{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:02:53Z","timestamp":1740153773138,"version":"3.37.3"},"reference-count":74,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,7,10]],"date-time":"2023-07-10T00:00:00Z","timestamp":1688947200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,10]],"date-time":"2023-07-10T00:00:00Z","timestamp":1688947200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100018755","name":"Universit\u00e4t Trier","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100018755","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Prog. Comp."],"published-print":{"date-parts":[[2023,12]]},"abstract":"Abstract<\/jats:title>Developing solution methods for discrete bilevel problems is known to be a challenging task\u2014even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a $$\\varGamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-robust approach, we present an extended formulation and a multi-follower formulation to model this type of problem. For both settings, we provide a generic branch-and-cut framework. Specifically, we investigate interdiction problems with a monotone $$\\varGamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-robust follower and we derive problem-tailored cuts, which extend existing techniques that have been proposed for the deterministic case. For the $$\\varGamma $$<\/jats:tex-math>\n \u0393<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-robust knapsack interdiction problem, we computationally evaluate and compare the performance of the proposed algorithms for both modeling approaches.<\/jats:p>","DOI":"10.1007\/s12532-023-00244-6","type":"journal-article","created":{"date-parts":[[2023,7,10]],"date-time":"2023-07-10T10:01:42Z","timestamp":1688983302000},"page":"733-782","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Exact methods for discrete $${\\varGamma }$$-robust interdiction problems with an application to the bilevel knapsack problem"],"prefix":"10.1007","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5735-0149","authenticated-orcid":false,"given":"Yasmine","family":"Beck","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4834-6284","authenticated-orcid":false,"given":"Ivana","family":"Ljubi\u0107","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6208-5677","authenticated-orcid":false,"given":"Martin","family":"Schmidt","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,10]]},"reference":[{"issue":"2","key":"244_CR1","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1049\/iet-gtd.2009.0098","volume":"4","author":"JM Arroyo","year":"2010","unstructured":"Arroyo, J.M.: Bilevel programming applied to power system vulnerability analysis under multiple contingencies. IET Gener. Trans. Distrib. 4(2), 178\u2013190 (2010). https:\/\/doi.org\/10.1049\/iet-gtd.2009.0098","journal-title":"IET Gener. Trans. Distrib."},{"key":"244_CR2","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1007\/s10878-011-9449-4","volume":"26","author":"C Bazgan","year":"2013","unstructured":"Bazgan, C., Toubaline, S., Vanderpooten, D.: Critical edges\/nodes for the minimum spanning tree problem: complexity and approximation. J. Comb. Optim. 26, 178\u2013189 (2013). https:\/\/doi.org\/10.1007\/s10878-011-9449-4","journal-title":"J. Comb. Optim."