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One approach to solving this problem is to identify those inputs to the objective functions that produce an output" /> <title>(PDF) Optimization and Engineering manuscript No. (will be inserted by the editor) Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement | Thomas Santner - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/73949557/Optimization_and_Engineering_manuscript_No_will_be_inserted_by_the_editor_Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '49879c2402910372f4abc62630a427bbe033d190'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1732399208000); window.Aedu.timeDifference = new Date().getTime() - 1732399208000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"Many engineering design optimization problems contain multiple objective functions all of which it is desired to minimize, say. One approach to solving this problem is to identify those inputs to the objective functions that produce an output (vector) on the Pareto Front; the inputs that produce outputs on the Pareto Front form the Pareto Set. This paper proposes a method for identifying the Pareto Front and the Pareto Set when the objective functions are expensive to compute. The method replaces the objective function evaluations by a rapidly computable approximator based on an interpolating Gaussian process (GP) model. It sequentially selects new input sites guided by an improvement function; the next input to evaluate each output is that vector which maximizes the conditional expected value of this improvement function given the current data. The method introduced in this paper provides two advances within this framework. First, it proposes an improvement function based on the mo...","author":[{"@context":"https://schema.org","@type":"Person","name":"Thomas Santner"}],"contributor":[],"dateCreated":"2022-03-17","dateModified":null,"datePublished":"2014-01-01","headline":"Optimization and Engineering manuscript No. (will be inserted by the editor) Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement","inLanguage":"en","keywords":[],"locationCreated":null,"publication":null,"publisher":{"@context":"https://schema.org","@type":"Organization","name":null},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/73949557/Optimization_and_Engineering_manuscript_No_will_be_inserted_by_the_editor_Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}]}</script><link rel="stylesheet" media="all" 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{"work":{"id":73949557,"created_at":"2022-03-17T07:00:43.085-07:00","from_world_paper_id":198761403,"updated_at":"2022-03-17T07:22:50.406-07:00","_data":{"abstract":"Many engineering design optimization problems contain multiple objective functions all of which it is desired to minimize, say. One approach to solving this problem is to identify those inputs to the objective functions that produce an output (vector) on the Pareto Front; the inputs that produce outputs on the Pareto Front form the Pareto Set. This paper proposes a method for identifying the Pareto Front and the Pareto Set when the objective functions are expensive to compute. The method replaces the objective function evaluations by a rapidly computable approximator based on an interpolating Gaussian process (GP) model. It sequentially selects new input sites guided by an improvement function; the next input to evaluate each output is that vector which maximizes the conditional expected value of this improvement function given the current data. The method introduced in this paper provides two advances within this framework. First, it proposes an improvement function based on the mo...","publication_date":"2014,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Optimization and Engineering manuscript No. (will be inserted by the editor) Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [37978259]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":82281281,"attachmentType":"pdf"}"><img alt="First page of “Optimization and Engineering manuscript No. (will be inserted by the editor) Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/82281281/mini_magick20220317-25876-lskzig.png?1647525710" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Optimization and Engineering manuscript No. (will be inserted by the editor) Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="37978259" href="https://independent.academia.edu/ThomasSantner"><img alt="Profile image of Thomas Santner" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Thomas Santner</a></div><p class="ds-work-card--detail ds2-5-body-sm">2014</p><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Many engineering design optimization problems contain multiple objective functions all of which it is desired to minimize, say. One approach to solving this problem is to identify those inputs to the objective functions that produce an output (vector) on the Pareto Front; the inputs that produce outputs on the Pareto Front form the Pareto Set. This paper proposes a method for identifying the Pareto Front and the Pareto Set when the objective functions are expensive to compute. The method replaces the objective function evaluations by a rapidly computable approximator based on an interpolating Gaussian process (GP) model. It sequentially selects new input sites guided by an improvement function; the next input to evaluate each output is that vector which maximizes the conditional expected value of this improvement function given the current data. The method introduced in this paper provides two advances within this framework. First, it proposes an improvement function based on the mo...</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":82281281,"attachmentType":"pdf","workUrl":"https://www.academia.edu/73949557/Optimization_and_Engineering_manuscript_No_will_be_inserted_by_the_editor_Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":82281281,"attachmentType":"pdf","workUrl":"https://www.academia.edu/73949557/Optimization_and_Engineering_manuscript_No_will_be_inserted_by_the_editor_Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="82281281" data-landing_url="https://www.academia.edu/73949557/Optimization_and_Engineering_manuscript_No_will_be_inserted_by_the_editor_Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="73949410" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/73949410/Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement">Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="37978259" href="https://independent.academia.edu/ThomasSantner">Thomas Santner</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2011</p><p class="ds-related-work--abstract ds2-5-body-sm">Many engineering design optimization problems contain multiple objective functions all of which it is desired to minimize, say. One approach to solving this problem is to identify those inputs to the objective functions that produce an output (vector) on the Pareto Front; the inputs that produce outputs on the Pareto Front form the Pareto Set. This paper proposes a method for identifying the Pareto Front and the Pareto Set when the objective functions are expensive to compute. The method replaces the objective function evaluations by a rapidly computable approximator based on an interpolating Gaussian process (GP) model. It sequentially selects new input sites guided by an improvement function; the next input to evaluate each output is that vector which maximizes the conditional expected value of this improvement function given the current data. The method introduced in this paper provides two advances within this framework. First, it proposes an improvement function based on the mo...