Document Zbl 1470.90060

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1470.90060 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1470.90060?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1470.90060" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/chen.xi.6" title="Author Profile">Chen, Xi</a>; <a href="/authors/he.simai" title="Author Profile">He, Simai</a>; <a href="/authors/jiang.bo.2" title="Author Profile">Jiang, Bo</a>; <a href="/authors/ryan.christopher-thomas" title="Author Profile">Ryan, Christopher Thomas</a>; <a href="/authors/zhang.teng" title="Author Profile">Zhang, Teng</a></div> <h2 class="title"> <strong>The discrete moment problem with nonconvex shape constraints.</strong> <i>(English)</i> <a class="label nowrap" href="/1470.90060">Zbl 1470.90060</a> </h2> <div class="source"> <a href="/serials/611" title="Journal Profile">Oper. Res.</a> <a href="/?q=in%3A465138" title="Articles in this Issue">69, No. 1, 279-296 (2021)</a>. </div> <div class="abstract">Summary: The discrete moment problem is a foundational problem in distribution-free robust optimization, where the goal is to find a worst-case distribution that satisfies a given set of moments. This paper studies the discrete moment problems with additional shape constraints that guarantee the worst-case distribution is either log-concave (LC), has an increasing failure rate (IFR), or increasing generalized failure rate (IGFR). These classes of shape constraints have not previously been studied in the literature, in part due to their inherent nonconvexities. Nonetheless, these classes are useful in practice, with applications in revenue management, reliability, and inventory control. We characterize the structure of optimal extreme point distributions under these constraints. We show, for example, that an optimal extreme point solution to a moment problem with $m$ moments and LC shape constraints is piecewise geometric with at most $m$ pieces. This optimality structure allows us to design an exact algorithm for computing optimal solutions in a low-dimensional space of parameters. We also leverage this structure to study a robust newsvendor problem with shape constraints and compute optimal solutions.<br class="zbmathjax-paragraph">The e-companion is available at <a href="https://doi.org/10.1287/opre.2020.1990">https://doi.org/10.1287/opre.2020.1990</a>.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7360277">Cited in <strong>6</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90C17" title="MSC2020">90C17</a> </td> <td class="space"> Robustness in mathematical programming </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aprogramming">programming</a>; <a href="/?q=ut%3Astochastic+and+probability">stochastic and probability</a>; <a href="/?q=ut%3Adistributions">distributions</a>; <a href="/?q=ut%3Aoptimization">optimization</a>; <a href="/?q=ut%3Arobust+optimization">robust optimization</a>; <a href="/?q=ut%3Amoment+problem">moment problem</a>; <a href="/?q=ut%3Ashape+constraints">shape constraints</a>; <a href="/?q=ut%3Areverse+convex+programming">reverse convex programming</a></div> <div class="software"> <h3>Software:</h3><a href="/software/4733">ROME</a>; <a href="/software/29674">logcondiscr</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1470.90060" data-ciurl="/ci/07360277" data-biburl="/bibtex/07360277.bib" data-amsurl="/amsrefs/07360277.bib" data-xmlurl="/xml/07360277.