[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123135-en":3,"doc-seo-123135-105":30,"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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},123135,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Global Optimization - A Machine Learning Approach","Global Optimization focuses on minimizing an objective under nonlinear inequality and equality constraints where variables can be continuous and integer. Many solvers depend on relaxations tailored to specific constraint structures, which becomes limiting for black-box, implicit, or more general primitives. The proposed OCTHaGOn framework approximates nonlinear constraints via hyperplane-based decision trees to build a unified mixed integer optimization approximation. Extensions include alternative MIO-representable ML models, adaptive sampling, robust optimization for training uncertainty, and a family of relaxations to handle infeasibilities. Extensive tests on 81 instances show improved feasibility and optimality, with BARON comparisons highlighting better gaps or times in several cases.","arXiv :2311 .01742v1 [math .OC] 3 Nov 2023  \nGlobal Optimization: A Machine Learning  \nApproach  \nDimitris Bertsimas 1 and Georgios Margaritis2  \n1 Sloan School of Management, Massachusetts Institute of Technology, Cambridge, 02139, MA.  \n2 Operations Research Center, Massachusetts Institute of Technology, Cambridge, 02139, MA.  \nContributing authors: [dbertsim@mit.edu](dbertsim@mit.edu) ; [geomar@mit.edu](geomar@mit.edu) ;  \nAbstract  \nMany approaches for addressing Global Optimization problems typically rely on relaxations of nonlinear constraints over specific mathematical primitives. This is restricting in applications with constraints that are black-box, implicit or consist of more general primitives. Trying to address such limitations, Bertsimas and Ozturk (2023) proposed OCTHaGOn as a way of solving black-box global optimization problems by approximating the nonlinear constraints using hyperplane-based Decision-Trees and then using those trees to construct a unified mixed integer optimization (MIO) approximation of the original problem. We provide extensions to this approach, by (i) approximating the original problem using other MIO-representable ML models besides Decision Trees, such as Gradient Boosted Trees, Multi Layer Perceptrons and Suport Vector Machines (ii) proposing adaptive sampling procedures for more accurate machine learningbased constraint approximations, (iii) utilizing robust optimization to account for the uncertainty of the sample-dependent training of the ML models, (iv) leveraging a family of relaxations to address the infeasibilities of the final MIO approximation. We then test the enhanced framework in 81 Global Optimization instances. We show improvements in solution feasibility and optimality in the majority of instances. We also compare against BARON, showing improved optimality gaps or solution times in 11 instances.  \nKeywords: global optimization; machine learning; mixed integer optimization; robust optimization  \n1  \n1 Introduction  \nGlobal optimizers aim to solve problems of the following form:  \nmin f (x)  \n[s.t.](s.t. gi)[ g](s.t. gi)[i](s.t. gi)(x) ≤ 0, i ∈ ¯I, hj (x) = 0, j ∈ ¯J, x ∈ Zm × Rn−m ,  \n(1)  \nwhere f, gi , hi represent the objective function, the inequality constraints and the equality constraints respectively. The objective function and constraints may lack desirable mathematical properties like linearity or convexity, and the decision variables may be continuous or integer.  \nMost approaches in the Global Optimization literature, attempt to solve Problem (1) by approximating it with more tractable optimization forms. For this purpose, they often use a combination of gradient-based methods, outer approximations, relaxationsand Mixed Integer Optimization (MIO) . For instance, the popular nonlinear Optimizer CONOPT uses a gradient-based approach in its solution process. As noted in [1], CONOPT finds an initial feasible solution using heuristics, performs a series of gradient descent iterations and then confirms optimality via bound projections. During the gradient-based part of the algorithm, CONOPT linearizes the constraints and performsa series of linear-search gradient-based iterations, while preserving feasibility at each step.  \nA different approach is the one detailed by [2], which uses outer approximations. This approach simplifies the problem by approximating the constraints via linear and nonlinear cuts, while preserving the initial feasible set of the problem. Such approach can only be used with constraints that obey a particular mathematical structure, such as linearity of integer variables and convexity of the nonlinear functions [3], or concavity and bilinearity [4], where the functions involved are amenable to efficient outer approximations. Although such approaches are effective in some scenarios, they have not found extensive use as they they are restricted to certain classes of problems. Another approach is the one used by the well-established commercial opti","cbCainz5Q51H3Eb0","https://ap.wps.com/l/cbCainz5Q51H3Eb0","pdf",2242552,1,35,"English","en",105,"# Introduction\n## Problem formulation and global optimizers\n## Limitations of existing solvers and relaxations\n## Motivation for ML-based constraint approximation\n# Proposed framework and extensions\n## Decision-tree hyperplane approximation (OCTHaGOn)\n## Alternative ML models for MIO approximation\n## Adaptive sampling for improved accuracy\n## Robust optimization for training uncertainty\n## Relaxation families to mitigate infeasibilities\n# Experimental evaluation","[{\"question\":\"What problem class does the document address?\",\"answer\":\"It addresses global optimization problems with nonlinear objective and constraints, including inequality and equality constraints, with decision variables that may be continuous or integer.\"},{\"question\":\"Why are standard global optimization approaches limited?\",\"answer\":\"Many methods rely on relaxations or outer approximations that require specific mathematical structure in constraints; they often fail or become inefficient for black-box, implicit, or more general primitives.\"},{\"question\":\"How does the proposed approach use machine learning to improve optimization?\",\"answer\":\"It approximates nonlinear constraints using hyperplane-based decision trees, then builds a unified mixed integer optimization approximation. The extensions also use other ML models, adaptive sampling, robust optimization to handle uncertainty, and relaxations to reduce infeasibilities.\"}]","Global Optimization - A Machine Learning Approach | PDF",1785814800,88,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"global-optimization-a-machine-learning-approach","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/global-optimization-a-machine-learning-approach/123135/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem class does the document address?","Question",{"text":75,"@type":76},"It addresses global optimization problems with nonlinear objective and constraints, including inequality and equality constraints, with decision variables that may be continuous or integer.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are standard global optimization approaches limited?",{"text":80,"@type":76},"Many methods rely on relaxations or outer approximations that require specific mathematical structure in constraints; they often fail or become inefficient for black-box, implicit, or more general primitives.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed approach use machine learning to improve optimization?",{"text":84,"@type":76},"It approximates nonlinear constraints using hyperplane-based decision trees, then builds a unified mixed integer optimization approximation. 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