[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82075-en":3,"doc-seo-82075-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82075,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Parallel Versions of the Mesh Adaptive Direct Search Algorithm","This work surveys parallel variants of the mesh adaptive direct search (MADS) algorithm for constrained blackbox optimization, where objective and constraint functions are evaluated via simulation and derivatives are unavailable. The study focuses on cases with large numbers of variables and multimodal search spaces that make computation costly, and where blackbox evaluation time is heterogeneous. By leveraging multi-core computer architectures, the reviewed methods apply different parallelism levels and strategies, report computational results, and discuss advantages and limitations with practical implementation guidance.","arXiv :2607 .08872v1 [math .OC] 9 Jul 2026  \nParallel versions of the mesh adaptive direct search  \nalgorithm  \nS´ebastien Le Digabel2,3 , Antoine Lesage-Landry 1,3 ,  \nSamuel Mendoza 1,3 , Christophe Tribes2,3  \n1 Department of Electrical Engineering, Polytechnique Montr´eal, 2500 Chemin de Polytechnique, Montr´eal, H3T 1J4, Qu´ebec, Canada.  \n2 Department of Mathematics and Industrial Engineering, Polytechnique Montr´eal, 2500 Chemin de Polytechnique, Montr´eal, H3T 1J4, Qu´ebec,  \nCanada.  \n3 Group for Research in Decision Analysis (GERAD), 3000 Chemin dela Cˆote-Sainte-Catherine, Montr´eal, H3T 2A7, Qu´ebec, Canada.  \nContributing authors: [sebastien.le-digabel@polymtl.ca](sebastien.le-digabel@polymtl.ca) ; [antoine.lesage-landry@polymtl.ca](antoine.lesage-landry@polymtl.ca) ; [samuel.mendoza@polymtl.ca](samuel.mendoza@polymtl.ca) ;  \n[christophe.tribes@polymtl.ca](christophe.tribes@polymtl.ca) ;  \nAbstract  \nThis work surveys the different parallel variants of the mesh adaptive direct search (MADS) algorithm for constrained blackbox optimization. These problems can inherently imply high computational costs due to the possible large number of variables and multi-modality of the search space. In addition, the potential time-intensive nature and time heterogeneity of the blackboxes defining the problem prompts the need for efficient implementations. Parallelism emerges as an actionable solution to mitigate computation time, as modern computer systems rely on multi-core architecture. The reviewed methods employ diverse levels of parallelism and distinct parallel strategies to effectively tackle each aspect outlined above. The manuscript details the practical implementations, provides computational results, and offers insights into the advantages and limitations of each MADS parallel method.  \nKeywords: Blackbox optimization, derivative-free optimization, mesh adaptive direct search, parallel computing.  \n1  \n1 Introduction  \nThis work considers optimization problems of the form  \nmin f(x), (P)  \nx∈Ω  \nwhere Ω = { x ∈ X | cj (x) ≤ 0, j ∈ J = {1, 2 ,..., m}} ⊆ Rn , m, n ∈ N, is the feasible set and X is a subset of Rn , typically defined by bound constraints. The functions f : X → R ∪ {∞} and cj : X → R ∪ {∞}, j ∈ J , are typically evaluated through a computer simulation with no available derivatives, and is considered as a blackbox. We consider that evaluating such a blackbox is time-consuming and that only a limited budget of evaluations is available. We further assume that the blackbox may be subject to heterogeneity in the time required to evaluate a point, as illustrated in Figure 1 .  \nBlackbox(x1 )  \ntime  \nBlackbox(x2 )  \nFigure 1: Illustration of a time-heterogeneous blackbox.  \nProblem (P) is a blackbox optimization (BBO) problem, for which derivativefree optimization (DFO) algorithms are considered. These techniques are described in [1, 2], and some practical applications are illustrated in [3] .  \nWith DFO methods, a rule of thumb is to allow a budget of the order of 1,000n evaluations to obtain satisfactory results. This is not conceivable when blackboxes are time-consuming because this process would lead to impractical resolution times. The situation worsens when dealing with large-scale problems, typically with more than 50 variables, as the evaluations budget escalates significantly.  \nParallel computing appears as a solution because most of computational resources exploits multi-core architectures available through various levels, e.g. , workstations, servers, and supercomputers. In addition, with the end of Dennard scaling, parallelization has become much more important than before, as the alternative of increasing processor clock speed is no longer possible due to heat dissipation and power consumption issues.  \nAn obvious way of using parallelism is to “open” the blackbox and to modify it to exploit the available parallel resources. However, this is not always possible due to many reasons: The source code of","cbCaijlVVRV0cync","https://ap.wps.com/l/cbCaijlVVRV0cync","pdf",972074,1,23,"English","en",105,"# Introduction\n## Problem setting: constrained blackbox optimization\n## Motivation: time cost, limited evaluation budget, and heterogeneity\n## Parallelism as a solution\n# Parallelism in BBO\n## Benchmarking principles and efficiency considerations\n# Mesh adaptive direct search (MADS)\n# Parallel versions of MADS\n# Computational experiments and recommendations\n# Concluding remarks","[{\"question\":\"What problem does the document address?\",\"answer\":\"It addresses constrained blackbox optimization where objective and constraint functions must be evaluated through simulations without available derivatives.\"},{\"question\":\"Why is parallel computing important for this setting?\",\"answer\":\"Because blackbox evaluations are time-consuming and evaluation budgets are limited, parallelism reduces total execution time by running multiple evaluations concurrently on multi-core systems.\"},{\"question\":\"How does the document organize its discussion of parallel MADS methods?\",\"answer\":\"It reviews related literature, introduces MADS, presents different parallel versions, and then reports computational experiments with recommendations and concluding remarks.\"}]",1784178064,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"parallel-versions-of-the-mesh-adaptive-direct-search-algorithm","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/parallel-versions-of-the-mesh-adaptive-direct-search-algorithm/82075/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address?","Question",{"text":75,"@type":76},"It addresses constrained blackbox optimization where objective and constraint functions must be evaluated through simulations without available derivatives.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is parallel computing important for this setting?",{"text":80,"@type":76},"Because blackbox evaluations are time-consuming and evaluation budgets are limited, parallelism reduces total execution time by running multiple evaluations concurrently on multi-core systems.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the document organize its discussion of parallel MADS methods?",{"text":84,"@type":76},"It reviews related literature, introduces MADS, presents different parallel versions, and then reports computational experiments with recommendations and concluding 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