Sufficient Optimality Conditions in Bilevel Programming

Published Online:https://doi.org/10.1287/moor.2021.1122

References

  • [1] Aboussoror A, Adly S (2018) New necessary and sufficient optimality conditions for strong bilevel programming problems. J. Global Optim. 70(2):309–327.Google Scholar
  • [2] Andreani R, Haeser G, Schuverdt ML, Silva PJS (2012) A relaxed constant positive linear dependence constraint qualification and applications. Math. Programming 135(1):255–273.Google Scholar
  • [3] Aubin JP, Frankowska H (2009) Set-Valued Analysis (Modern Birkhäuser Classics, Boston).Google Scholar
  • [4] Bard JF (1998) Practical Bilevel Optimization: Algorithms and Applications (Kluwer Academic Publishers, London).Google Scholar
  • [5] Bazaraa MS, Sherali HD, Shetty CM (1993) Nonlinear Programming: Theory and Algorithms (Wiley & Sons, New York).Google Scholar
  • [6] Ben-Tal A (1980) Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. 31(2):143–165.Google Scholar
  • [7] Ben-Tal A, Zowe J (1982) Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Math. Programming 24(1):70–91.Google Scholar
  • [8] Bonnans JF, Shapiro A (2000) Perturbation Analysis of Optimization Problems (Springer, New York).CrossrefGoogle Scholar
  • [9] Bonnans JF, Cominetti R, Shapiro A (1999) Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9(2):466–492.Google Scholar
  • [10] Cambini A, Martein L, Vlach M (1999) Second order tangent sets and optimality conditions. Math. Japonicae 49(3):451–461.Google Scholar
  • [11] Clarke FH (1983) Optimization and Nonsmooth Analysis (Wiley & Sons, New York).Google Scholar
  • [12] Dempe S (1992) A necessary and a sufficient optimality condition for bilevel programming problems. Optimization 25(4):341–354.Google Scholar
  • [13] Dempe S (2002) Foundations of Bilevel Programming (Kluwer Academic Publishers, London).Google Scholar
  • [14] Dempe S, Dutta J (2012) Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Programming 131(1):37–48.Google Scholar
  • [15] Dempe S, Gadhi N (2010) Second order optimality conditions for bilevel set optimization problems. J. Global Optim. 47(2):233–245.Google Scholar
  • [16] Dempe S, Zemkoho AB (2013) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Programming 138(1):447–473.Google Scholar
  • [17] Dempe S, Kalashnikov VV, Kalashnykova N (2006) Optimality conditions for bilevel programming problems. Dempe S, Kalashnikov V, eds. Optimization with Multivalued Mappings: Theory, Applications, and Algorithms (Springer, Boston), 3–28.CrossrefGoogle Scholar
  • [18] Dempe S, Mordukhovich BS, Zemkoho AB (2012) Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22(4):1309–1343.Google Scholar
  • [19] Dempe S, Kalashnikov V, Pérez-Valdéz G, Kalashnykova N (2015) Bilevel Programming Problems—Theory, Algorithms and Applications to Energy Networks (Springer, Berlin).Google Scholar
  • [20] Fiacco AV (1983) Introduction to Sensitivity and Stability Analysis in Nonlinear Programming (Academic Press Inc., New York).Google Scholar
  • [21] Flegel ML, Kanzow C (2005) Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124(3):595–614.Google Scholar
  • [22] Gadhi N, Hamdaoui K, El Idrissi M (2020) Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a Ψ reformulation. Optimization 69(4):681–702.Google Scholar
  • [23] Geiger C, Kanzow C (2002) Theorie und Numerik restringierter Optimierungsaufgaben (Springer, Berlin).CrossrefGoogle Scholar
  • [24] Gfrerer H, Klatte D (2016) Lipschitz and Hölder stability of optimization problems and generalized equations. Math. Programming 158(1):35–75.Google Scholar
  • [25] Gfrerer H, Mordukhovich BS (2019) Second-order variational analysis of parametric constraint and variational systems. SIAM J. Optim. 29(1):423–453.Google Scholar
  • [26] Gfrerer H, Outrata JV (2016) On computation of generalized derivatives of the normal-cone mapping and their applications. Math. Oper. Res. 41(4):1535–1556.Google Scholar
  • [27] Guo L, Lin GH, Ye JJ, Zhang J (2014) Sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints. SIAM J. Optim. 24(3):1206–1237.Google Scholar
  • [28] Han SP, Mangasarian OL (1979) Exact penalty functions in nonlinear programming. Math. Programming 17(1):251–269.Google Scholar
