[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125960-en":3,"doc-seo-125960-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},125960,137451207643,"Noah","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Adaptive Stabilization Based on Machine Learning for Column Generation","Column generation (CG) is a standard approach for solving large-scale linear programs by iteratively optimizing a restricted subproblem, using its dual solution to generate new columns with negative reduced costs until dual values converge. Dual oscillations often slow convergence, and existing stabilization methods still struggle to produce sufficiently accurate dual values early. This work proposes machine-learning prediction of optimal dual solutions combined with an adaptive stabilization mechanism that leverages those predictions. Experiments on graph coloring show a substantially improved convergence rate.","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  \narXiv :2405 . 11198v1 [math .OC] 18 May 2024  \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","cbCaiiPN70TMseHb","https://ap.wps.com/l/cbCaiiPN70TMseHb","pdf",560203,7,1,18,"English","en",105,"# Introduction\n## Column generation and dual oscillations\n## Stabilization techniques and remaining gap\n## Proposed ML-assisted adaptive stabilization\n# Method Overview\n## ML prediction of optimal dual solutions\n## Adaptive stabilization using prediction deviation penalty\n# Experiments\n## Graph coloring problem results","[{\"question\":\"What problem does the document address in column generation?\",\"answer\":\"It addresses heavy oscillations of dual values during CG iterations, which can delay convergence and cause redundant column generation.\"},{\"question\":\"How does the proposed method use machine learning?\",\"answer\":\"It introduces a machine learning approach to predict the optimal dual solution and then guides CG by penalizing dual variables that deviate from the prediction.\"},{\"question\":\"What are the reported results and where are they tested?\",\"answer\":\"The method is evaluated on the graph coloring problem, where it achieves a significantly improved convergence rate compared with standard CG and a traditional stabilization approach.\"}]","Adaptive Stabilization Based on Machine Learning for Column Generation | 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problem does the document address in column generation?","Question",{"text":77,"@type":78},"It addresses heavy oscillations of dual values during CG iterations, which can delay convergence and cause redundant column generation.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the proposed method use machine learning?",{"text":82,"@type":78},"It introduces a machine learning approach to predict the optimal dual solution and then guides CG by penalizing dual variables that deviate from the prediction.",{"name":84,"@type":75,"acceptedAnswer":85},"What are the reported results and where are they tested?",{"text":86,"@type":78},"The method is evaluated on the graph coloring problem, where it achieves a significantly improved convergence rate compared with standard CG and a traditional stabilization 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