},{"issue":"2","key":"244_CR3","first-page":"1","volume":"30","author":"Y Beck","year":"2022","unstructured":"Beck, Y., Ljubi\u0107, I., Schmidt, M.: A brief introduction to robust bilevel optimization. SIAG Optimiz. Views News 30(2), 1\u201310 (2022)","journal-title":"SIAG Optimiz. Views News"},{"key":"244_CR4","doi-asserted-by":"publisher","unstructured":"Beck, Y., Ljubi\u0107, I., Schmidt, M.: Gamma-robust-knapsack-interdiction-solver (2023). https:\/\/doi.org\/10.5281\/zenodo.7965281","DOI":"10.5281\/zenodo.7965281"},{"key":"244_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2023.01.008","author":"Y Beck","year":"2023","unstructured":"Beck, Y., Ljubi\u0107, I., Schmidt, M.: A survey on bilevel optimization under uncertainty. Eur. J. Oper. Res. (2023). https:\/\/doi.org\/10.1016\/j.ejor.2023.01.008","journal-title":"Eur. J. Oper. Res."},{"issue":"5","key":"244_CR6","doi-asserted-by":"publisher","first-page":"752","DOI":"10.1016\/j.orl.2021.07.010","volume":"49","author":"Y Beck","year":"2021","unstructured":"Beck, Y., Schmidt, M.: A robust approach for modeling limited observability in bilevel optimization. Oper. Res. Lett. 49(5), 752\u2013758 (2021). https:\/\/doi.org\/10.1016\/j.orl.2021.07.010","journal-title":"Oper. Res. Lett."},{"key":"244_CR7","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/BF02098181","volume":"34","author":"O Ben-Ayed","year":"1992","unstructured":"Ben-Ayed, O., Blair, C., Boyce, D., LeBlanc, L.: Construction of a real-world bilevel linear programming model of the highway network design problem. Ann. Oper. Res. 34, 219\u2013254 (1992). https:\/\/doi.org\/10.1007\/BF02098181","journal-title":"Ann. Oper. Res."},{"key":"244_CR8","doi-asserted-by":"publisher","DOI":"10.1515\/9781400831050","author":"A Ben-Tal","year":"2009","unstructured":"Ben-Tal, A., Ghaoui, L., Nemirovski, A.: Distributed control of robotic networks. Robust. Optim. (2009). https:\/\/doi.org\/10.1515\/9781400831050","journal-title":"Robust. Optim."},{"key":"244_CR9","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1007\/BF01386316","volume":"4","author":"JF Benders","year":"1962","unstructured":"Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4, 238\u2013252 (1962). https:\/\/doi.org\/10.1007\/BF01386316","journal-title":"Numer. Math."},{"key":"244_CR10","doi-asserted-by":"publisher","DOI":"10.1137\/080734510","author":"D Bertsimas","year":"2010","unstructured":"Bertsimas, D., Brown, D., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. (2010). https:\/\/doi.org\/10.1137\/080734510","journal-title":"SIAM Rev."},{"key":"244_CR11","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/s10107-003-0396-4","volume":"98","author":"D Bertsimas","year":"2003","unstructured":"Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98, 49\u201371 (2003). https:\/\/doi.org\/10.1007\/s10107-003-0396-4","journal-title":"Math. Program."},{"issue":"1","key":"244_CR12","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1287\/opre.1030.0065","volume":"52","author":"D Bertsimas","year":"2004","unstructured":"Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35\u201353 (2004). https:\/\/doi.org\/10.1287\/opre.1030.0065","journal-title":"Oper. Res."},{"key":"244_CR13","unstructured":"Besan\u00e7on, M., Anjos, M.F., Brotcorne, L.: Near-optimal robust bilevel optimization (2019). https:\/\/arxiv.org\/pdf\/1908.04040.pdf"},{"key":"244_CR14","doi-asserted-by":"publisher","first-page":"2597","DOI":"10.1007\/s11590-021-01754-9","volume":"15","author":"M Besan\u00e7on","year":"2021","unstructured":"Besan\u00e7on, M., Anjos, M.F., Brotcorne, L.: Complexity of near-optimal robust versions of multilevel optimization problems. Optim. Lett. 15, 2597\u20132610 (2021). https:\/\/doi.org\/10.1007\/s11590-021-01754-9","journal-title":"Optim. Lett."