{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,1]],"date-time":"2025-03-01T05:39:23Z","timestamp":1740807563232,"version":"3.38.0"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T00:00:00Z","timestamp":1716768000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T00:00:00Z","timestamp":1716768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100020884","name":"Agencia Nacional de Investigaci\u00f3n y Desarrollo","doi-asserted-by":"publisher","award":["11190515","11220586","FB210005","3210735"],"award-info":[{"award-number":["11190515","11220586","FB210005","3210735"]}],"id":[{"id":"10.13039\/501100020884","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2025,3]]},"DOI":"10.1007\/s10107-024-02097-w","type":"journal-article","created":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T14:04:26Z","timestamp":1716818666000},"page":"695-730","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exploiting the polyhedral geometry of stochastic linear bilevel programming"],"prefix":"10.1007","volume":"210","author":[{"given":"Gonzalo","family":"Mu\u00f1oz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3924-366X","authenticated-orcid":false,"given":"David","family":"Salas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anton","family":"Svensson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,27]]},"reference":[{"issue":"1","key":"2097_CR1","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1016\/j.amc.2006.01.043","volume":"181","author":"C Audet","year":"2006","unstructured":"Audet, C., Haddad, J., Savard, G.: A note on the definition of a linear bilevel programming solution. Appl. Math. Comput. 181(1), 351\u2013355 (2006). https:\/\/doi.org\/10.1016\/j.amc.2006.01.043","journal-title":"Appl. Math. Comput."},{"key":"2097_CR2","doi-asserted-by":"crossref","unstructured":"Beck, Y., Ljubi\u0107, I., Schmidt, M.: A survey on bilevel optimization under uncertainty. Eur. J. Oper. Res. (2023)","DOI":"10.1016\/j.ejor.2023.01.008"},{"issue":"1","key":"2097_CR3","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","volume":"59","author":"J Bezanson","year":"2017","unstructured":"Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59(1), 65\u201398 (2017). https:\/\/doi.org\/10.1137\/141000671","journal-title":"SIAM Rev."},{"issue":"6","key":"2097_CR4","doi-asserted-by":"publisher","first-page":"618","DOI":"10.1016\/j.orl.2023.10.006","volume":"51","author":"C Buchheim","year":"2023","unstructured":"Buchheim, C.: Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations. Oper. Res. Lett. 51(6), 618\u2013622 (2023)","journal-title":"Oper. Res. Lett."},{"issue":"4","key":"2097_CR5","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1007\/s10898-021-01117-9","volume":"83","author":"C Buchheim","year":"2022","unstructured":"Buchheim, C., Henke, D.: The robust bilevel continuous knapsack problem with uncertain coefficients in the follower\u2019s objective. J. Global Optim. 83(4), 803\u2013824 (2022)","journal-title":"J. Global Optim."},{"issue":"5","key":"2097_CR6","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1016\/j.orl.2021.07.009","volume":"49","author":"C Buchheim","year":"2021","unstructured":"Buchheim, C., Henke, D., Hommelsheim, F.: On the complexity of robust bilevel optimization with uncertain follower\u2019s objective. Oper. Res. Lett. 49(5), 703\u2013707 (2021)","journal-title":"Oper. Res. Lett."},{"key":"2097_CR7","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1007\/s10957-022-02037-8","volume":"194","author":"C Buchheim","year":"2022","unstructured":"Buchheim, C., Henke, D., Irmai, J.: The stochastic bilevel continuous knapsack problem with uncertain follower\u2019s objective. J. Optim. Theory Appl. 194, 521\u2013542 (2022). https:\/\/doi.org\/10.1007\/s10957-022-02037-8","journal-title":"J. Optim. Theory Appl."},{"key":"2097_CR8","doi-asserted-by":"publisher","unstructured":"Burtscheidt, J., Claus, M.: Bilevel linear optimization under uncertainty. In: Bilevel Optimization\u2014Advances and Next Challenges, Springer Optim. Appl., vol. 161, pp. 485\u2013511. Springer, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-52119-6_17","DOI":"10.1007\/978-3-030-52119-6_17"},{"issue":"3","key":"2097_CR9","doi-asserted-by":"publisher","first-page":"412","DOI":"10.1016\/j.orl.2021.04.007","volume":"49","author":"M Claus","year":"2021","unstructured":"Claus, M.: On continuity in risk-averse bilevel stochastic linear programming with random lower level objective function. Oper. Res. Lett. 49(3), 412\u2013417 (2021). https:\/\/doi.org\/10.1016\/j.orl.2021.04.007","journal-title":"Oper. Res. Lett."