  • [29] Henrion R, Outrata JV (2005) Calmness of constraint systems with applications. Math. Programming 104(2):437–464.Google Scholar
  • [30] Henrion R, Surowiec T (2011) On calmness conditions in convex bilevel programming. Applicable Anal. 90(6):951–970.Google Scholar
  • [31] Janin R (1984) Directional derivative of the marginal function in nonlinear programming. Fiacco AV, ed. Sensitivity, Stability and Parametric Analysis (Springer, Berlin) , 110–126.CrossrefGoogle Scholar
  • [32] Kyparisis J (1985) On uniqueness of Kuhn–Tucker multipliers in nonlinear programming. Math. Programming 32(2):242–246.Google Scholar
  • [33] McCormick G (1967) Second order conditions for constrained minima. SIAM J. Appl. Math. 15(3):641–652.Google Scholar
  • [34] Mehlitz P (2020) On the linear independence constraint qualification in disjunctive programming. Optimization 69(10):2241–2277.Google Scholar
  • [35] Mehlitz P, Minchenko L (2020) R-regularity of set-valued mappings under the relaxed constant positive linear dependence constraint qualification with applications to parametric and bilevel optimization. Preprint, submitted May 14, 2020, updated December 16, 2020, https://arxiv.org/abs/2005.06768.Google Scholar
  • [36] Mehlitz P, Minchenko L, Zemkoho AB (2020) A note on partial calmness for bilevel optimization problems with linearly structured lower level. Optim. Lett. 1–15, doi:10.1007/s11590-020-01636-6.Google Scholar
  • [37] Mohammadi A, Mordukhovich BS, Sarabi ME (2021) Parabolic regularity in geometric variational analysis. Trans. Amer. Math. Soc. 374:1711-1763.Google Scholar
  • [38] Mordukhovich BS (2018) Variational Analysis and Applications (Springer, Berlin).CrossrefGoogle Scholar
  • [39] Penot J (1998) Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37(1):303–318.Google Scholar
  • [40] Ralph D, Dempe S (1995) Directional derivatives of the solution of a parametric nonlinear program. Math. Programming 70(1):159–172.Google Scholar
  • [41] Robinson SM (1981) Some continuity properties of polyhedral multifunctions. König H, Korte B, Ritter K, eds. Mathematical Programming at Oberwolfach (Springer, Berlin), 206–214.CrossrefGoogle Scholar
  • [42] Robinson SM (1982) Generalized equations and their solutions, part II: Applications to nonlinear programming. Guignard M, ed. Optimality and Stability in Mathematical Programming (Springer, Berlin), 200–221.Google Scholar
  • [43] Rockafellar RT, Wets RJB (1998) Variational Analysis. Grundlehren der mathematischen Wissenschaften, vol. 317 (Springer, Berlin).CrossrefGoogle Scholar
  • [44] Rückmann JJ, Shapiro A (2001) Second-order optimality conditions in generalized semi-infinite programming. Set-Valued Analysis 9(1):169–186.Google Scholar
  • [45] Scheel S, Scholtes S (2000) Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25(1):1–22.Google Scholar
  • [46] Shapiro A (1988) Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton. Appl. Math. Optim 18(1):215–229.Google Scholar
  • [47] Shapiro A (1988) Sensitivity analysis of nonlinear programs and differentiability properties of metric projections. SIAM J. Control Optim. 26(3):628–645.Google Scholar
  • [48] Shapiro A (1990) On concepts of directional differentiability. J. Optim. Theory Appl. 66(3):477–487.Google Scholar
  • [49] Shimizu K, Ishizuka Y, Bard JF (1997) Nondifferentiable and Two-level Mathematical Programming (Kluwer Academic Publishers, Dordrecht, Germany).CrossrefGoogle Scholar
  • [50] Studniarski M (1986) Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24(5):1044–1049.Google Scholar
  • [51] Wachsmuth G (2013) On LICQ and the uniqueness of Lagrange multipliers. Oper. Res. Lett. 41(1):78–80.Google Scholar
  • [52] Ye JJ (2004) Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15(1):252–274.Google Scholar
  • [53] Ye JJ, Wu SY (2008) First order optimality conditions for generalized semi-infinite programming problems. J. Optim. Theory Appl. 137(2):419–434.Google Scholar
  • [54] Ye JJ, Zhu DL (1995) Optimality conditions for bilevel programming problems. Optimization 33(1):9–27.Google Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.