <li><a href="/wiki/Software_engineering" title="Software engineering">Software engineering</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Category:Systems_engineering" title="Category:Systems engineering">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q141495#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4043664-0">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Programming (Mathematics)"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85107312">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Programmation (mathématiques)"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11932649z">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Programmation (mathématiques)"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11932649z">BnF data</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="matematická optimalizace"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph122672&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX533091">Spain</a></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007538691105171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Mathematical_optimization&oldid=1257438818">https://en.wikipedia.org/w/index.php?title=Mathematical_optimization&oldid=1257438818</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mathematical_optimization" title="Category:Mathematical optimization">Mathematical optimization</a></li><li><a href="/wiki/Category:Operations_research" title="Category:Operations research">Operations research</a></li><li><a href="/wiki/Category:Mathematical_and_quantitative_methods_(economics)" title="Category:Mathematical and quantitative methods (economics)">Mathematical and quantitative methods (economics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Commons_category_link_from_Wikidata" title="Category:Commons category link from Wikidata">Commons category link from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 November 2024, at 22:12<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Mathematical_optimization&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-jlfln","wgBackendResponseTime":159,"wgPageParseReport":{"limitreport":{"cputime":"0.960","walltime":"1.188","ppvisitednodes":{"value":5520,"limit":1000000},"postexpandincludesize":{"value":188402,"limit":2097152},"templateargumentsize":{"value":4548,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":28,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":181434,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 932.554 1 -total"," 36.30% 338.484 1 Template:Reflist"," 14.42% 134.455 22 Template:Cite_journal"," 13.97% 130.314 8 Template:Cite_web"," 13.02% 121.451 1 Template:Optimization_algorithms"," 12.70% 118.450 1 Template:Navbox_with_collapsible_groups"," 8.67% 80.881 1 Template:Short_description"," 8.50% 79.306 1 Template:Commons_category"," 8.26% 76.994 1 Template:Sister_project"," 8.05% 75.041 1 Template:Side_box"]},"scribunto":{"limitreport-timeusage":{"value":"0.582","limit":"10.000"},"limitreport-memusage":{"value":7095882,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-pdn6j","timestamp":"20241122140956","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Mathematical optimization","url":"https:\/\/en.wikipedia.org\/wiki\/Mathematical_optimization","sameAs":"http:\/\/www.wikidata.org\/entity\/Q141495","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q141495","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-05-14T14:07:38Z","dateModified":"2024-11-14T22:12:45Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/72\/Max_paraboloid.svg","headline":"study of mathematical algorithms for optimization problems"}</script> </body> </html>