Higher-order weakly generalized epiderivatives and applications to optimality conditions. (English) Zbl 1245.49030

Wang, Qilin; Yu, Guolin

J. Appl. Math. 2012, Article ID 691018, 19 p. (2012).

Summary: The notions of higher-order weakly generalized contingent epiderivative and higher-order weakly generalized adjacent epiderivative for set-valued maps are proposed. By virtue of the higher-order weakly generalized contingent (adjacent) epiderivatives, both necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. The imposed assumptions are relaxed in comparison with those of recent results in the literature. Examples are provided to show some advantages of our notions and results.

MSC:

49J53	Set-valued and variational analysis
49K27	Optimality conditions for problems in abstract spaces

Keywords:

higher-order weakly generalized contingent epiderivative; higher-order weakly generalized adjacent epiderivative; set-valued maps; necessary and sufficient optimality conditions; Henig efficient solution

References:

[1]	J.-P. Aubin and H. Frankowska, Set-Valued Analysis, vol. 2 of Systems & Control: Foundations & Applications, Birkhäuser Boston, Boston, Mass, USA, 1990. · Zbl 0745.17013
[2]	H. W. Corley, “Optimality conditions for maximizations of set-valued functions,” Journal of Optimization Theory and Applications, vol. 58, no. 1, pp. 1-10, 1988. · Zbl 0956.90509
[3]	J. Jahn and R. Rauh, “Contingent epiderivatives and set-valued optimization,” Mathematical Methods of Operations Research, vol. 46, no. 2, pp. 193-211, 1997. · Zbl 0889.90123
[4]	G. Y. Chen and J. Jahn, “Optimality conditions for set-valued optimization problems,” Mathematical Methods of Operations Research, vol. 48, no. 2, pp. 187-200, 1998. · Zbl 0927.90095
[5]	C. S. Lalitha and R. Arora, “Weak Clarke epiderivative in set-valued optimization,” Journal of Mathematical Analysis and Applications, vol. 342, no. 1, pp. 704-714, 2008. · Zbl 1149.49018
[6]	J.-P. Aubin, “Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,” in Mathematical Analysis and Applications, Part A, L. Nachbin, Ed., vol. 7, pp. 160-229, Academic Press, New York, NY, USA, 1981. · Zbl 0484.47034
[7]	J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1984. · Zbl 0641.47066
[8]	J.-P. Penot, “Differentiability of relations and differential stability of perturbed optimization problems,” SIAM Journal on Control and Optimization, vol. 22, no. 4, pp. 529-551, 1984. · Zbl 0552.58006
[9]	A. D. Ioffe, “Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 8, no. 5, pp. 517-539, 1984. · Zbl 0542.46023
[10]	Z.-B. Liu, J. K. Kim, and N.-J. Huang, “Existence of solutions for nonconvex and nonsmooth vector optimization problems,” Journal of Inequalities and Applications, vol. 2008, Article ID 678014, 7 pages, 2008. · Zbl 1357.49041
[11]	D. S. Shi, “Contingent derivative of the perturbation map in multiobjective optimization,” Journal of Optimization Theory and Applications, vol. 70, no. 2, pp. 385-396, 1991. · Zbl 0743.90092
[12]	X. Q. Yang, “Directional derivatives for set-valued mappings and applications,” Mathematical Methods of Operations Research, vol. 48, no. 2, pp. 273-285, 1998. · Zbl 0929.90077
[13]	X.-H. Gong, H.-B. Dong, and S.-Y. Wang, “Optimality conditions for proper efficient solutions of vector set-valued optimization,” Journal of Mathematical Analysis and Applications, vol. 284, no. 1, pp. 332-350, 2003. · Zbl 1160.90649
[14]	L. Rodríguez-Marín and M. Sama, “About contingent epiderivatives,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 745-762, 2007. · Zbl 1122.49012
[15]	J. Jahn and A. A. Khan, “Generalized contingent epiderivatives in set-valued optimization: optimality conditions,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 807-831, 2002. · Zbl 1134.90466
[16]	B. Jiménez and V. Novo, “Second order necessary conditions in set constrained differentiable vector optimization,” Mathematical Methods of Operations Research, vol. 58, no. 2, pp. 299-317, 2003. · Zbl 1049.90083
[17]	J. Jahn, A. A. Khan, and P. Zeilinger, “Second-order optimality conditions in set optimization,” Journal of Optimization Theory and Applications, vol. 125, no. 2, pp. 331-347, 2005. · Zbl 1071.49015
[18]	S. J. Li, K. L. Teo, and X. Q. Yang, “Higher-order optimality conditions for set-valued optimization,” Journal of Optimization Theory and Applications, vol. 137, no. 3, pp. 533-553, 2008. · Zbl 1149.49020
[19]	S. J. Li, K. L. Teo, and X. Q. Yang, “Higher-order Mond-Weir duality for set-valued optimization,” Journal of Computational and Applied Mathematics, vol. 217, no. 2, pp. 339-349, 2008. · Zbl 1171.90011
[20]	S. J. Li and C. R. Chen, “Higher order optimality conditions for Henig efficient solutions in set-valued optimization,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1184-1200, 2006. · Zbl 1144.90481
[21]	Q. L. Wang and S. J. Li, “Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency,” Numerical Functional Analysis and Optimization, vol. 30, no. 7-8, pp. 849-869, 2009. · Zbl 1180.90254
[22]	C. R. Chen, S. J. Li, and K. L. Teo, “Higher order weak epiderivatives and applications to duality and optimality conditions,” Computers & Mathematics with Applications, vol. 57, no. 8, pp. 1389-1399, 2009. · Zbl 1186.90122
[23]	J. M. Borwein and D. Zhuang, “Super efficiency in vector optimization,” Transactions of the American Mathematical Society, vol. 338, no. 1, pp. 105-122, 1993. · Zbl 0796.90045
[24]	D. T. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 1989. · Zbl 0688.90051