},{"issue":"2","key":"2097_CR10","doi-asserted-by":"publisher","first-page":"542","DOI":"10.1016\/j.ejor.2021.12.004","volume":"299","author":"M Claus","year":"2022","unstructured":"Claus, M.: Existence of solutions for a class of bilevel stochastic linear programs. Eur. J. Oper. Res. 299(2), 542\u2013549 (2022). https:\/\/doi.org\/10.1016\/j.ejor.2021.12.004","journal-title":"Eur. J. Oper. Res."},{"key":"2097_CR11","unstructured":"Coral bilevel optimization problem library. https:\/\/coral.ise.lehigh.edu\/data-sets\/bilevel-instances\/. Accessed 3 Nov 2022"},{"key":"2097_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-12971-1","volume-title":"Triangulations: Structures for Algorithms and Applications","author":"J De Loera","year":"2010","unstructured":"De Loera, J., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications, vol. 25. Springer, Berlin (2010)"},{"key":"2097_CR13","volume-title":"Foundations of Bilevel Programming","author":"S Dempe","year":"2002","unstructured":"Dempe, S.: Foundations of Bilevel Programming. Springer, Berlin (2002)"},{"key":"2097_CR14","doi-asserted-by":"publisher","unstructured":"Dempe, S., Kalashnikov, V., P\u00e9rez-Vald\u00e9s, G.A., Kalashnykova, N.: Bilevel programming problems. In: Energy Systems. Theory, Algorithms and Applications to Energy Networks. Springer, Heidelberg (2015). https:\/\/doi.org\/10.1007\/978-3-662-45827-3","DOI":"10.1007\/978-3-662-45827-3"},{"key":"2097_CR15","doi-asserted-by":"publisher","unstructured":"Dempe, S., Zemkoho, A. (eds.): Bilevel Optimization\u2014Advances and Next Challenges. Springer Optimization and Its Applications, vol. 161. Springer, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-52119-6","DOI":"10.1007\/978-3-030-52119-6"},{"issue":"1","key":"2097_CR16","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1137\/22M1508005","volume":"34","author":"M Forcier","year":"2024","unstructured":"Forcier, M., Gaubert, S., Lecl\u00e8re, V.: Exact quantization of multistage stochastic linear problems. SIAM J. Optim. 34(1), 533\u2013562 (2024). https:\/\/doi.org\/10.1137\/22M1508005","journal-title":"SIAM J. Optim."},{"key":"2097_CR17","unstructured":"Forcier, M., Gaubert, S., Lecl\u00e8re, V.: Exact quantization of multistage stochastic linear problems (2021). Preprint. arXiv:2107.09566"},{"key":"2097_CR18","doi-asserted-by":"crossref","unstructured":"Gawrilow, E., Joswig, M.: polymake: a framework for analyzing convex polytopes. In: Polytopes\u2014combinatorics and computation (Oberwolfach, 1997), DMV Sem., vol.\u00a029, pp. 43\u201373. Birkh\u00e4user, Basel (2000)","DOI":"10.1007\/978-3-0348-8438-9_2"},{"key":"2097_CR19","unstructured":"Gurobi Optimization, LLC: Gurobi Optimizer Reference Manual (2022). https:\/\/www.gurobi.com"},{"issue":"5","key":"2097_CR20","doi-asserted-by":"publisher","first-page":"1194","DOI":"10.1137\/0913069","volume":"13","author":"P Hansen","year":"1992","unstructured":"Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13(5), 1194\u20131217 (1992)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"2097_CR21","doi-asserted-by":"crossref","unstructured":"Henke, D., Lefebvre, H., Schmidt, M., Th\u00fcrauf, J.: On coupling constraints in linear bilevel optimization, 2024. Preprint arXiv:2402.12191","DOI":"10.1007\/s11590-024-02156-3"},{"issue":"1","key":"2097_CR22","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/j.sorms.2014.05.001","volume":"19","author":"T Homem-de Mello","year":"2014","unstructured":"Homem-de Mello, T., Bayraksan, G.: Monte Carlo sampling-based methods for stochastic optimization. Surv. Oper. Res. Manag. Sci. 19(1), 56\u201385 (2014). https:\/\/doi.org\/10.1016\/j.sorms.2014.05.001","journal-title":"Surv. Oper. Res. Manag. Sci."},{"issue":"4","key":"2097_CR23","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1134\/S1990478918040063","volume":"25","author":"SV Ivanov","year":"2018","unstructured":"Ivanov, S.V.: A bilevel programming problem with random parameters in the follower\u2019s objective function. Diskretn. Anal. Issled. Oper. 25(4), 27\u201345 (2018). https:\/\/doi.org\/10.1134\/S1990478918040063","journal-title":"Diskretn. Anal. Issled. Oper."},{"key":"2097_CR24","doi-asserted-by":"crossref","unstructured":"Khachiyan, L., Boros, E., Borys, K., Gurvich, V., Elbassioni, K.: Generating all vertices of a polyhedron is hard. In: 20th Anniversary Volume, pp. 1\u201317. Springer (2009)","DOI":"10.1007\/978-0-387-87363-3_17"},{"key":"2097_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejco.2021.100007","volume":"9","author":"T Kleinert","year":"2021","unstructured":"Kleinert, T., Labb\u00e9, M., Ljubi\u0107, I., Schmidt, M.: A survey on mixed-integer programming techniques in bilevel optimization. EURO J. Comput. Optim. 