{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T15:20:55Z","timestamp":1759332055267},"reference-count":41,"publisher":"EDP Sciences","issue":"2","license":[{"start":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T00:00:00Z","timestamp":1679616000000},"content-version":"vor","delay-in-days":23,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2023,2,22]]},"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:p>Robust bi-level programming problems are a newborn branch of optimization theory. In this study, we have considered a bi-level model with constraint-wise uncertainty at the upper-level, and the lower-level problem is fully convex. We use the optimal value reformulation to transform the given bi-level problem into a single-level mathematical problem and the concept of robust counterpart optimization to deal with uncertainty in the upper-level problem. Necessary optimality conditions are beneficial because any local minimum must satisfy these conditions. As a result, one can only look for local (or global) minima among points that hold the necessary optimality conditions. Here we have introduced an extended non-smooth robust constraint qualification (RCQ) and developed the KKT type necessary optimality conditions in terms of convexifactors and subdifferentials for the considered uncertain two-level problem. Further, we establish as an application the robust bi-level Mond-Weir dual (MWD) for the considered problem and produce the duality results. Moreover, an example is proposed to show the applicability of necessary optimality conditions.<\/jats:p>","DOI":"10.1051\/ro\/2023026","type":"journal-article","created":{"date-parts":[[2023,2,28]],"date-time":"2023-02-28T20:05:26Z","timestamp":1677614726000},"page":"525-539","source":"Crossref","is-referenced-by-count":1,"title":["Optimality conditions and duality results for a robust bi-level programming problem"],"prefix":"10.1051","volume":"57","author":[{"given":"Shivani","family":"Saini","sequence":"first","affiliation":[]},{"given":"Navdeep","family":"Kailey","sequence":"additional","affiliation":[]},{"given":"Izhar","family":"Ahmad","sequence":"additional","affiliation":[]}],"member":"250","published-online":{"date-parts":[[2023,3,24]]},"reference":[{"key":"R1","unstructured":"Abadie J.M., On the Kuhn-Tucker theorem. Operations Research Center University of Calif Berkeley (1967)."},{"key":"R2","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1007\/s10957-011-9831-5","volume":"149","author":"Aboussoror","year":"2011","journal-title":"J. Optim. Theory Appl."},{"key":"R3","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1016\/j.orl.2021.07.010","volume":"49","author":"Beck","year":"2021","journal-title":"Oper. Res. Lett."},{"key":"R4","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/PL00011380","volume":"88","author":"Ben-Tal","year":"2000","journal-title":"Math. Program."},{"key":"R5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1287\/opre.21.1.37","volume":"21","author":"Bracken","year":"1973","journal-title":"Oper. Res."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"Buchheim C., Henke D. and Hommelsheim F., On the complexity of robust bilevel optimization with uncertain follower\u2019s objective. Preprint arXiv:2105.08378 (2021).","DOI":"10.1016\/j.orl.2021.07.009"},{"key":"R7","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1080\/02331939508844048","volume":"32","author":"Chen","year":"1995","journal-title":"Optimization"},{"key":"R8","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s10957-018-1437-8","volume":"181","author":"Chen","year":"2019","journal-title":"J. Optim. Theory Appl."},{"key":"R9","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1007\/s10957-020-01753-3","volume":"187","author":"Chuong","year":"2020","journal-title":"J. Optim. Theory Appl."},{"key":"R10","unstructured":"Clarke F.H., Optimization and Nonsmooth Analysis. Wiley, New York (1983)."},{"key":"R11","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1016\/j.na.2011.05.097","volume":"75","author":"Dempe","year":"2012","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"R12","doi-asserted-by":"crossref","first-page":"1639","DOI":"10.1137\/130917715","volume":"24","author":"Dempe","year":"2014","journal-title":"SIAM J. Optim."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Dempe S. and Zemkoho A.B., Bilevel Optimization. Springer Cham (2020).","DOI":"10.1007\/978-3-030-52119-6"},{"key":"R14","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1080\/02331930600816189","volume":"55","author":"Dempe","year":"2006","journal-title":"Optimization"},{"key":"R15","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1080\/02331930701617551","volume":"56","author":"Dempe","year":"2007","journal-title":"Optimization"},{"key":"R16","doi-asserted-by":"crossref","first-page":"1811","DOI":"10.1080\/02331934.2020.1750610","volume":"69","author":"Dempe","year":"2020","journal-title":"Optimization"},{"key":"R17","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1023\/A:1014853129484","volume":"113","author":"Dutta","year":"2002","journal-title":"J. Optim. Theory Appl."