[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117559-en":3,"doc-seo-117559-105":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},117559,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Adaptive Stabilization Based on Machine Learning for Column Generation","Column generation (CG) solves large-scale linear programs by iteratively optimizing a subproblem over a subset of columns, then using the dual solution to generate new columns with negative reduced costs until the dual values converge. However, dual iterates often exhibit heavy oscillations, delaying convergence and causing redundant column generation. Stabilization methods use extra problem data or historical iterates, yet accurate dual values early in the process remain challenging. This paper proposes machine learning prediction of optimal dual solutions combined with an adaptive stabilization strategy to accelerate convergence.","Adaptive Stabilization Based on Machine Learning for Column Generation  \nYunzhuang Shen 1 Yuan Sun 2 Xiaodong Li 3 Zhiguang Cao 4 Andrew Eberhard 3 Guangquan Zhang 1  \nAbstract  \nColumn generation (CG) is a well-established method for solving large-scale linear programs.  \nIt involves iteratively optimizing a subproblem containing a subset of columns and using its dual solution to generate new columns with negative reduced costs. This process continues until the dual values converge to the optimal dual solution to the original problem. A natural phenomenon in CG is the heavy oscillation of the dual values during iterations, which can lead to a substantial slowdown in the convergence rate. Stabilization techniques are devised to accelerate the convergence of dual values by using information beyond the state of the current subproblem. However, there remains a significant gap in obtaining more accurate dual values at an earlier stage. To further narrow this gap, this paper introduces a novel approach consisting of 1) a machine learning approach for accurate prediction of optimal dual solutions and 2) an adaptive stabilization technique that effectively capitalizes on accurate predictions.  \nOn the graph coloring problem, we show that our method achieves a significantly improved convergence rate compared to traditional methods.  \n1. Introduction  \nColumn generation (CG) is an effective method for solving linear programs (LP) with a large number of variables (or columns) (L¨ubbecke & Desrosiers, 2005) . It has many applications in solving combinatorial optimization problems with a decomposable structure (Vanderbeck, 2000), such as the vehicle routing problem (Agarwal et al., 1989), the  \n1Australian Artificial Intelligence Institute, University of Technology Sydney, Australia 2La Trobe Business School, La Trobe University, Australia 3 School of Computing Technologies, Royal Melbourne Institute of Technology, Australia 4 School of Computing and Information Systems, Singapore Management University, Singapore. Correspondence to: Yunzhuang Shen \u003Cshenyun[zhuang@outlook.com](zhuang@outlook.com) >.  \nProceedings of the 41 st International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024 . Copyright 2024 by the author(s) .  \ncutting stock problem (Gilmore & Gomory, 1961), and the graph coloring problem (Mehrotra & Trick, 1996) .  \nCG solves a large-scale LP in iterative steps. In an iteration, a set of dual values is obtained by solving the LP that contains a small subset of columns and is then used to generate new columns with negative reduced costs. As this process repeats, the dual values converge to optimal dual values, i.e., an optimal dual solution to the original LP. This point of convergence is identified when no column with a negative reduced cost can be further generated.  \nAs CG updates dual values by iteratively re-optimizing an evolving subproblem, this method may lead to significant oscillations in the dual iterates within the high-dimensional dual space. In this context, a dual iterate refers to the set of dual values obtained in a specific iteration of CG. This phenomenon is depicted in Figure 1, which shows the trajectory of dual iterates for CG marked in green. Notably, the dual iterates tend to stay far from the optimal dual solution until a later stage. These issues can lead to the generation of redundant columns, causing a significant slowdown in the convergence rate.  \nVarious techniques, termed stabilization, have been devised to overcome this challenge. These techniques succeed in deriving dual values that are more closely aligned with the optimal dual solution by using information beyond the current state of the subproblem. Typically, this additional information includes the problem data (Agarwal et al., 1989 ; Briant et al., 2008 ; Kraul et al., 2023) and/or historical dual iterates collected during the solution process (Du Merleet al., 1999 ; Pessoa et al., 2018) . However, there is still significa","cbCairzAth4sdFeS","https://ap.wps.com/l/cbCairzAth4sdFeS","pdf",462052,1,18,"English","en",105,"# Introduction\n## Column generation and dual oscillations\n## Stabilization techniques and the early-stage accuracy gap\n## Proposed adaptive stabilized CG with ML\n## Empirical results on graph coloring","[{\"question\":\"What causes slow convergence in column generation?\",\"answer\":\"Dual values may oscillate heavily across CG iterations, so dual iterates remain far from the optimal dual solution for a long time. This can generate redundant columns and substantially slow convergence.\"},{\"question\":\"How does the proposed method use machine learning in CG?\",\"answer\":\"It predicts the optimal dual solution with a machine learning approach, then uses this prediction during subproblem optimization to guide dual variables toward the predicted values.\"},{\"question\":\"How is the stabilization strength adjusted during the algorithm?\",\"answer\":\"The method adaptively changes the attraction strength in line with CG progression, motivated by improving prediction accuracy over time, and it pulls dual values toward the ML-predicted region.\"}]","Adaptive Stabilization Based on Machine Learning for Column Generation | PDF",1785676973,45,{"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},"adaptive-stabilization-based-on-machine-learning-for-column-generation","",{"@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/adaptive-stabilization-based-on-machine-learning-for-column-generation/117559/",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","2026-08-02",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 causes slow convergence in column generation?","Question",{"text":75,"@type":76},"Dual values may oscillate heavily across CG iterations, so dual iterates remain far from the optimal dual solution for a long time. This can generate redundant columns and substantially slow convergence.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method use machine learning in CG?",{"text":80,"@type":76},"It predicts the optimal dual solution with a machine learning approach, then uses this prediction during subproblem optimization to guide dual variables toward the predicted values.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the stabilization strength adjusted during the algorithm?",{"text":84,"@type":76},"The method adaptively changes the attraction strength in line with CG progression, motivated by improving prediction accuracy over time, and it pulls dual values toward the ML-predicted region.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"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":53,"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"]