[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120649-en":3,"doc-seo-120649-105":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},120649,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Machine Learning Approach to Solving Large Bilevel and Stochastic Programs - Application to Cycling Network Design","A machine learning-based optimization method addresses bilevel programs with many independent followers, covering two-stage stochastic programming as a special case. The model samples a subset of followers, uses a learning component to estimate objective values for unsampled followers, and embeds training within the optimization to leverage general follower features. The work derives bounds on the leader’s optimality gap under the full follower set, introduces sampling schemes and representation learning, and validates performance on cycling network design, including a real-world infrastructure case with over one million followers.","arXiv :2209 .09404v1 [math .OC] 20 Sep 2022  \nA Machine Learning Approach to Solving Large Bilevel and Stochastic Programs: Application to  \nCycling Network Design  \nTimothy C. Y. Chan, Bo Lin  \nDepartment of Mechanical & Industrial Engineering, University of Toronto, ftcychan, [blin](bling@mie.utoronto.ca)[g](bling@mie.utoronto.ca)[@mie.utoronto.ca](bling@mie.utoronto.ca),  \nShoshanna Saxe  \nDepartment of Civil & Mineral Engineering, University of Toronto, [s.saxe@utoronto.ca](s.saxe@utoronto.ca),  \nWe present a novel machine learning-based approach to solving bilevel programs that involve a large number of independent followers, which as a special case include two-stage stochastic programming. We propose an optimization model that explicitly considers a sampled subset of followers and exploits a machine learning model to estimate the objective values of unsampled followers. Unlike existing approaches, we embed machine learning model training into the optimization problem, which allows us to employ general follower features that can not be represented using leader decisions. We prove bounds on the optimality gap of the generated leader decision as measured by the original objective function that considers the full follower set. We then develop follower sampling algorithms to tighten the bounds and a representation learning approach to learn follower features, which can be used as inputs to the embedded machine learning model. Using synthetic instances of a cycling network design problem, we compare the computational performance of our approach versus baseline methods. Our approach provides more accurate predictions for follower objective values, and more importantly, generates leader decisions of higher quality. Finally, we perform a real-world case study on cycling infrastructure planning, where we apply our approach to solve a network design problem with over one million followers. Our approach presents favorable performance compared to the current cycling network expansion practices.  \nKey words : bilevel optimization; two-stage stochastic programming; machine learning; model reduction; cycling network design.  \n1 . Introduction  \nThis paper is concerned with solving bilevel optimization problems with a large number of followers and where the feasible region of the leader is independent of the followers. A wide range of decision problems can be modeled this way, including active transportation network design (Liu et al. 2019 , 2021b, Lim et al. 2021), network pricing (Van Hoesel 2008 , Alizadeh et al. 2013), energy pricing (Fampa et al. 2008 , Zugno et al. 2013), and portfolio optimization (Carri􀀓on et al. 2009 , Leal et al. 2020) . This model also generalizes two-stage stochastic programming. Indeed, if the objectives of the leader and followers  \nare identical, then this model becomes a two-stage stochastic program where the set of followers represent scenarios. Thus, in this paper, the reader should think of \\leader\" in abilevel program and \\􀀌rst-stage decision maker\" in a stochastic program as synonymous, and similarly for \\follower\" and \\second-stage decision maker\". As we discuss the bilevel or stochastic programming literature below, we use the corresponding terminology.  \nThe main challenge of solving a bilevel problem with a large set of followers S stems from having to solve a large number of follower problems to evaluate the quality of the leader's decision. For the bilevel problem we consider, given its relationship to stochastic programming, we can draw on approaches to deal with large S from both communities. Two predominant strategies are: i) solving the problem with a small sample of S, and ii) approximating the followers' cost without explicitly solving the followers' problems.  \nSampling a smaller follower set can be done via random sampling (Liu et al. 2021b, Limet al. 2021) or clustering (Dupa􀀔cov􀀓a et al. 2003 , Hewitt et al. 2021 , Bertsimas and Mundru 2022) . Given a sample T 􀀒 S, we can obtain a fea","cbCaivfkXGSVyhBU","https://ap.wps.com/l/cbCaivfkXGSVyhBU","pdf",8305993,1,59,"English","en",105,"# Introduction\n## Problem setting and challenge\n## Sampling strategies for large follower sets\n## Approximating follower costs and ML-based alternatives\n## Paper contribution overview","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses bilevel optimization problems where the leader’s feasible region is independent of a large set of followers, and it generalizes two-stage stochastic programming by interpreting followers as scenarios.\"},{\"question\":\"How does the proposed approach use machine learning?\",\"answer\":\"It samples a subset of followers and uses an embedded machine learning model to estimate objective values for unsampled followers, integrating model training directly into the optimization problem.\"},{\"question\":\"What theoretical results are provided?\",\"answer\":\"The paper proves bounds on the optimality gap of the generated leader decision measured against the original objective considering the full follower set.\"}]","A Machine Learning Approach to Solving Large Bilevel and Stochastic Programs - Application to Cycling Network Design | PDF",1785731157,149,{"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},"a-machine-learning-approach-to-solving-large-bilevel-and-stochastic-programs-application-to-cycling-network-design","",{"@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/a-machine-learning-approach-to-solving-large-bilevel-and-stochastic-programs-application-to-cycling-network-design/120649/",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","2026-08-03",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 problem does the paper address?","Question",{"text":75,"@type":76},"It addresses bilevel optimization problems where the leader’s feasible region is independent of a large set of followers, and it generalizes two-stage stochastic programming by interpreting followers as scenarios.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed approach use machine learning?",{"text":80,"@type":76},"It samples a subset of followers and uses an embedded machine learning model to estimate objective values for unsampled followers, integrating model training directly into the optimization problem.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical results are provided?",{"text":84,"@type":76},"The paper proves bounds on the optimality gap of the generated leader decision measured against the original objective considering the full follower set.","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"]