},{"key":"244_CR15","doi-asserted-by":"publisher","unstructured":"Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer-Verlag New York (2011). https:\/\/doi.org\/10.1007\/978-1-4614-0237-4","DOI":"10.1007\/978-1-4614-0237-4"},{"key":"244_CR16","doi-asserted-by":"publisher","unstructured":"Bolusani, S., Coniglio, S., Ralphs, T.K., Tahernejad, S.: A Unified Framework for Multistage Mixed Integer Linear Optimization, pp. 513\u2013560. Springer International Publishing (2020). https:\/\/doi.org\/10.1007\/978-3-030-52119-6_18","DOI":"10.1007\/978-3-030-52119-6_18"},{"key":"244_CR17","doi-asserted-by":"publisher","first-page":"530","DOI":"10.1287\/inte.1060.0252","volume":"36","author":"G Brown","year":"2006","unstructured":"Brown, G., Carlyle, M., Salmer\u00f3n, J., Wood, R.: Defending critical infrastructure. Interfaces 36, 530\u2013544 (2006). https:\/\/doi.org\/10.1287\/inte.1060.0252","journal-title":"Interfaces"},{"key":"244_CR18","unstructured":"Buchheim, C., Henke, D.: The bilevel continuous knapsack problem with uncertain follower\u2019s objective (2020). https:\/\/arxiv.org\/abs\/1903.02810"},{"key":"244_CR19","unstructured":"Buchheim, C., Henke, D., Hommelsheim, F.: On the complexity of robust bilevel optimization with uncertain follower\u2019s objective (2021). https:\/\/arxiv.org\/abs\/2105.08378"},{"key":"244_CR20","doi-asserted-by":"publisher","unstructured":"Burtscheidt, J., Claus, M.: Bilevel linear optimization under uncertainty, pp. 485\u2013511. Springer International Publishing (2020). https:\/\/doi.org\/10.1007\/978-3-030-52119-6_17","DOI":"10.1007\/978-3-030-52119-6_17"},{"issue":"1","key":"244_CR21","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1137\/19M1242240","volume":"30","author":"J Burtscheidt","year":"2020","unstructured":"Burtscheidt, J., Claus, M., Dempe, S.: Risk-averse models in bilevel stochastic linear programming. SIAM J. Optim. 30(1), 377\u2013406 (2020). https:\/\/doi.org\/10.1137\/19M1242240","journal-title":"SIAM J. Optim."},{"key":"244_CR22","doi-asserted-by":"publisher","unstructured":"Caprara, A., Carvalho, M., Lodi, A., Woeginger, G.J.: A complexity and approximability study of the bilevel knapsack problem. In: M.\u00a0Goemans, J.\u00a0Correa (eds.) Integer Programming and Combinatorial Optimization, IPCO 2013, vol. 7801, pp. 98\u2013109. Springer, Berlin, Heidelberg (2013). https:\/\/doi.org\/10.1007\/978-3-642-36694-9_9","DOI":"10.1007\/978-3-642-36694-9_9"},{"issue":"2","key":"244_CR23","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1287\/ijoc.2015.0676","volume":"28","author":"A Caprara","year":"2016","unstructured":"Caprara, A., Carvalho, M., Lodi, A., Woeginger, G.J.: Bilevel knapsack with interdiction constraints. INFORMS J. Comput. 28(2), 319\u2013333 (2016). https:\/\/doi.org\/10.1287\/ijoc.2015.0676","journal-title":"INFORMS J. Comput."},{"issue":"2","key":"244_CR24","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1007\/s10957-017-1069-4","volume":"173","author":"TD Chuong","year":"2017","unstructured":"Chuong, T.D., Jeyakumar, V.: Finding robust global optimal values of bilevel polynomial programs with uncertain linear constraints. J. Optim. Theory Appl. 173(2), 683\u2013703 (2017). https:\/\/doi.org\/10.1007\/s10957-017-1069-4","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"244_CR25","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1287\/opre.46.2.184","volume":"46","author":"KJ Cormican","year":"1998","unstructured":"Cormican, K.J., Morton, D.P., Wood, R.K.: Stochastic network interdiction. Oper. Res. 46(2), 184\u2013197 (1998). https:\/\/doi.org\/10.1287\/opre.46.2.184","journal-title":"Oper. Res."},{"key":"244_CR26","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10107-006-0086-0","volume":"112","author":"G Cornu\u00e9jols","year":"2008","unstructured":"Cornu\u00e9jols, G.: Valid inequalities for mixed integer linear programs. Math. Program. 