},{"key":"R18","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1080\/02331934.2019.1625901","volume":"69","author":"Gadhi","year":"2020","journal-title":"Optimization"},{"key":"R19","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1007\/s10957-021-01864-5","volume":"189","author":"Gadhi","year":"2021","journal-title":"J. Optim. Theory Appl."},{"key":"R20","unstructured":"Goerigk M., Adam K. and Pawel Z., Robust two-stage combinatorial optimization problems under convex uncertainty. Preprint arXiv:1905.02469 (2019)."},{"key":"R21","unstructured":"Hiriart-Urruty J.B. and Lemar\u00e9chal C., Fundamentals of Convex Analysis, Springer Science & Business Media (2004)."},{"key":"R22","doi-asserted-by":"crossref","first-page":"3041","DOI":"10.1051\/ro\/2021139","volume":"55","author":"Jennane","year":"2021","journal-title":"RAIRO: OR"},{"key":"R23","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1023\/A:1021790120780","volume":"101","author":"Jeyakumar","year":"1999","journal-title":"J. Optim. Theory Appl."},{"key":"R24","first-page":"413","volume":"94","author":"Kenneth","year":"2018","journal-title":"Math. Methods Oper. Res."},{"key":"R25","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s00186-022-00787-9","volume":"96","author":"Kesarwani","year":"2022","journal-title":"Math. Methods Oper. Res."},{"key":"R26","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1007\/s10957-011-9941-0","volume":"152","author":"Kohli","year":"2012","journal-title":"J. Optim. Theory Appl."},{"key":"R27","doi-asserted-by":"crossref","first-page":"2179","DOI":"10.1080\/02331934.2018.1523402","volume":"67","author":"Lafhim","year":"2018","journal-title":"Optimization"},{"key":"R28","doi-asserted-by":"crossref","first-page":"287","DOI":"10.4134\/BKMS.2014.51.1.287","volume":"51","author":"Lee","year":"2014","journal-title":"Bull. Korean Math. Soc."},{"key":"R29","doi-asserted-by":"crossref","first-page":"7521","DOI":"10.1002\/er.6332","volume":"45","author":"Lotfi","year":"2021","journal-title":"Int. J. Energy Res."},{"key":"R30","unstructured":"Mordukhovich B.S., Variational Analysis and Generalized Differentiation I: Basic Theory, Springer Science & Business Media (2006)."},{"key":"R31","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s10107-007-0120-x","volume":"116","author":"Mordukhovich","year":"2009","journal-title":"Math. Program."},{"key":"R32","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.apm.2017.12.003","volume":"56","author":"Safaei","year":"2018","journal-title":"Appl. Math. Model"},{"key":"R33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-011-9819-1","volume":"150","author":"Suneja","year":"2011","journal-title":"J. Optim. Theory Appl."},{"key":"R34","doi-asserted-by":"crossref","first-page":"2941","DOI":"10.1051\/ro\/2021129","volume":"55","author":"Swain","year":"2021","journal-title":"RAIRO:OR"},{"key":"R35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-021-01431-8","volume":"40","author":"Van Su","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"R36","unstructured":"Von Stackelberg H., Marktform und Gleichgewicht. Springer (1934)."},{"key":"R37","unstructured":"Von Stackelberg H., Marktform and Gleichgewicht, Springer-verlag berlin (1934) [Engl. Transl.: The Theory of the Market Economy (1954)]."},{"key":"R38","doi-asserted-by":"crossref","first-page":"2962","DOI":"10.1111\/poms.13413","volume":"30","author":"Xiong","year":"2021","journal-title":"Prod. Oper. Manag."},{"key":"R39","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF02192648","volume":"89","author":"Yezza","year":"1996","journal-title":"J. Optim. Theory Appl."},{"key":"R40","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1007\/s10589-020-00250-7","volume":"78","author":"Zemkoho","year":"2021","journal-title":"Comput. Optim. Appl."},{"key":"R41","doi-asserted-by":"crossref","first-page":"5836","DOI":"10.1109\/TIA.2020.2984741","volume":"56","author":"Zeng","year":"2020","journal-title":"IEEE Trans. Ind. Appl."}],"container-title":["RAIRO - Operations Research"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2023026\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T09:17:36Z","timestamp":1679649456000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ro.org\/10.1051\/ro\/2023026"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3]]},"references-count":41,"journal-issue":{"issue":"2"},"alternative-id":["ro220496"],"URL":"https:\/\/doi.org\/10.1051\/ro\/2023026","relation":{},"ISSN":["0399-0559","2804-7303"],"issn-type":[{"type":"print","value":"0399-0559"},{"type":"electronic","value":"2804-7303"}],"subject":[],"published":{"date-parts":[[2023,3]]}}}