[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86369-en":3,"doc-seo-86369-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86369,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Decoupling Constraints from Two Directions for Evolutionary Constrained Multi-objective Optimization","Real-world constrained multi-objective optimization problems (CMOPs) involve many constraints whose coupling relationships strongly affect efficiency. Existing decoupling methods treat each constraint separately yet often approximate only the evolutionary direction related to single-constraint Pareto fronts. This work shows parts of the constrained Pareto front may be shaped by infeasible-region boundaries instead. The study defines independent CPF (ICPF) and reverse CPF (RCPF), then proposes DCF2D, a bidirectional coevolution algorithm dynamically activating constraint-specific auxiliary populations and combining global exploration, event-driven coevolution, and final refinement, validated on 87 benchmarks and 28 engineering instances.","Decoupling Constraints from Two Directions for Evolutionary Constrained Multi-objective  \nOptimization  \nRuiqing Sun, Dawei Feng, Xing Zhou, Lianghao Li, Sheng Qi, Bo Ding, Yijie Wang, Rui Wang Senior  \nMember, IEEE, Huaimin Wang  \narXiv :2512 .23945v2 [ cs .NE] 12 Jul 2026  \nAbstract—Real-world constrained multi-objective optimization problems (CMOPs) commonly involve multiple constraints, and understanding and exploiting their coupling relationships is crucial for efficient optimization. Recent constraintdecoupling methods handle individual constraints separately, but they generally search only in the evolutionary direction to approximate single-constraint Pareto fronts (SCPFs) . In this study, we show that part or all of the constrained Pareto front (CPF) may be unrelated to any SCPF and instead be shaped by the boundaries of infeasible regions. We refer to such a portion as the independent CPF (ICPF) and introduce thereverse CPF (RCPF) to characterize its associated informative infeasible boundaries. Based on these observations, we propose a bidirectional constraint-decoupling coevolutionary algorithm named DCF2D. DCF2D dynamically identifies the constraints obstructing the main population and activates constraint-specific auxiliary populations. These populations adaptively search in the evolutionary direction for the corresponding SCPFs or in the reverse evolutionary direction for the corresponding RCPFs. Its three-stage framework integratesunconstrained global exploration, event-driven bidirectional coevolution, and final convergence refinement. Experiments on 87 benchmark instances from seven test suites and 28 realworld engineering CMOPs demonstrate that DCF2D achieves the best overall performance among nine algorithms. Code available at: [https://github.com/RuiqingS/DCF2D](https://github.com/RuiqingS/DCF2D).  \nIndex Terms—Constraint handling, Coevolutionary algorithm, Constraint decoupling.  \nI. Introduction  \nMany engineering and scientific applications involve optimizing multiple conflicting objectives under multiple constraints [1], [2], [3] . Such problems are known as constrained multi-objective optimization problems (CMOPs) .  \nRuiqing Sun, Dawei Feng, Bo Ding, Yijie Wang, and Huaimin Wang are with the College of Computer Science and Technology, National University of Defense Technology, Changsha 410000, P.R. China (e-mail: [sunny0331@foxmail.com](sunny0331@foxmail.com), [wangyijie@nudt.edu.cn](wangyijie@nudt.edu.cn)).  \nXing Zhou is with the College of Intelligence Science and Technology, National University of Defense Technology, Changsha, 410000, P.R. China.  \nLianghao Li is with the State Key Laboratory of Complex & Critical Software Environment, College of Information and Communication, National University of Defense Technology, Wuhan 430019, P.R. China.  \nSheng Qi and Rui Wang are with the College of System Engineering, National University of Defense Technology, Changsha 410000, P.R. China.  \nA CMOP can be formulated as follows [4]:  \n􀀾􀀸 min F (X) = (f1 (X), . . . , fM (X)),  \n􀀾  \n sug ibje(X)cto0, Xi  Ω1,,. . . , p, (1)  \n􀀾  \n􀀺 hj (X) = 0, j = 1 , . . . , q,  \nwhere X ∈ Ω ⊆ RD is a decision vector and F(X) contains M objective functions. The total number of constraints isncon = p + q.  \nThe violation of the k-th constraint is defined as [5] ck (X) = { maxmax((00,, g| khk()()X, ) | − τ), kk  1p,+. .1. ,, p. ., , ncon ,  \n(2)  \nwhere τ is a small tolerance for equality constraints. The  \ntotal constraint violation is  \nCV (X) = n1 ck (X) . (3)  \nA solution is feasible if CV (X) = 0 . Let  \nF = {X ∈ Ω | CV (X) = 0} . (4)  \nUsing ≺ to denote Pareto dominance, the constrained Pareto front (CPF) and unconstrained Pareto front (UPF) are respectively defined as  \nCPF = {F(X) | X ∈ F, ∄ Y ∈ F : F (Y ) ≺ F (X)} , (5) and  \nUPF = {F(X) | X ∈ Ω, ∄ Y ∈ Ω : F (Y ) ≺ F (X)} . (6)  \nConstrained multi-objective evolutionary algorithms (CMOEAs) combine multi-objective evolutionary algorithms with constraint-handling techniq","cbCaipcBmiUw5tMs","https://ap.wps.com/l/cbCaipcBmiUw5tMs","pdf",1468280,3,1,14,"English","en",105,"# Abstract\n# Introduction\n## Constrained multi-objective optimization formulation\n## Constraint violation and feasibility\n## Constrained Pareto front and unconstrained Pareto front\n## Constraint-decoupling background and limitations","[{\"question\":\"Why is constraint coupling important in constrained multi-objective optimization?\",\"answer\":\"In CMOPs, multiple constraints interact and together determine the final constrained Pareto front. Treating constraints only as independent elements can misrepresent which constraints actually shape the solution set.\"},{\"question\":\"What are ICPF and RCPF in the proposed framework?\",\"answer\":\"ICPF refers to the portion of the constrained Pareto front that lies outside all single-constraint Pareto fronts, being driven by joint constraint effects. RCPF characterizes the informative infeasible boundaries by describing the associated reverse direction related to infeasibility boundaries.\"},{\"question\":\"How does the DCF2D algorithm perform bidirectional constraint decoupling?\",\"answer\":\"DCF2D identifies constraints obstructing the main population and activates constraint-specific auxiliary populations. These populations search either in the evolutionary direction for corresponding single-constraint Pareto fronts or in the reverse direction for corresponding reverse constrained Pareto fronts, within a three-stage framework culminating in convergence refinement.\"}]",1784211104,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"decoupling-constraints-from-two-directions-for-evolutionary-constrained-multi-objective-optimization","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/decoupling-constraints-from-two-directions-for-evolutionary-constrained-multi-objective-optimization/86369/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-22","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},"Why is constraint coupling important in constrained multi-objective optimization?","Question",{"text":75,"@type":76},"In CMOPs, multiple constraints interact and together determine the final constrained Pareto front. Treating constraints only as independent elements can misrepresent which constraints actually shape the solution set.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are ICPF and RCPF in the proposed framework?",{"text":80,"@type":76},"ICPF refers to the portion of the constrained Pareto front that lies outside all single-constraint Pareto fronts, being driven by joint constraint effects. RCPF characterizes the informative infeasible boundaries by describing the associated reverse direction related to infeasibility boundaries.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the DCF2D algorithm perform bidirectional constraint decoupling?",{"text":84,"@type":76},"DCF2D identifies constraints obstructing the main population and activates constraint-specific auxiliary populations. These populations search either in the evolutionary direction for corresponding single-constraint Pareto fronts or in the reverse direction for corresponding reverse constrained Pareto fronts, within a three-stage framework culminating in convergence refinement.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]