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New optimality conditions for bilevel programs by using an exact separation principle. (English) Zbl 07238345
Summary: In this paper, we are concerned with a bilevel optimization problem \((P)\). Reducing it into a one-level nonlinear and nonsmooth program using an exact separation theorem we give necessary optimality conditions in terms of Fréchet subdifferentials and Fréchet normal cones.
MSC:
90C46 Optimality conditions and duality in mathematical programming
90C30 Nonlinear programming
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[1] Babahadda, H.; Gadhi, N., Necessary optimality conditions for bilevel optimization problems using convexificators, J. Glob. Optim., 34, 535-549 (2006) · Zbl 1090.49021
[2] Bard, JF, Some properities of the bilevel programming problem, J. Optim. Theory Appl., 68, 371-378 (1991) · Zbl 0696.90086
[3] Dempe, S., A necessary and a sufficient optimality condition for bilevel programming problem, Optimization, 25, 341-354 (1992) · Zbl 0817.90104
[4] Dempe, S.; Gadhi, N., Necessary optimality conditions for bilevel set optimization problems, J. Glob. Optim., 39, 529-542 (2007) · Zbl 1190.90178
[5] Dempe, S.; Gadhi, N., A new equivalent single-level problem for bilevel problems, Optimization, 63, 5, 789-798 (2014) · Zbl 1291.90216
[6] Kruger, A., Ya, On Fréchet subdifferentials, J. Math. Sci., 116, 3, 3325-3358 (2003) · Zbl 1039.49021
[7] Mangasarian, OL; Fromovitz, S., The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, J. Math. Anal. Appl., 17, 37-47 (1967) · Zbl 0149.16701
[8] Mordukhovich, BS, Variational Analysis and Generalized Differentiation, I: Basic Theory (2006), Berlin: Springer, Berlin
[9] Mordukhovich, BS; Shao, Y., Nonsmooth sequential analysis in Asplund spaces, Trans. Am. Math. Soc., 348, 1235-1280 (1996) · Zbl 0881.49009
[10] Outrata, JV, On necessary optimality conditions for Stackelberg problems, J. Optim. Theory Appl., 76, 306-320 (1993) · Zbl 0802.49007
[11] Ye, JJ; Zhu, DL, Optimality conditions for bilevel programming problems, Optimization, 33, 9-27 (1995) · Zbl 0820.65032
[12] Zhang, R., Problems of hierarchical optimization in finite dimensions, SIAM J. Optim., 4, 521-536 (1993) · Zbl 0819.90107
[13] Zheng, XY; Yang, Z.; Zou, J., Exact separation theorem for closed sets in Asplund spaces, Optimization, 66, 7, 1065-1077 (2017) · Zbl 1407.90344
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