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement","attachmentId":82281253,"attachmentType":"pdf","work_url":"https://www.academia.edu/73949410/Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/73949410/Multiobjective_Optimization_of_Expensive_Black_Box_Functions_via_Expected_Maximin_Improvement"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="66455908" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/66455908/An_Efficient_Pareto_Set_Identification_Approach_for_Multiobjective_Optimization_on_Black_Box_Functions">An Efficient Pareto Set Identification Approach for Multiobjective Optimization on Black-Box Functions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="11579" href="https://sfu.academia.edu/GaryWang">Gary Wang</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mechanical Design, 2005</p><p class="ds-related-work--abstract ds2-5-body-sm">Both multiple objectives and computation-intensive black-box functions often exist simultaneously in engineering design problems. Few of existing multiobjective optimization approaches addresses problems with expensive black-box functions. In this paper, a new method called the Pareto set pursuing (PSP) method is developed. By developing sampling guidance functions based on approximation models, this approach progressively provides a designer with a rich and evenly distributed set of Pareto optimal points. This work describes PSP procedures in detail. From testing and design application, PSP demonstrates considerable promises in efficiency, accuracy, and robustness. Properties of PSP and differences between PSP and other approximation-based methods are also discussed. It is believed that PSP has a great potential to be a practical tool for multiobjective optimization problems.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"An Efficient Pareto Set Identification Approach for Multiobjective Optimization on Black-Box Functions","attachmentId":77640126,"attachmentType":"pdf","work_url":"https://www.academia.edu/66455908/An_Efficient_Pareto_Set_Identification_Approach_for_Multiobjective_Optimization_on_Black_Box_Functions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/66455908/An_Efficient_Pareto_Set_Identification_Approach_for_Multiobjective_Optimization_on_Black_Box_Functions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="74054783" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/74054783/Evolutionary_Multiobjective_Optimization_Based_on_Gaussian_Process_Modeling">Evolutionary Multiobjective Optimization Based on Gaussian Process Modeling</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="49440025" href="https://independent.academia.edu/MlakarMiha">Miha Mlakar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Informatica (Slovenia), 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">This paper presents a summary of the doctoral dissertation of the author, which addresses the task of evolutionary multiobjective optimization using surrogate models. The main contributions are done for the optimization problems, where solutions are presented with uncertainty. To compare solutions under uncertainty and improve the optimization results the new relations for comparing solutions under uncertainty are defined. These relations reduce the possibility of incorrect comparisons due to the inaccurate approximations. The relations under uncertainty are then used in the new surrogate-model-based multiobjective evolutionary algorithm called GP-DEMO. 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class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/87206557/PAINT_Pareto_front_interpolation_for_nonlinear_multiobjective_optimization"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="7520604" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/7520604/Response_surface_approximation_of_Pareto_optimal_front_in_multi_objective_optimization">Response surface approximation of Pareto optimal front in multi-objective optimization</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="13469505" href="https://independent.academia.edu/GoelTushar">Tushar Goel</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Computer Methods in Applied Mechanics and Engineering, 2007</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Response surface approximation of Pareto optimal front in multi-objective optimization","attachmentId":48438199,"attachmentType":"pdf","work_url":"https://www.academia.edu/7520604/Response_surface_approximation_of_Pareto_optimal_front_in_multi_objective_optimization","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/7520604/Response_surface_approximation_of_Pareto_optimal_front_in_multi_objective_optimization"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="30440556" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/30440556/Pareto_surface_construction_for_multi_objective_optimization_under_uncertainty">Pareto surface construction for multi-objective optimization under uncertainty</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="24934662" href="https://ford.academia.edu/ChenLiang">Chen Liang</a></div><p class="ds-related-work--abstract ds2-5-body-sm">This paper presents a novel approach for multi-objective optimization under both aleatory and epistemic sources of uncertainty. Given paired samples of the inputs and outputs from the system analysis model, a Bayesian network (BN) is built to represent the joint probability distribution of the inputs and outputs. In each design iteration, the optimizer provides the values of the design variables to the BN, and copula-based sampling is used to rapidly generate samples of the output variables conditioned on the input values. Samples from the conditional distributions are used to evaluate the objectives and constraints, which are fed back to the optimizer for further iteration. The proposed approach is formulated in the context of reliability-based design optimization (RBDO). The joint probability of multiple objectives and constraints is included in the formulation. The Bayesian network along with conditional sampling is exploited to select training points that enable effective construction of the Pareto front. A vehicle side impact problem is employed to demonstrate the proposed methodology.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Pareto surface construction for multi-objective optimization under uncertainty","attachmentId":59918832,"attachmentType":"pdf","work_url":"https://www.academia.edu/30440556/Pareto_surface_construction_for_multi_objective_optimization_under_uncertainty","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/30440556/Pareto_surface_construction_for_multi_objective_optimization_under_uncertainty"><span class="ds2-5-text-link__content">View PDF</span><span 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href="https://www.academia.edu/27399840/Towards_Efficient_Multiobjective_Optimization_Multiobjective_statistical_criterions">Towards Efficient Multiobjective Optimization: Multiobjective statistical criterions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51488929" href="https://independent.academia.edu/TDhaene">Tom Dhaene</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012 IEEE Congress on Evolutionary Computation, 2012</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Towards Efficient Multiobjective Optimization: Multiobjective statistical 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