112, 3\u201344 (2008). https:\/\/doi.org\/10.1007\/s10107-006-0086-0","journal-title":"Math. Program."},{"key":"244_CR27","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s10107-020-01482-5","volume":"183","author":"F Della Croce","year":"2020","unstructured":"Della Croce, F., Scatamacchia, R.: An exact approach for the bilevel knapsack problem with interdiction constraints and extensions. Math. Program. 183, 249\u2013281 (2020). https:\/\/doi.org\/10.1007\/s10107-020-01482-5","journal-title":"Math. Program."},{"key":"244_CR28","doi-asserted-by":"publisher","unstructured":"Dempe, S.: Foundations of Bilevel Programming. Springer US (2002). https:\/\/doi.org\/10.1007\/b101970","DOI":"10.1007\/b101970"},{"issue":"5","key":"244_CR29","doi-asserted-by":"publisher","first-page":"544","DOI":"10.1002\/asmb.2254","volume":"33","author":"S Dempe","year":"2017","unstructured":"Dempe, S., Ivanov, S., Naumov, A.: Reduction of the bilevel stochastic optimization problem with quantile objective function to a mixed-integer problem. Appl. Stoch. Model. Bus. Ind. 33(5), 544\u2013554 (2017). https:\/\/doi.org\/10.1002\/asmb.2254","journal-title":"Appl. Stoch. Model. Bus. Ind."},{"key":"244_CR30","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s10479-011-1023-z","volume":"196","author":"S Dempe","year":"2012","unstructured":"Dempe, S., Zemkoho, A.B.: Bilevel road pricing: theoretical analysis and optimality conditions. Ann. Oper. Res. 196, 223\u2013240 (2012). https:\/\/doi.org\/10.1007\/s10479-011-1023-z","journal-title":"Ann. Oper. Res."},{"key":"244_CR31","unstructured":"DeNegre, S.T.: Interdiction and discrete bilevel linear programming. Ph.D. thesis (2011). https:\/\/coral.ise.lehigh.edu\/~ted\/files\/papers\/ScottDeNegreDissertation11.pdf"},{"key":"244_CR32","doi-asserted-by":"publisher","unstructured":"DeNegre, S.T., Ralphs, T.K.: A branch-and-cut algorithm for integer bilevel linear programs. In: Operations research and cyber-infrastructure, pp. 65\u201378. Springer (2009). https:\/\/doi.org\/10.1007\/978-0-387-88843-9_4","DOI":"10.1007\/978-0-387-88843-9_4"},{"key":"244_CR33","doi-asserted-by":"publisher","unstructured":"Fischetti, M., Ljubi\u0107, I., Monaci, M., Sinnl, M.: Intersection cuts for bilevel optimization. In: Q.\u00a0Louveaux, M.\u00a0Skutella (eds.) Integer Programming and Combinatorial Optimization, IPCO 2016, pp. 77\u201388. Springer (2016). https:\/\/doi.org\/10.1007\/978-3-319-33461-5_7","DOI":"10.1007\/978-3-319-33461-5_7"},{"issue":"6","key":"244_CR34","doi-asserted-by":"publisher","first-page":"1615","DOI":"10.1287\/opre.2017.1650","volume":"65","author":"M Fischetti","year":"2017","unstructured":"Fischetti, M., Ljubi\u0107, I., Monaci, M., Sinnl, M.: A new general-purpose algorithm for mixed-integer bilevel linear programs. Oper. Res. 65(6), 1615\u20131637 (2017). https:\/\/doi.org\/10.1287\/opre.2017.1650","journal-title":"Oper. Res."},{"key":"244_CR35","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/s10107-017-1189-5","volume":"172","author":"M Fischetti","year":"2018","unstructured":"Fischetti, M., Ljubi\u0107, I., Monaci, M., Sinnl, M.: On the use of intersection cuts for bilevel optimization. Math. Program. 172, 77\u2013103 (2018). https:\/\/doi.org\/10.1007\/s10107-017-1189-5","journal-title":"Math. Program."},{"issue":"2","key":"244_CR36","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1287\/ijoc.2018.0831","volume":"31","author":"M Fischetti","year":"2019","unstructured":"Fischetti, M., Ljubi\u0107, I., Monaci, M., Sinnl, M.: Interdiction games and monotonicity, with application to knapsack problems. INFORMS J. Comput. 31(2), 390\u2013410 (2019). https:\/\/doi.org\/10.1287\/ijoc.2018.0831","journal-title":"INFORMS J. Comput."