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07360277.pdf" title="Zbl 1470.90060 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1287/opre.2020.1990" aria-label="DOI for “The discrete moment problem with nonconvex shape constraints”" title="10.1287/opre.2020.1990">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1708.02079"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Aliprantis C, Border K (2006) <span class="zbmathjax-textit">Infinite Dimensional Analysis: A Hitchhiker’s Guide</span>, 3rd ed. (Springer Verlag, Berlin, Heidelberg).Google Scholar · <a href="/1156.46001" class="nowrap">Zbl 1156.46001</a></td> </tr><tr> <td>[2]</td> <td class="space">An M (1997) Log-concave probability distributions: Theory and statistical testing. Working paper, Duke University, Durham, NC.Google Scholar</td> </tr><tr> <td>[3]</td> <td class="space">Bagnoli M, Bergstrom T (2005) Log-concave probability and its applications. <span class="zbmathjax-textit">Econom. Theory</span> 26(2):445-469.Crossref, Google Scholar · <a href="/1077.60012" class="nowrap">Zbl 1077.60012</a> · <a href="https://doi.org/10.1007/s00199-004-0514-4" class="nowrap">doi:10.1007/s00199-004-0514-4</a></td> </tr><tr> <td>[4]</td> <td class="space">Bagnoli M, Salant SW, Swierzbinski JE (1989) Durable-goods monopoly with discrete demand. <span class="zbmathjax-textit">J. Political Econom.</span> 97(6):1459-1478.Crossref, Google Scholar · <a href="https://doi.org/10.1086/261663" class="nowrap">doi:10.1086/261663</a></td> </tr><tr> <td>[5]</td> <td class="space">Balabdaoui F, Jankowski H, Rufibach K, Pavlides M (2011) Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. <span class="zbmathjax-textit">J. Royal Statist. Soc. Ser. A</span> 75(4):769-790.Crossref, Google Scholar · <a href="/1411.62119" class="nowrap">Zbl 1411.62119</a> · <a href="https://doi.org/10.1111/rssb.12011" class="nowrap">doi:10.1111/rssb.12011</a></td> </tr><tr> <td>[6]</td> <td class="space">Banciu M, Mirchandani P (2013) New results concerning probability distributions with increasing generalized failure rates. <span class="zbmathjax-textit">Oper. Res.</span> 61(4):925-931.Link, Google Scholar · <a href="/1291.90078" class="nowrap">Zbl 1291.90078</a></td> </tr><tr> <td>[7]</td> <td class="space">Bandi C, Bertsimas D (2012) Tractable stochastic analysis in high dimensions via robust optimization. <span class="zbmathjax-textit">Math. Programming</span> 134:23-70.Crossref, Google Scholar · <a href="/1263.90001" class="nowrap">Zbl 1263.90001</a> · <a href="https://doi.org/10.1007/s10107-012-0567-2" class="nowrap">doi:10.1007/s10107-012-0567-2</a></td> </tr><tr> <td>[8]</td> <td class="space">Barlow R, Proschan F (1996) <span class="zbmathjax-textit">Mathematical Theory of Reliability</span>, volume 17 (SIAM, Philadelphia).Crossref, Google Scholar · <a href="/0132.39302" class="nowrap">Zbl 0132.39302</a> · <a href="https://doi.org/10.1137/1.9781611971194" class="nowrap">doi:10.1137/1.9781611971194</a></td> </tr><tr> <td>[9]</td> <td class="space">Baron DP, Myerson RB (1982) Regulating a monopolist with unknown costs. <span class="zbmathjax-textit">Econometrica</span> 50(4):911-930.Google Scholar · <a href="/0483.90019" class="nowrap">Zbl 0483.90019</a></td> </tr><tr> <td>[10]</td> <td class="space">Bertsimas D, Popescu I (2005) Optimal inequalities in probability theory: A convex optimization approach. <span class="zbmathjax-textit">SIAM J. Optim.