9, 100007 (2021)","journal-title":"EURO J. Comput. Optim."},{"key":"2097_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-5361-0","volume-title":"Probability Theory: A Comprehensive Course","author":"A Klenke","year":"2014","unstructured":"Klenke, A.: Probability Theory: A Comprehensive Course. Springer, Berlin (2014)"},{"key":"2097_CR27","doi-asserted-by":"publisher","unstructured":"Leobacher, G., Pillichshammer, F.: Introduction to Quasi-Monte Carlo Integration and Applications. Compact Textbooks in Mathematics. Birkh\u00e4user\/Springer, Cham (2014). https:\/\/doi.org\/10.1007\/978-3-319-03425-6","DOI":"10.1007\/978-3-319-03425-6"},{"key":"2097_CR28","doi-asserted-by":"publisher","unstructured":"Mallozzi, L., Morgan, J.: Hierarchical Systems with Weighted Reaction Set, pp. 271\u2013282. Springer, Boston (1996). https:\/\/doi.org\/10.1007\/978-1-4899-0289-4_19","DOI":"10.1007\/978-1-4899-0289-4_19"},{"issue":"1","key":"2097_CR29","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.amc.2005.11.134","volume":"180","author":"AG Mersha","year":"2006","unstructured":"Mersha, A.G., Dempe, S.: Linear bilevel programming with upper level constraints depending on the lower level solution. Appl. Math. Comput. 180(1), 247\u2013254 (2006). https:\/\/doi.org\/10.1016\/j.amc.2005.11.134","journal-title":"Appl. Math. Comput."},{"key":"2097_CR30","doi-asserted-by":"crossref","unstructured":"Mu\u00f1oz, G., Salas, D., Svensson, A.: Exploiting the polyhedral geometry of stochastic linear bilevel programming. In: International Conference on Integer Programming and Combinatorial Optimization, pp. 363\u2013377. Springer, Cham (2023)","DOI":"10.1007\/978-3-031-32726-1_26"},{"issue":"3","key":"2097_CR31","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1007\/BF02711510","volume":"16","author":"J Rambau","year":"1996","unstructured":"Rambau, J., Ziegler, G.M.: Projections of polytopes and the generalized Baues conjecture. Discrete Comput. Geometry 16(3), 215\u2013237 (1996)","journal-title":"Discrete Comput. Geometry"},{"issue":"3","key":"2097_CR32","doi-asserted-by":"publisher","first-page":"2311","DOI":"10.1137\/21M1442164","volume":"33","author":"D Salas","year":"2023","unstructured":"Salas, D., Svensson, A.: Existence of solutions for deterministic bilevel games under a general Bayesian approach. SIAM J. Optim. 33(3), 2311\u20132340 (2023). https:\/\/doi.org\/10.1137\/21M1442164","journal-title":"SIAM J. Optim."},{"key":"2097_CR33","doi-asserted-by":"publisher","unstructured":"Shapiro, A., Dentcheva, D., Ruszczy\u0144ski, A.: Lectures on stochastic programming\u2014modeling and theory. In: MOS-SIAM Series on Optimization, vol.\u00a028. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Mathematical Optimization Society, Philadelphia, PA (2021). https:\/\/doi.org\/10.1137\/1.9781611976595. Third edition [of 2562798]","DOI":"10.1137\/1.9781611976595"},{"issue":"2","key":"2097_CR34","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/BF02191670","volume":"81","author":"L Vicente","year":"1994","unstructured":"Vicente, L., Savard, G., J\u00fadice, J.: Descent approaches for quadratic bilevel programming. J. Optim. Theory Appl. 81(2), 379\u2013399 (1994)","journal-title":"J. Optim. Theory Appl."},{"key":"2097_CR35","volume-title":"Marktform und Gleichgewitch","author":"H Von Stackelberg","year":"1934","unstructured":"Von Stackelberg, H.: Marktform und Gleichgewitch. Springer, Berlin (1934)"},{"key":"2097_CR36","doi-asserted-by":"crossref","unstructured":"Zhou, S., Zemkoho, A.B., Tin, A.: Bolib: Bilevel optimization library of test problems. In: Bilevel Optimization, pp. 563\u2013580. Springer, Berlin (2020)","DOI":"10.1007\/978-3-030-52119-6_19"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-024-02097-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-024-02097-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-024-02097-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,28]],"date-time":"2025-02-28T15:51:01Z","timestamp":1740757861000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-024-02097-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,27]]},"references-count":36,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["2097"],"URL":"https:\/\/doi.org\/10.1007\/s10107-024-02097-w","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"type":"print","value":"0025-5610"},{"type":"electronic","value":"1436-4646"}],"subject":[],"published":{"date-parts":[[2024,5,27]]},"assertion":[{"value":"25 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 May 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}