},{"issue":"16","key":"244_CR37","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.ejor.2017.11.043","volume":"267","author":"M Fischetti","year":"2018","unstructured":"Fischetti, M., Monaci, M., Sinnl, M.: A dynamic reformulation heuristic for generalized interdiction problems. Eur. J. Oper. Res. 267(16), 40\u201351 (2018). https:\/\/doi.org\/10.1016\/j.ejor.2017.11.043","journal-title":"Eur. J. Oper. Res."},{"issue":"9","key":"244_CR38","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1057\/jors.1981.156","volume":"32","author":"J Fortuny-Amat","year":"1981","unstructured":"Fortuny-Amat, J., McCarl, B.: A representation and economic interpretation of a two-level programming problem. J. Oper. Res. Soc. 32(9), 783\u2013792 (1981). https:\/\/doi.org\/10.1057\/jors.1981.156","journal-title":"J. Oper. Res. Soc."},{"key":"244_CR39","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/s12532-019-00167-1","volume":"12","author":"F Furini","year":"2020","unstructured":"Furini, F., Ljubi\u0107, I., Malaguti, E., Paronuzzi, P.: On integer and bilevel formulations for the $$k$$-vertex cut problem. Math. Program. Comput. 12, 133\u2013164 (2020). https:\/\/doi.org\/10.1007\/s12532-019-00167-1","journal-title":"Math. Program. Comput."},{"key":"244_CR40","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2021.2110","author":"F Furini","year":"2021","unstructured":"Furini, F., Ljubi\u0107, I., Malaguti, E., Paronuzzi, P.: Casting light on the hidden bilevel combinatorial structure of the capacitated vertex separator problem. Oper. Res. (2021). https:\/\/doi.org\/10.1287\/opre.2021.2110","journal-title":"Oper. Res."},{"issue":"1","key":"244_CR41","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/j.ejor.2021.01.030","volume":"294","author":"F Furini","year":"2021","unstructured":"Furini, F., Ljubi\u0107, I., Segundo, P.S., Zhao, Y.: A branch-and-cut algorithm for the edge interdiction clique problem. Eur. J. Oper. Res. 294(1), 54\u201369 (2021). https:\/\/doi.org\/10.1016\/j.ejor.2021.01.030","journal-title":"Eur. J. Oper. Res."},{"key":"244_CR42","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/BF00934810","volume":"10","author":"AM Geoffrion","year":"1972","unstructured":"Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237\u2013260 (1972). https:\/\/doi.org\/10.1007\/BF00934810","journal-title":"J. Optim. Theory Appl."},{"issue":"4\u2013part\u20131","key":"244_CR43","doi-asserted-by":"publisher","first-page":"902","DOI":"10.1287\/opre.1090.0795","volume":"58","author":"J Goh","year":"2010","unstructured":"Goh, J., Sim, M.: Distributionally robust optimization and its tractable approximations. Oper. Res. 58(4\u2013part\u20131), 902\u2013917 (2010). https:\/\/doi.org\/10.1287\/opre.1090.0795","journal-title":"Oper. Res."},{"key":"244_CR44","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1002\/nav.3800250412","volume":"4","author":"B Golden","year":"1978","unstructured":"Golden, B.: A problem in network interdiction. Nav. Res. Logist. Q. 4, 711\u20133 (1978)","journal-title":"Nav. Res. Logist. Q."},{"issue":"2","key":"244_CR45","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s00186-018-0647-z","volume":"89","author":"V Grimm","year":"2019","unstructured":"Grimm, V., Schewe, L., Schmidt, M., Z\u00f6ttl, G.: A multilevel model of the European entry-exit gas market. Math. Methods Oper. Res. 89(2), 223\u2013255 (2019). https:\/\/doi.org\/10.1007\/s00186-018-0647-z","journal-title":"Math. Methods Oper. Res."