</span> 15(3):780-804.Crossref, Google Scholar · <a href="/1077.60020" class="nowrap">Zbl 1077.60020</a> · <a href="https://doi.org/10.1137/S1052623401399903" class="nowrap">doi:10.1137/S1052623401399903</a></td> </tr><tr> <td>[11]</td> <td class="space">Bertsimas D, Gupta V, Kallus N (2018) Data-driven robust optimization. <span class="zbmathjax-textit">Math. Programming</span> 167(2):235-292.Crossref, Google Scholar · <a href="/1397.90298" class="nowrap">Zbl 1397.90298</a> · <a href="https://doi.org/10.1007/s10107-017-1125-8" class="nowrap">doi:10.1007/s10107-017-1125-8</a></td> </tr><tr> <td>[12]</td> <td class="space">Bertsimas D, Natarajan K, Teo CP (2006) Persistence in discrete optimization under data uncertainty. <span class="zbmathjax-textit">Math. Programming</span> 108(2):251-274.Crossref, Google Scholar · <a href="/1130.90365" class="nowrap">Zbl 1130.90365</a> · <a href="https://doi.org/10.1007/s10107-006-0710-z" class="nowrap">doi:10.1007/s10107-006-0710-z</a></td> </tr><tr> <td>[13]</td> <td class="space">Burer S (2009) On the copositive representation of binary and continuous nonconvex quadratic programs. <span class="zbmathjax-textit">Math. Programming</span> 120(2):479-495.Crossref, Google Scholar · <a href="/1180.90234" class="nowrap">Zbl 1180.90234</a> · <a href="https://doi.org/10.1007/s10107-008-0223-z" class="nowrap">doi:10.1007/s10107-008-0223-z</a></td> </tr><tr> <td>[14]</td> <td class="space">Canonne CL, Diakonikolas I, Gouleakis T, Rubinfeld R (2018) Testing shape restrictions of discrete distributions. <span class="zbmathjax-textit">Theory Comput. Systems</span> 62(1):4-62.Crossref, Google Scholar · <a href="/1386.68215" class="nowrap">Zbl 1386.68215</a> · <a href="https://doi.org/10.1007/s00224-017-9785-6" class="nowrap">doi:10.1007/s00224-017-9785-6</a></td> </tr><tr> <td>[15]</td> <td class="space">Chakraborty S (2015) Generating discrete analogues of continuous probability distributions—A survey of methods and constructions. <span class="zbmathjax-textit">J. Statist. Distributed Appl.</span> 2(1):article 6.Google Scholar · <a href="/1359.62053" class="nowrap">Zbl 1359.62053</a></td> </tr><tr> <td>[16]</td> <td class="space">Chen Z, Sim M, Xu H (2019) Distributionally robust optimization with infinitely constrained ambiguity sets. Oper. Res. 67(5):1328-1344.Google Scholar · <a href="/1455.90117" class="nowrap">Zbl 1455.90117</a></td> </tr><tr> <td>[17]</td> <td class="space">Dai T, Jerath K (2016) Impact of inventory on quota-bonus contracts with rent sharing. <span class="zbmathjax-textit">Oper. Res.</span> 64(1):94-98.Link, Google Scholar · <a href="/1338.90024" class="nowrap">Zbl 1338.90024</a></td> </tr><tr> <td>[18]</td> <td class="space">Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. <span class="zbmathjax-textit">Oper. Res.</span> 58(3):595-612.Link, Google Scholar · <a href="/1228.90064" class="nowrap">Zbl 1228.90064</a></td> </tr><tr> <td>[19]</td> <td class="space">Dennis J, Jr, Schnabel R (1996) <span class="zbmathjax-textit">Numerical Methods for Unconstrained Optimization and Nonlinear Equations</span>, volume 16 (SIAM, Philadelphia).Crossref, Google Scholar · <a href="/0847.65038" class="nowrap">Zbl 0847.65038</a> · <a href="https://doi.org/10.1137/1.9781611971200" class="nowrap">doi:10.1137/1.9781611971200</a></td> </tr><tr> <td>[20]</td> <td class="space">Duembgen L, Huesler A, Rufibach K (2011) Active set and EM algorithms for log-concave densities based on complete and censored data. arXiv preprint arXiv:0707.4643v4.Google Scholar</td> </tr><tr> <td>[21]</td> <td class="space">Gao R, Kleywegt AJ (2016) Distributionally robust stochastic optimization with Wasserstein distance. arXiv preprint arXiv:1604.02199.Google Scholar</td> </tr><tr> <td>[22]</td> <td class="space">Gao R, Chen X, Kleywegt AJ (2017) Wasserstein distributional robustness and regularization in statistical learning. arXiv preprint arXiv:1712.06050.Google Scholar</td> </tr><tr> <td>[23]</td> <td class="space">Gavirneni S, Kapuscinski R, Tayur S (1999) Value of information in capacitated supply chains. <span class="zbmathjax-textit">Management Sci.