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New optimality conditions and a scalarization approach for a nonconvex semi-vectorial bilevel optimization problem. (English) Zbl 07193703
Summary: In this paper, we are concerned with the optimistic formulation of a semivectorial bilevel optimization problem. Introducing a new scalarization technique for multiobjective programs, we transform our problem into a scalar-objective optimization problem by means of the optimal value reformulation and establish its theoretical properties. Detailed necessary conditions, to characterize local optimal solutions of the problem, were then provided, while using the weak basic CQ together with the generalized differentiation calculus of Mordukhovich. Our approach is applicable to nonconvex problems and is different from the classical scalarization techniques previously used in the literature and the conditions obtained are new.

90C29 Multi-objective and goal programming
90C26 Nonconvex programming, global optimization
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
49K99 Optimality conditions
Full Text: DOI
[1] Bard, JF, Optimality conditions for the bilevel programming problem, Nav. Res. Logist. Q., 31, 13-26 (1984) · Zbl 0537.90087
[2] Bard, JF, Practical Bilevel Optimization: Algorithms and Applications (1998), Dordrecht: Kluwer Academic Publishers, Dordrecht
[3] Bonnel, H.; Morgan, J., Semivectorial bilevel optimization problem: penalty approach, J. Optim. Theory Appl., 131, 3, 365-382 (2006) · Zbl 1205.90258
[4] Bonnel, H., Optimality conditions for the semivectorial bilevel optimization problem, Pac. J. Optim., 2, 3, 447-467 (2006) · Zbl 1124.90028
[5] Clarke, F.H.: Optimization and nonsmooth analysis. In: Classics in Applied Mathematics, vol. 5, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1990) · Zbl 0696.49002
[6] Dempe, S., A necessary and sufficient optimality condition for bilevel programming problem, Optimization, 25, 4, 341-354 (1992) · Zbl 0817.90104
[7] Dempe, S., Foundations of Bilevel Programming (2002), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 1038.90097
[8] Dempe, S.; Dutta, J.; Mordukhovich, BS, New necessary optimality conditions in optimistic bilevel programming, Optimization, 56, 5-6, 577-604 (2007) · Zbl 1172.90481
[9] Dempe, S.; Gadhi, N.; Lafhim, L., Fuzzy and exact optimality conditions for a bilevel set-valued problem via extremal principles, Numer. Func. Anal. Opt., 31, 8, 907-920 (2010) · Zbl 1251.90385
[10] Dempe, S.; Gadhi, N.; Zemkoho, A., New optimality conditions for the semivectoriel bilevel optimization problem, J. Optim. Theory Appl., 157, 54-74 (2013) · Zbl 1266.90160
[11] Dempe, S., Mehlitz, P.: Semivectorial bilevel programming versus scalar bilevel programming. Preprint SPP 1962, 2018. submitted to Optimization. https://spp1962.wias-berlin.de/preprints/082.pdf
[12] Dempe, S.; Pilecka, M., Optimality conditions for set-valued optimisation problems using a modified Demyanov difference, J. Optim. Theory Appl., 171, 2, 402-421 (2016) · Zbl 1353.49028
[13] Dempe, S.; Zemkoho, A., The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs, J. Optim. Theory Appl., 148, 1, 46-68 (2011) · Zbl 1223.90061
[14] Eichfelder, G., Multiobjective bilevel optimization, Math. Program., 123, 2, 419-449 (2010) · Zbl 1198.90347
[15] Gadhi, N.; El idrissi, M., An equivalent one level optimization problem to a semivectorial bilevel problem, Positivity, 22, 1, 261-274 (2018) · Zbl 06861663
[16] Huy, NQ; Mordukhovich, BS; Yao, JC, Coderivatives of frontier and solution maps in parametric multiobjective optimization, Taiwan. J. Math., 12, 2083-2111 (2008) · Zbl 1194.90082
[17] Mordukhovich, BS, Variational Analysis and Generalized Differentiation. I: Basic Theory (2006), Berlin: Springer, Berlin
[18] Mordukhovich, BS, Variational Analysis and Generalized Differentiation. II. Applications (2006), Berlin: Springer, Berlin
[19] Mordukhovich, BS, Subgradient of marginal functions in parametric mathematical programming, Math. Program., 116, 1-2, 369-396 (2009) · Zbl 1177.90377
[20] Mordukhovich, BS; Nam, NM, Variational stability and marginal functions via generalized differentiation, Math. Oper. Res., 30, 4, 800-816 (2005) · Zbl 1284.90083
[21] Outrata, JV, A note on the usage of nondifferentiable exact penalties in some special optimization problems, Kybernetika, 24, 4, 251-258 (1988) · Zbl 0649.90092
[22] Outrata, JV, On necessary optimality conditions for Stackelberg problems, J. Optim. Theory Appl., 76, 2, 306-320 (1993)
[23] Rochafellar, RT; Wets, RJ-B, Variational Analysis (1998), Berlin: Springer, Berlin
[24] Stackelberg, HV, Marktform und Gleichgewicht (1934), Berlin: Springer, Berlin
[25] Ye, JJ; Zhu, DL, Optimality conditions for bilevel programming problems, Optimization, 33, 1, 9-27 (1995) · Zbl 0820.65032
[26] Zemkoho, AB, Solving illposed bilevel programs, Set Valued Var. Anal., 24, 423-448 (2016) · Zbl 1362.90346
[27] Zhang, R.; Truong, B.; Zhang, Q., Multistage hierarchical optimization problems with multi-criterion objectives, J. Ind. Manag. Optim., 7, 1, 103-115 (2011) · Zbl 1232.49026
[28] Zheng, Y.; Wan, Z., A solution method for semivectorial bilevel programming problem via penalty method, J. Appl. Math. Comput., 37, 1-2, 207-219 (2011) · Zbl 1297.90131
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