},{"key":"244_CR46","doi-asserted-by":"publisher","first-page":"1070","DOI":"10.1016\/j.ijepes.2014.05.049","volume":"63","author":"H Haghighat","year":"2014","unstructured":"Haghighat, H.: Strategic offering under uncertainty in power markets. Int. J. Electr. Power Energy Syst. 63, 1070\u20131077 (2014). https:\/\/doi.org\/10.1016\/j.ijepes.2014.05.049","journal-title":"Int. J. Electr. Power Energy Syst."},{"issue":"5","key":"244_CR47","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). https:\/\/doi.org\/10.1137\/0913069","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"244_CR48","unstructured":"Israeli, E.: System interdiction and defense. Ph.D. thesis (1999). https:\/\/apps.dtic.mil\/sti\/pdfs\/ADA361997.pdf"},{"issue":"2","key":"244_CR49","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1002\/net.10039","volume":"40","author":"E Israeli","year":"2002","unstructured":"Israeli, E., Wood, R.K.: Shortest-path network interdiction. Networks 40(2), 97\u2013111 (2002). https:\/\/doi.org\/10.1002\/net.10039","journal-title":"Networks"},{"issue":"4","key":"244_CR50","doi-asserted-by":"publisher","first-page":"658","DOI":"10.1134\/S1990478918040063","volume":"12","author":"S Ivanov","year":"2018","unstructured":"Ivanov, S.: A bilevel stochastic programming problem with random parameters in the follower\u2019s objective function. J. Appl. Ind. Math. 12(4), 658\u2013667 (2018). https:\/\/doi.org\/10.1134\/S1990478918040063","journal-title":"J. Appl. Ind. Math."},{"key":"244_CR51","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejco.2021.100007","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. (2021). https:\/\/doi.org\/10.1016\/j.ejco.2021.100007","journal-title":"EURO J. Comput. Optim."},{"issue":"12","key":"244_CR52","volume":"44","author":"M Labb\u00e9","year":"1998","unstructured":"Labb\u00e9, M., Marcotte, P., Savard, G.: A bilevel model of taxation and its application to optimal highway pricing. Manag. Sci. 44(12), 160822 (1998)","journal-title":"Manag. Sci."},{"key":"244_CR53","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s10288-014-0270-7","volume":"12","author":"T Lee","year":"2014","unstructured":"Lee, T., Kwon, C.: A short note on the robust combinatorial optimization problems with cardinality constrained uncertainty. 4OR 12, 373\u2013378 (2014). https:\/\/doi.org\/10.1007\/s10288-014-0270-7","journal-title":"4OR"},{"issue":"3","key":"244_CR54","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1287\/mnsc.45.3.414","volume":"45","author":"S Martello","year":"1999","unstructured":"Martello, S., Pisinger, D., Toth, P.: Dynamic programming and strong bounds for the 0\u20131 knapsack problem. Manag. Sci. 45(3), 414\u201324 (1999). https:\/\/doi.org\/10.1287\/mnsc.45.3.414","journal-title":"Manag. Sci."},{"key":"244_CR55","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/BF01099649","volume":"7","author":"A Migdalas","year":"1995","unstructured":"Migdalas, A.: Bilevel programming in traffic planning: models, methods and challenge. J. Global Optim. 7, 381\u2013405 (1995). https:\/\/doi.org\/10.1007\/BF01099649","journal-title":"J. Global Optim."},{"key":"244_CR56","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.trb.2015.06.001","volume":"79","author":"E \u00c1lvarez Miranda","year":"2015","unstructured":"\u00c1lvarez Miranda, E., Fern\u00e1ndez, E., Ljubi\u0107, I.: The recoverable robust facility location problem. Transp. Res. Part B Methodol. 79, 93\u2013120 (2015). https:\/\/doi.org\/10.1016\/j.trb.2015.06.001","journal-title":"Transp. Res. Part B Methodol."},{"key":"244_CR57","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s10288-013-0231-6","volume":"11","author":"E \u00c1lvarez Miranda","year":"2013","unstructured":"\u00c1lvarez Miranda, E., Ljubi\u0107, I., Toth, P.: A note on the bertsimas & sim algorithm for robust combinatorial optimization problems. 