</span> 45(1):16-24.Link, Google Scholar · <a href="/1231.90088" class="nowrap">Zbl 1231.90088</a></td> </tr><tr> <td>[24]</td> <td class="space">Goh J, Sim M (2010) Distributionally robust optimization and its tractable approximations. <span class="zbmathjax-textit">Oper. Res.</span> 58(4-part-1):902-917.Google Scholar · <a href="/1228.90067" class="nowrap">Zbl 1228.90067</a></td> </tr><tr> <td>[25]</td> <td class="space">Han Q, Du D, Zuluaga LF (2014) Technical note: A risk and ambiguity-averse extension of the max-min newsvendor order formula. <span class="zbmathjax-textit">Oper. Res.</span> 62(3):535-542.Link, Google Scholar · <a href="/1304.90012" class="nowrap">Zbl 1304.90012</a></td> </tr><tr> <td>[26]</td> <td class="space">Hanasusanto GA, Kuhn D (2018) Conic programming reformulations of two-stage distributionally robust linear programs over Wasserstein balls. <span class="zbmathjax-textit">Oper. Res.</span> 66(3):849-869.Link, Google Scholar · <a href="/1455.90121" class="nowrap">Zbl 1455.90121</a></td> </tr><tr> <td>[27]</td> <td class="space">Hanasusanto GA, Kuhn D, Wallace SW, Zymler S (2015) Distributionally robust multi-item newsvendor problems with multimodal demand distributions. <span class="zbmathjax-textit">Math. Programming</span> 152(1):1-32.Crossref, Google Scholar · <a href="/1327.90153" class="nowrap">Zbl 1327.90153</a> · <a href="https://doi.org/10.1007/s10107-014-0776-y" class="nowrap">doi:10.1007/s10107-014-0776-y</a></td> </tr><tr> <td>[28]</td> <td class="space">He S, Zhang J, Zhang S (2010) Bounding probability of small deviation: A fourth moment approach. <span class="zbmathjax-textit">Math. Oper. Res.</span> 35(1):208-232.Link, Google Scholar · <a href="/1217.60015" class="nowrap">Zbl 1217.60015</a></td> </tr><tr> <td>[29]</td> <td class="space">Hillestad R, Jacobsen S (1980) Reverse convex programming. <span class="zbmathjax-textit">Appl. Math. Optim.</span> 6(1):63-78.Crossref, Google Scholar · <a href="/0448.90044" class="nowrap">Zbl 0448.90044</a> · <a href="https://doi.org/10.1007/BF01442883" class="nowrap">doi:10.1007/BF01442883</a></td> </tr><tr> <td>[30]</td> <td class="space">Horst R, Thoai N (1999) DC programming: Overview. <span class="zbmathjax-textit">J. Optim. Theory Appl.</span> 103(1):1-43.Crossref, Google Scholar · <a href="/1073.90537" class="nowrap">Zbl 1073.90537</a> · <a href="https://doi.org/10.1023/A:1021765131316" class="nowrap">doi:10.1023/A:1021765131316</a></td> </tr><tr> <td>[31]</td> <td class="space">Jiang R, Wang J, Guan Y (2012) Robust unit commitment with wind power and pumped storage hydro. <span class="zbmathjax-textit">IEEE Trans. Power Systems</span> 27(2):800-810.Crossref, Google Scholar · <a href="https://doi.org/10.1109/TPWRS.2011.2169817" class="nowrap">doi:10.1109/TPWRS.2011.2169817</a></td> </tr><tr> <td>[32]</td> <td class="space">Johnson O, Goldschmidt C (2006) Preservation of log-concavity on summation. <span class="zbmathjax-textit">ESAIM Probab. Statist.</span> 10:206-215.Crossref, Google Scholar · <a href="/1185.60016" class="nowrap">Zbl 1185.60016</a> · <a href="https://doi.org/10.1051/ps:2006008" class="nowrap">doi:10.1051/ps:2006008</a></td> </tr><tr> <td>[33]</td> <td class="space">Johnson O, Kontoyiannis I, Madiman M (2013) Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures. <span class="zbmathjax-textit">Discrete Appl. Math.