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Optimality and duality results for bilevel programming problem using convexifactors. (English) Zbl 1229.90207
The bilevel programming problem is reformulated as a single mathematical programming problem by using the nonconvex value function of the lower level problem. Sufficient optimality conditions for the bilevel programming problem are obtained, as well as duality results corresponding to Wolfe and Mond-Weier type duals, respectively. To this end the authors consider convexificators and define $$\partial^*$$-convex, $$\partial^*$$-pseudoconvex and $$\partial^*$$-quasiconvex bifunctions based on the work of J. Dutta and S. Chandra [Optimization 53, No. 1, 77–94 (2004; Zbl 1079.90104)] and X. F. Li and J. Z. Zhang [J. Optim. Theory Appl. 131, No. 3, 429–452 (2006; Zbl 1143.90035)].

##### MSC:
 90C30 Nonlinear programming 90C46 Optimality conditions and duality in mathematical programming
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##### References:
 [1] Dutta, J., Chandra, S.: Convexifactor, generalized convexity and vector optimization. Optimization 53, 77–94 (2004) · Zbl 1079.90104 · doi:10.1080/02331930410001661505 [2] Li, X.F., Zhang, J.Z.: Necessary optimality conditions in terms of convexificators in Lipschitz optimization. J. Optim. Theory Appl. 131, 429–452 (2006) · Zbl 1143.90035 · doi:10.1007/s10957-006-9155-z [3] Bard, J.F.: Optimality conditions for the bilevel programming problem. Nav. Res. Logist. Q. 31, 13–26 (1984) · Zbl 0537.90087 · doi:10.1002/nav.3800310104 [4] Bard, J.F.: Some properties of the bilevel programming problem. J. Optim. Theory Appl. 68, 371–378 (1991) · Zbl 0696.90086 · doi:10.1007/BF00941574 [5] Dempe, S.: A necessary and sufficient optimality condition for bilevel programming problem. Optimization 25, 341–354 (1992) · Zbl 0817.90104 · doi:10.1080/02331939208843831 [6] Dempe, S.: First order necessary optimality conditions for general bilevel programming problems. J. Optim. Theory Appl. 95, 735–739 (1997) · Zbl 0903.90148 · doi:10.1023/A:1022646611097 [7] Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic, Dordrecht (2002) · Zbl 1038.90097 [8] Outrata, J.V.: On necessary optimality conditions for Stackelberg problems. J. Optim. Theory Appl. 76, 306–320 (1993) · Zbl 0802.49007 · doi:10.1007/BF00939610 [9] Wang, S., Wang, Q., Romano-Rodriguez, S.: Optimality conditions and an algorithm for linear-quadratic bilevel programs. Optimization 4, 521–536 (1993) [10] Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997) · Zbl 1088.90042 · doi:10.1287/moor.22.4.977 [11] Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33, 9–27 (1995) · Zbl 0820.65032 · doi:10.1080/02331939508844060 [12] Zhang, R.: Problems of hierarchical optimization in finite dimensions. SIAM J. Optim. 4, 521–536 (1995) · Zbl 0819.90107 · doi:10.1137/0804029 [13] Demyanov, V.F.: Convexification and concavification of a positively homogeneous function by the same family of linear functions. Report 3, 208, 802, Universita di Pisa (1994) [14] Demyanov, V.F., Jeyakumar, V.: Hunting for a smaller convex subdifferential. J. Glob. Optim. 10, 305–326 (1997) · Zbl 0872.90083 · doi:10.1023/A:1008246130864 [15] Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, maximality and monotonicity of convexificators. J. Optim. Theory Appl. 101, 599–621 (1999) · Zbl 0956.90033 · doi:10.1023/A:1021790120780 [16] Dutta, J., Chandra, S.: Convexifactors, generalized convexity and optimality conditions. J. Optim. Theory Appl. 113, 41–65 (2002) · Zbl 1172.90500 · doi:10.1023/A:1014853129484 [17] Babahadda, H., Gadhi, N.: Necessary optimality conditions for bilevel optimization problems using convexificators. J. Glob. Optim. 34, 535–549 (2006) · Zbl 1090.49021 · doi:10.1007/s10898-005-1650-5 [18] Amahroq, T., Gadhi, N.: On the regularity condition for vector programming problems. J. Glob. Optim. 21, 435–443 (2001) · Zbl 1175.90409 · doi:10.1023/A:1012748412618 [19] Yezza, A.: First order necessary optimality conditions for general bilevel programming problems. J. Optim. Theory Appl. 89, 189–219 (1996) · Zbl 0866.90119 · doi:10.1007/BF02192648
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