[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122800-en":3,"doc-seo-122800-105":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},122800,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1786009248482753345",8,"Research & Report","Machine Learning for Cutting Planes in Integer Programming - A Survey","The paper surveys recent machine learning (ML) methods for selecting cutting planes in mixed-integer linear programming (MILP). While many cut families exist, deciding which set of cuts to add to an LP relaxation at a given node of the branch-and-bound (B&B) tree remains difficult for both formal and heuristic approaches. The work reviews data collection practices, evaluation protocols, and ML model architectures, analyzes reported empirical results to gauge progress, and outlines directions for future research.","Machine Learning for Cutting Planes in Integer Programming: A Survey  \nArnaud Deza 1 , Elias B. Khalil 1 ,2  \n1Department of Mechanical & Industrial Engineering, University of Toronto  \n2 SCALE AI Research in Data-Driven Algorithms for Modern Supply Chains  \n[arnaud.deza@mail.utoronto.ca](arnaud.deza@mail.utoronto.ca), [khalil@mie.utoronto.ca](khalil@mie.utoronto.ca)  \narXiv :2302 .09166v2 [math .OC] 31 Oct 2023  \nAbstract  \nWe survey recent work on machine learning (ML) techniques for selecting cutting planes (or cuts) in mixed-integer linear programming (MILP) . Despite the availability of various classes of cuts, the task of choosing a set of cuts to add to the linear programming (LP) relaxation at a given node of the branch-and-bound (B&B) tree has defied both formal and heuristic solutions to date. ML offers a promising approach for improving the cut selection process by using data to identify promising cuts that accelerate the solution of MILP instances.  \nThis paper presents an overview of the topic, highlighting recent advances in the literature, common approaches to data collection, evaluation, and ML model architectures. We analyze the empirical results in the literature in an attempt to quantify the progress that has been made and conclude by suggesting avenues for future research.  \n1 Introduction  \nML has recently been applied to accelerate the solution of optimization problems, with MILP being one of the most active research areas [Bengio et al., 2021; Kotary et al., 2021; Mazyavkina et al., 2021] . A MILP is an optimization problem that involves both continuous and discrete variables, and aims to minimize or maximize a linear objective function c⊺ x, over its decision variables x ∈ Z |J| × Rn−|J| while satisfying a set of m linear constraints Ax ≤ b. Here, J ⊆ {1, ··· , n}, |J| ≥ 1, corresponds to the set of indices of integer variables. Similarly, Integer programming (IP) problems are of the same form only with discrete variables, i.ex ∈ Zn. The MILP problem is written as:  \nzIP = min{c⊺ x | Ax ≤ b, x ∈ Z |J| × Rn−|J| } (1)  \nThe MILP formalism is widely used in supply chain and logistics, production planning, etc. While the MILP (1) problem is NP-hard in general, modern solvers are able to effectively tackle large-scale instances, often to global optimality, using a combination of exact search and heuristic techniques. The backbone of MILP solving is the implementation of a tree search algorithm, Branch and Bound (B&B) [Land and  \nDoig, 2010], which relies on repeatedly solving computationally tractable versions of the original problem where discrete variables are relaxed to take on continuous values. Formally, we denote the linear programming (LP) relaxation of problem (1):  \nz LP = min{c⊺ x | Ax ≤ b, x ∈ Rn } (2)  \nTo render the B&B search more efficient, valid linear inequalities (or tightening constraints) to problem (1)– cutting planes –are added to LP relaxations ofthe MILP with the aim of producing tighter relaxations and thus better lower bounds to problem (1), as illustrated in figure 1; this approach is referred to as the Branch and Cut (B&C) algorithm. Cuts are essential for MILP solving, as they can significantly reduce the feasible region of the B&B algorithm and exploit the structure of the underlying combinatorial problem, which a pure B&B approach does not do. The incorporation of cutting planes in the B&B search necessitates appropriate filtering and selection as there can be many such valid inequalities and adding them to a MILP comes at a cost in computation time. As such, cut selection has been an area of active research in recent years.  \nVarious families of general-purpose and problem-specific cuts have been studied theoretically and implemented in modern-day solvers [Dey and Molinaro, 2018] . However, there is an overall lack of scientific understanding regarding many of the key design decisions when it comes to incorporating cuts in B&B. Traditionally, the management of cutting plane ge","cbCaif73A57g9UZR","https://ap.wps.com/l/cbCaif73A57g9UZR","pdf",530810,1,9,"English","en",105,"# Introduction\n## MILP and LP relaxations in branch-and-bound\n## Cutting planes and the branch-and-cut framework\n## Cut selection strategies in modern solvers\n## ML approaches for cut selection","[{\"question\":\"Why is cut selection challenging in MILP branch-and-bound?\",\"answer\":\"Even though many valid cuts exist, choosing a subset of cuts for each B\\u0026B node is computationally costly and lacks universally effective formal or heuristic rules.\"},{\"question\":\"How do cutting planes improve the branch-and-bound process?\",\"answer\":\"Cuts tighten LP relaxations, reduce the feasible region, and produce better lower bounds, which can accelerate convergence in branch-and-cut methods.\"},{\"question\":\"What does the survey cover about machine learning for cut selection?\",\"answer\":\"It reviews recent ML techniques, typical data collection and evaluation setups, and common model architectures, then analyzes empirical findings and proposes future research directions.\"}]","Machine Learning for Cutting Planes in Integer Programming - A Survey | PDF",1785812963,23,{"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},"machine-learning-for-cutting-planes-in-integer-programming-a-survey","",{"@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/machine-learning-for-cutting-planes-in-integer-programming-a-survey/122800/",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},"Why is cut selection challenging in MILP branch-and-bound?","Question",{"text":75,"@type":76},"Even though many valid cuts exist, choosing a subset of cuts for each B&B node is computationally costly and lacks universally effective formal or heuristic rules.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do cutting planes improve the branch-and-bound process?",{"text":80,"@type":76},"Cuts tighten LP relaxations, reduce the feasible region, and produce better lower bounds, which can accelerate convergence in branch-and-cut methods.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the survey cover about machine learning for cut selection?",{"text":84,"@type":76},"It reviews recent ML techniques, typical data collection and evaluation setups, and common model architectures, then analyzes empirical findings and proposes future research directions.","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},[93,97,101,105,110,115,120,123,127,130,134],{"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":21,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]