</span> 161(9):1232-1250.Crossref, Google Scholar · <a href="/1282.60016" class="nowrap">Zbl 1282.60016</a> · <a href="https://doi.org/10.1016/j.dam.2011.08.025" class="nowrap">doi:10.1016/j.dam.2011.08.025</a></td> </tr><tr> <td>[34]</td> <td class="space">Laffont JJ, Tirole J (1988) The dynamics of incentive contracts. <span class="zbmathjax-textit">Econometrica</span> 56(5):1153-1175.Crossref, Google Scholar · <a href="/0663.90014" class="nowrap">Zbl 0663.90014</a> · <a href="https://doi.org/10.2307/1911362" class="nowrap">doi:10.2307/1911362</a></td> </tr><tr> <td>[35]</td> <td class="space">Lam H, Mottet C (2017) Tail analysis without parametric models: A worst-case perspective. <span class="zbmathjax-textit">Oper. Res.</span> 65(6):1696-1711.Link, Google Scholar · <a href="/1405.62145" class="nowrap">Zbl 1405.62145</a></td> </tr><tr> <td>[36]</td> <td class="space">Lariviere M (2006) A note on probability distributions with increasing generalized failure rates. <span class="zbmathjax-textit">Oper. Res.</span> 54(3):602-604.Link, Google Scholar · <a href="/1167.60309" class="nowrap">Zbl 1167.60309</a></td> </tr><tr> <td>[37]</td> <td class="space">Lewis TR, Sappington DEM (1988) Regulating a monopolist with unknown demand. <span class="zbmathjax-textit">Amer. Econom. Rev.</span> 78(5):986-998.Google Scholar</td> </tr><tr> <td>[38]</td> <td class="space">Li B, Jiang R, Mathieu JL (2017) Ambiguous risk constraints with moment and unimodality information. Math. Programming 173.(1-2):151-192.Google Scholar · <a href="/1410.90139" class="nowrap">Zbl 1410.90139</a></td> </tr><tr> <td>[39]</td> <td class="space">Long DZ, Qi J (2014) Distributionally robust discrete optimization with entropic value-at-risk. <span class="zbmathjax-textit">Oper. Res. Lett.</span> 42(8):532-538.Crossref, Google Scholar · <a href="/1408.90218" class="nowrap">Zbl 1408.90218</a> · <a href="https://doi.org/10.1016/j.orl.2014.09.004" class="nowrap">doi:10.1016/j.orl.2014.09.004</a></td> </tr><tr> <td>[40]</td> <td class="space">Matthews S (1987) Comparing auctions for risk averse buyers: A buyer’s point of view. <span class="zbmathjax-textit">Econometrica</span> 55(3):633-646.Crossref, Google Scholar · <a href="/0612.90018" class="nowrap">Zbl 0612.90018</a> · <a href="https://doi.org/10.2307/1913603" class="nowrap">doi:10.2307/1913603</a></td> </tr><tr> <td>[41]</td> <td class="space">Meyer R (1970) The validity of a family of optimization methods. <span class="zbmathjax-textit">SIAM J. Control</span> 8(1):41-54.Crossref, Google Scholar · <a href="/0194.20501" class="nowrap">Zbl 0194.20501</a> · <a href="https://doi.org/10.1137/0308003" class="nowrap">doi:10.1137/0308003</a></td> </tr><tr> <td>[42]</td> <td class="space">Myerson RB, Satterthwaite MA (1983) Efficient mechanisms for bilateral trading. <span class="zbmathjax-textit">J. Econom. Theory</span> 29(2):265-281.Crossref, Google Scholar · <a href="/0523.90099" class="nowrap">Zbl 0523.90099</a> · <a href="https://doi.org/10.1016/0022-0531(83)90048-0" class="nowrap">doi:10.1016/0022-0531(83)90048-0</a></td> </tr><tr> <td>[43]</td> <td class="space">Natarajan K, Sim M, Uichanco J (2017) Asymmetry and ambiguity in newsvendor models. <span class="zbmathjax-textit">Management Sci.</span> 64(7):3146-3167.Link, Google Scholar</td> </tr><tr> <td>[44]</td> <td class="space">Natarajan K, Teo CP, Zheng Z (2011) Mixed 0-1 linear programs under objective uncertainty: A completely positive representation. <span class="zbmathjax-textit">Oper. Res.</span> 59(3):713-728.Link, Google Scholar · <a href="/1231.90304" class="nowrap">Zbl 1231.90304</a></td> </tr><tr> <td>[45]</td> <td class="space">Ninh A, Hu H, Allen D (2019) Robust newsvendor problems: Effect of discrete demands. <span class="zbmathjax-textit">Ann. Oper. Res.