4OR Q. J. Oper. Res. 11, 349\u2013360 (2013). https:\/\/doi.org\/10.1007\/s10288-013-0231-6","journal-title":"4OR Q. J. Oper. Res."},{"issue":"5","key":"244_CR58","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1287\/opre.38.5.911","volume":"38","author":"JT Moore","year":"1990","unstructured":"Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Oper. Res. 38(5), 911\u2013921 (1990). https:\/\/doi.org\/10.1287\/opre.38.5.911","journal-title":"Oper. Res."},{"key":"244_CR59","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s10479-019-03315-x","volume":"294","author":"FM Pajouh","year":"2020","unstructured":"Pajouh, F.M.: Minimum cost edge blocker clique problem. Ann. Oper. Res. 294, 345\u2013376 (2020). https:\/\/doi.org\/10.1007\/s10479-019-03315-x","journal-title":"Ann. Oper. Res."},{"issue":"1","key":"244_CR60","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1002\/net.21556","volume":"64","author":"FM Pajouh","year":"2014","unstructured":"Pajouh, F.M., Boginski, V., Pasiliao, E.L.: Minimum vertex blocker clique problem. Networks 64(1), 48\u201364 (2014). https:\/\/doi.org\/10.1002\/net.21556","journal-title":"Networks"},{"issue":"1","key":"244_CR61","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.ejor.2015.05.037","volume":"247","author":"FM Pajouh","year":"2015","unstructured":"Pajouh, F.M., Walteros, J.L., Boginski, V., Pasiliao, E.L.: Minimum edge blocker dominating set problem. Eur. J. Oper. Res. 247(1), 16\u201326 (2015). https:\/\/doi.org\/10.1016\/j.ejor.2015.05.037","journal-title":"Eur. J. Oper. Res."},{"issue":"1","key":"244_CR62","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s12532-022-00227-z","volume":"15","author":"X Shi","year":"2020","unstructured":"Shi, X., Prokopyev, O., Ralphs, T.K.: Mixed integer bilevel optimization with k-optimal follower: a hierarchy of bounds. Math. Program. Comput. 15(1), 1\u201351 (2020)","journal-title":"Math. Program. Comput."},{"key":"244_CR63","unstructured":"Sim, M.: Robust optimization. Ph.D. thesis (2004). https:\/\/dspace.mit.edu\/handle\/1721.1\/17725"},{"issue":"3","key":"244_CR64","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1016\/j.ejor.2019.06.024","volume":"283","author":"JC Smith","year":"2020","unstructured":"Smith, J.C., Song, Y.: A survey of network interdiction models and algorithms. Eur. J. Oper. Res. 283(3), 797\u2013811 (2020). https:\/\/doi.org\/10.1016\/j.ejor.2019.06.024","journal-title":"Eur. J. Oper. Res."},{"issue":"5","key":"244_CR65","doi-asserted-by":"publisher","first-page":"1154","DOI":"10.1287\/opre.21.5.1154","volume":"21","author":"AL Soyster","year":"1973","unstructured":"Soyster, A.L.: Technical note-convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21(5), 1154\u20131157 (1973). https:\/\/doi.org\/10.1287\/opre.21.5.1154","journal-title":"Oper. Res."},{"key":"244_CR66","unstructured":"Tahernejad, S., Ralphs, T.K.: Valid inequalities for mixed integer bilevel linear optimization problems (2020). https:\/\/engineering.lehigh.edu\/sites\/engineering.lehigh.edu\/files\/_DEPARTMENTS\/ise\/pdf\/tech-papers\/20\/20T_013.pdf"},{"key":"244_CR67","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1007\/s12532-020-00183-6","volume":"12","author":"S Tahernejad","year":"2020","unstructured":"Tahernejad, S., Ralphs, T.K., DeNegre, S.T.: A branch-and-cut algorithm for mixed integer bilevel linear optimization problems and its implementation. Math. Program. Comput. 12, 529\u2013568 (2020). https:\/\/doi.org\/10.1007\/s12532-020-00183-6","journal-title":"Math. Program. Comput."