[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119690-en":3,"doc-seo-119690-105":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},119690,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","MACHINE LEARNING FOR K-ADAPTABILITY IN TWO-STAGE ROBUST OPTIMIZATION","Two-stage robust optimization tackles decisions under uncertainty but is among the most challenging classes of optimization problems. A key approach is K-adaptability, which partitions the uncertainty set into K subsets and assigns a decision to each subset while selecting the best feasible wait-and-see option. Typical solutions rely on K-adaptability branch-and-bound, whose search trees grow exponentially. This work introduces a machine learning node-selection strategy, using feature engineering built from two-stage robust optimization insights, trained on resolved B&B trees, and transferable to problems with different sizes and K-values. Experiments show higher-quality solutions than random node selection.","MACHINE LEARNING FOR K-ADAPTABILITY IN TWO-STAGE ROBUST OPTIMIZATION  \narXiv :2210 . 11152v2 [math .OC] 12 Dec 2022  \nEsther Julien  \nDelft University of Technology [e.a.t.julien@tudelft.nl](e.a.t.julien@tudelft.nl)  \nKrzysztof Postek  \nDeft University of Technology [k.s.postek@tudelft.nl](k.s.postek@tudelft.nl)  \n¸S. ˙Ilker Birbil  \nUniversity of Amsterdam[s.i.birbil@uva.nl](s.i.birbil@uva.nl)  \nDecember 13, 2022  \nABSTRACT  \nTwo-stage robust optimization problems constitute one of the hardest optimization problem classes.  \nOne of the solution approaches to this class of problems is K-adaptability. This approach simultaneously seeks the best partitioning of the uncertainty set of scenarios into K subsets, and optimizes decisions corresponding to each of these subsets. In general case, it is solved using the K-adaptability branch-and-bound algorithm, which requires exploration of exponentially-growing solution trees. To accelerate finding high-quality solutions in such trees, we propose a machine learning-based nodeselection strategy. In particular, we construct a feature engineering scheme based on general two-stage robust optimization insights that allows us to train our machine learning tool on a database of resolved B&B trees, and to apply it as-is to problems of different sizes and/or types. We experimentally show that using our learned node selection strategy outperforms a vanilla, random node selection strategy when tested on problems of the same type as the training problems, also in case the K-value or the problem size differs from the training ones.  \nKeywords node selection; clustering; two-stage robust optimization; K-adaptability; machine learning; tree search  \n1 Introduction  \nMany optimization problems are affected by data uncertainty caused by errors in the forecast, implementation, or measurement. Robust optimization (RO) is one of the key paradigms to solve such problems, where the goal is to find an optimal solution among the ones that remain feasible for all data realizations within an uncertainty set [Ben-Talet al., 2009] . This set includes all reasonable data outcomes.  \nA specific class of RO problems comprises two-stage robust optimization (2SRO) problems in which some decisions are implemented before the uncertain data is known (here-and-now decisions), and other decisions are implemented after the data is revealed (wait-and-see decisions) . Such a problem can be formulated as  \nmin max min 􀀈c(z)| x + d(z)| y : T(z)x + W(z)y 􀀔 h (z); 8z 2 Z􀀉 ; (1)  \nx2X z2Z y2Y  \nwhere x 2 X 􀀒 RNx and y 2 Y 􀀒 RNy are the here-and-now and wait-and-see decisions, respectively, and z is the vector of initially unknown data belonging to the uncertainty set Z 􀀒 RNz . Solving problem (1) is difficult in general, since Z might include an infinite number of scenarios, and hence different values of y might be optimal for different realizations of z. In fact, finding optimal x is an NP-hard problem [Guslitzer, 2002] . To address this difficulty, several approaches have been proposed. The first one is to use so-called decision rules which explicitly formulate the second-stage decision y as a function of z, and hence the function parameters become first-stage decisions next to x ; see Ben-Tal et al. [2004] . Another approach is to partition Z into subsets and to assign a separate copy of y to each of the subsets. The partitioning is then iteratively refined, and the decisions become increasingly customized to the outcomes of z.  \nIn this paper, we consider a third approach to (1) known as K-adaptability. There, at most K possible wait-and-see decisions y 1 ; : : : ; yK are allowed to be constructed, and the decision maker must select one of those. The values of the  \npossible yk 's become the first-stage variables, and the problem boils down to  \nmin max min 􀀈c(z)| x + d(z)| yk : T(z)x + W(z)y k 􀀔 h (z); 8z 2 Zk 􀀉 ; (2)  \nx2X ;y2Y K z2Z k2K  \nwhere K = f1; : : : ; Kg and YK = 􀀂  Y. Although the solution space of (2) is finite-dimensional, ","cbCainzfbktM2x45","https://ap.wps.com/l/cbCainzfbktM2x45","pdf",9255713,1,32,"English","en",105,"# Abstract\n# Introduction\n## Robust optimization under data uncertainty\n## Two-stage robust optimization formulation\n## K-adaptability and scenario clustering\n## K-adaptability branch-and-bound overview","[{\"question\":\"What problem class does the paper focus on?\",\"answer\":\"The paper focuses on two-stage robust optimization, where some decisions are made before uncertainty is observed and others after the data is revealed.\"},{\"question\":\"How does K-adaptability approach two-stage robust optimization?\",\"answer\":\"K-adaptability partitions the uncertainty set into K subsets and restricts the second-stage decisions to at most K options, selecting the decision that is optimal for the realized uncertainty.\"},{\"question\":\"What is the proposed contribution to accelerate solving K-adaptability problems?\",\"answer\":\"The paper proposes a machine learning-based node-selection strategy for the K-adaptability branch-and-bound algorithm, trained via feature engineering from robust optimization structure and evaluated against random node selection.\"}]","MACHINE LEARNING FOR K-ADAPTABILITY IN TWO-STAGE ROBUST OPTIMIZATION | 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problem class does the paper focus on?","Question",{"text":75,"@type":76},"The paper focuses on two-stage robust optimization, where some decisions are made before uncertainty is observed and others after the data is revealed.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does K-adaptability approach two-stage robust optimization?",{"text":80,"@type":76},"K-adaptability partitions the uncertainty set into K subsets and restricts the second-stage decisions to at most K options, selecting the decision that is optimal for the realized uncertainty.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the proposed contribution to accelerate solving K-adaptability problems?",{"text":84,"@type":76},"The paper proposes a machine learning-based node-selection strategy for the K-adaptability branch-and-bound algorithm, trained via feature engineering from robust optimization structure and evaluated against random node 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