</span> 275(2):607-621.Crossref, Google Scholar · <a href="/1423.90013" class="nowrap">Zbl 1423.90013</a> · <a href="https://doi.org/10.1007/s10479-018-3016-7" class="nowrap">doi:10.1007/s10479-018-3016-7</a></td> </tr><tr> <td>[46]</td> <td class="space">Peña J, Vera J, Zuluaga L (2015) Completely positive reformulations for polynomial optimization. <span class="zbmathjax-textit">Math. Programming</span> 151(2):405-431.Crossref, Google Scholar · <a href="/1328.90114" class="nowrap">Zbl 1328.90114</a> · <a href="https://doi.org/10.1007/s10107-014-0822-9" class="nowrap">doi:10.1007/s10107-014-0822-9</a></td> </tr><tr> <td>[47]</td> <td class="space">Perakis G, Roels G (2008) Regret in the newsvendor model with partial information. <span class="zbmathjax-textit">Oper. Res.</span> 56(1):188-203.Link, Google Scholar · <a href="/1167.90350" class="nowrap">Zbl 1167.90350</a></td> </tr><tr> <td>[48]</td> <td class="space">Popescu I (2005) A semidefinite programming approach to optimal-moment bounds for convex classes of distributions. <span class="zbmathjax-textit">Math. Oper. Res.</span> 30(3):632-657.Link, Google Scholar · <a href="/1082.60011" class="nowrap">Zbl 1082.60011</a></td> </tr><tr> <td>[49]</td> <td class="space">Prékopa A (1990) Sharp bounds on probabilities using linear programming. <span class="zbmathjax-textit">Oper. Res.</span> 38(2):227-239.Link, Google Scholar · <a href="/0727.90049" class="nowrap">Zbl 0727.90049</a></td> </tr><tr> <td>[50]</td> <td class="space">Prékopa A (2013) <span class="zbmathjax-textit">Stochastic Programming</span>, vol. 324 (Springer Science & Business Media, Berlin, Heidelberg).Google Scholar</td> </tr><tr> <td>[51]</td> <td class="space">Prékopa A, Boros E (1991) On the existence of a feasible flow in a stochastic transportation network. <span class="zbmathjax-textit">Oper. Res.</span> 39(1):119-129.Link, Google Scholar · <a href="/0747.90041" class="nowrap">Zbl 0747.90041</a></td> </tr><tr> <td>[52]</td> <td class="space">Prékopa A, Long J, Szantai T (2004) New bounds and approximations for the probability distribution of the length of the critical path. <span class="zbmathjax-textit">Dynamic Stochastic Optimization</span> (Springer Verlag, Berlin, Heidelberg), 293-320.Crossref, Google Scholar · <a href="/1190.90255" class="nowrap">Zbl 1190.90255</a> · <a href="https://doi.org/10.1007/978-3-642-55884-9_15" class="nowrap">doi:10.1007/978-3-642-55884-9_15</a></td> </tr><tr> <td>[53]</td> <td class="space">Prékopa A, Ninh A, Alexe G (2016) On the relationship between the discrete and continuous bounding moment problems and their numerical solutions. <span class="zbmathjax-textit">Ann. Oper. Res.</span> 238(1-2):521-575.Crossref, Google Scholar · <a href="/1334.90184" class="nowrap">Zbl 1334.90184</a> · <a href="https://doi.org/10.1007/s10479-015-1995-1" class="nowrap">doi:10.1007/s10479-015-1995-1</a></td> </tr><tr> <td>[54]</td> <td class="space">Rujeerapaiboon N, Kuhn D, Wiesemann W (2018) Chebyshev inequalities for products of random variables. <span class="zbmathjax-textit">Math. Oper. Res.</span> 43(3):887-918.Link, Google Scholar · <a href="/1433.60011" class="nowrap">Zbl 1433.60011</a></td> </tr><tr> <td>[55]</td> <td class="space">Saghafian S, Tomlin B (2016) The newsvendor under demand ambiguity: Combining data with moment and tail information. <span class="zbmathjax-textit">Oper. Res.