},{"key":"244_CR68","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s10898-015-0274-7","volume":"66","author":"Y Tang","year":"2016","unstructured":"Tang, Y., Richard, J.P., Smith, J.C.: A class of algorithms for mixed-integer bilevel min-max optimization. J. Global Optim. 66, 225\u2013262 (2016). https:\/\/doi.org\/10.1007\/s10898-015-0274-7","journal-title":"J. Global Optim."},{"key":"244_CR69","doi-asserted-by":"publisher","DOI":"10.1002\/9780470400531.eorms0932","author":"RK Wood","year":"2011","unstructured":"Wood, R.K.: Bilevel network interdiction models: formulations and solutions. Wiley Encycl. Oper. Res. Manag. Sci. (2011). https:\/\/doi.org\/10.1002\/9780470400531.eorms0932","journal-title":"Wiley Encycl. Oper. Res. Manag. Sci."},{"key":"244_CR70","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/j.cor.2013.07.016","volume":"41","author":"P Xu","year":"2014","unstructured":"Xu, P., Wang, L.: An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions. Comput. Oper. Res. 41, 309\u2013318 (2014). https:\/\/doi.org\/10.1016\/j.cor.2013.07.016","journal-title":"Comput. Oper. Res."},{"issue":"1","key":"244_CR71","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1137\/16M1098486","volume":"28","author":"I Yanikoglu","year":"2018","unstructured":"Yanikoglu, I., Kuhn, D.: Decision rule bounds for two-stage stochastic bilevel programs. SIAM J. Optim. 28(1), 198\u2013222 (2018). https:\/\/doi.org\/10.1137\/16M1098486","journal-title":"SIAM J. Optim."},{"issue":"1","key":"244_CR72","doi-asserted-by":"publisher","first-page":"7495","DOI":"10.1287\/deca.2019.0392","volume":"17","author":"MH Zare","year":"2020","unstructured":"Zare, M.H., Prokopyev, O.A., Saur\u00e9, D.: On bilevel optimization with inexact follower. Decis. Anal. 17(1), 7495 (2020). https:\/\/doi.org\/10.1287\/deca.2019.0392","journal-title":"Decis. Anal."},{"issue":"5","key":"244_CR73","doi-asserted-by":"publisher","first-page":"5836","DOI":"10.1109\/TIA.2020.2984741","volume":"56","author":"B Zeng","year":"2020","unstructured":"Zeng, B., Dong, H., Sioshansi, R., Xu, F., Zeng, M.: Bilevel robust optimization of electric vehicle charging stations with distributed energy resources. IEEE Trans. Ind. Appl. 56(5), 5836\u20135847 (2020). https:\/\/doi.org\/10.1109\/TIA.2020.2984741","journal-title":"IEEE Trans. Ind. Appl."},{"issue":"13","key":"244_CR74","doi-asserted-by":"publisher","first-page":"4306","DOI":"10.1016\/j.disc.2009.01.006","volume":"309","author":"R Zenklusen","year":"2009","unstructured":"Zenklusen, R., Ries, B., Picouleau, C., de Werra, D., Costa, M.C., Bentz, C.: Blockers and transversals. Discrete Math. 309(13), 4306\u20134314 (2009). https:\/\/doi.org\/10.1016\/j.disc.2009.01.006","journal-title":"Discrete Math."}],"container-title":["Mathematical Programming Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12532-023-00244-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12532-023-00244-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12532-023-00244-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,14]],"date-time":"2023-10-14T14:11:14Z","timestamp":1697292674000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12532-023-00244-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,10]]},"references-count":74,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["244"],"URL":"https:\/\/doi.org\/10.1007\/s12532-023-00244-6","relation":{},"ISSN":["1867-2949","1867-2957"],"issn-type":[{"type":"print","value":"1867-2949"},{"type":"electronic","value":"1867-2957"}],"subject":[],"published":{"date-parts":[[2023,7,10]]},"assertion":[{"value":"1 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 July 2023","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 declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}