</span> 64(1):167-185.Link, Google Scholar · <a href="/1336.90053" class="nowrap">Zbl 1336.90053</a></td> </tr><tr> <td>[56]</td> <td class="space">Saumard A, Wellner JA (2014) Log-concavity and strong log-concavity: A review. <span class="zbmathjax-textit">Statist. Surveys</span> 8:45-114.Crossref, Google Scholar · <a href="/1360.62055" class="nowrap">Zbl 1360.62055</a> · <a href="https://doi.org/10.1214/14-SS107" class="nowrap">doi:10.1214/14-SS107</a></td> </tr><tr> <td>[57]</td> <td class="space">Scarf H, Arrow K, Karlin S (1958) A min-max solution of an inventory problem. <span class="zbmathjax-textit">Studies in the Mathematical Theory of Inventory and Production</span> (RAND Corporation, Santa Monica, CA), 201-209.Google Scholar</td> </tr><tr> <td>[58]</td> <td class="space">Subasi E, Subasi M, Prékopa A (2009) Discrete moment problems with distributions known to be unimodal. <span class="zbmathjax-textit">Math. Inequalities Appl.</span> 12(3):587-610.Crossref, Google Scholar · <a href="/1168.60321" class="nowrap">Zbl 1168.60321</a> · <a href="https://doi.org/10.7153/mia-12-46" class="nowrap">doi:10.7153/mia-12-46</a></td> </tr><tr> <td>[59]</td> <td class="space">Sudheesh K, Anisha P, Deemat C (2015) A simple approach for testing constant failure rate against different ageing classes for discrete data. <span class="zbmathjax-textit">Statist. Methodology</span> 25:74-88.Crossref, Google Scholar · <a href="/1487.62128" class="nowrap">Zbl 1487.62128</a> · <a href="https://doi.org/10.1016/j.stamet.2015.02.001" class="nowrap">doi:10.1016/j.stamet.2015.02.001</a></td> </tr><tr> <td>[60]</td> <td class="space">Swaminathan JM, Shanthikumar JG (1999) Supplier diversification: Effect of discrete demand. <span class="zbmathjax-textit">Oper. Res. Lett.</span> 24(5):213-221.Crossref, Google Scholar · <a href="/0967.90006" class="nowrap">Zbl 0967.90006</a> · <a href="https://doi.org/10.1016/S0167-6377(99)00021-8" class="nowrap">doi:10.1016/S0167-6377(99)00021-8</a></td> </tr><tr> <td>[61]</td> <td class="space">Tian R, Cox SH, Zuluaga LF (2017) Moment problem and its applications to risk assessment. <span class="zbmathjax-textit">North Amer. Actuarial J.</span> 21(2):242-266.Crossref, Google Scholar · <a href="/1414.91236" class="nowrap">Zbl 1414.91236</a> · <a href="https://doi.org/10.1080/10920277.2017.1302805" class="nowrap">doi:10.1080/10920277.2017.1302805</a></td> </tr><tr> <td>[62]</td> <td class="space">Walther G (2000) Inference and modeling with log-concave distributions. <span class="zbmathjax-textit">Statist. Sci.</span> 24(3):319-327.Crossref, Google Scholar · <a href="/1329.62192" class="nowrap">Zbl 1329.62192</a> · <a href="https://doi.org/10.1214/09-STS303" class="nowrap">doi:10.1214/09-STS303</a></td> </tr><tr> <td>[63]</td> <td class="space">Wellner JA (2013) Strong log-concavity is preserved by convolution. <span class="zbmathjax-textit">High Dimensional Probability, vol. 6</span> (Springer Verlag, Berlin, Heidelberg), 95-102.Crossref, Google Scholar · <a href="/1271.60034" class="nowrap">Zbl 1271.60034</a> · <a href="https://doi.org/10.1007/978-3-0348-0490-5_7" class="nowrap">doi:10.1007/978-3-0348-0490-5_7</a></td> </tr><tr> <td>[64]</td> <td class="space">Xu G, Burer S (2018) A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides. <span class="zbmathjax-textit">Comput. Optim. Appl.</span> 70(1):33-59.Crossref, Google Scholar · <a href="/1417.90150" class="nowrap">Zbl 1417.90150</a> · <a href="https://doi.org/10.1007/s10589-017-9974-x" class="nowrap">doi:10.1007/s10589-017-9974-x</a></td> </tr><tr> <td>[65]</td> <td class="space">Ziya S, Ayhan H, Foley RD (2004) Relationships among three assumptions in revenue management. <span class="zbmathjax-textit">Oper. Res.</span> 52(5):804-809.Link, Google Scholar · <a href="/1165.90445" class="nowrap">Zbl 1165.90445</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1470.90060 - zbMATH Open