Optimality and duality in set-valued optimization using higher-order radial derivatives
AbstractThis paper is devoted to the study of optimality conditions and duality theory for a set-valued optimization problem. by using the higher-order radial derivative of a set-valued map, we establish Fritz John and Kuhn-Tucker types necessary and sufficient optimality conditions for a weak minimizer of a set-valued optimization problem under the assumption that set-valued maps in the formulation of objective and constraint maps are near cone-subconvexlike. As an application of the optimality conditions, we prove weak, strong and converse duality theorems for Mond-Weir and Wolfe types dual problems.
S. J. Li, K. L. Teo, X. Q. Yang, Higher-order optimality conditions for set-valued optimization, Journal of Optimization Theory and Applications, 137: 533-553, 2008.
Q. L. Wang, S. J. Li, K. L. Teo, Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization, Optimization Letters, 4: 425-437, 2010.
S. J. Li, K. L. Teo, X. Q. Yang, Higher-order Mond-Weir duality for set-valued optimization, Journal of Computational and Applied Mathematics, 217: 339-349, 2008.
S. J. Li, C. R. Chen, Higher order optimality conditions for Henig efficient solutions in set-valued optimization, Mathematical Analysis and Applications, 323: 1184-1200, 2006.
C. R. Chen, S. J. Li, K. L. Teo, Higher order weak epiderivatives and applications to duality and optimality conditions, Computers and Mathematics with Applications, 57: 1389-1399, 2009.
P. Q. Khanh, N. D. Tuan, Variational sets of multivalued mappings and a unified study of optimality conditions, Journal of Optimization Theory and Applications, 139: 45-67, 2008.
P. Q. Khanh, N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization, Journal of Optimization Theory and Applications, 139: 243-261, 2008.
Nguyen Le Hoang Anh, Phan Quoc Khanh, Le Thanh Tung, Variational sets: Calculus and applications to nonsmooth vector optimization, Nonlinear Analysis, 74: 2358-279, 2011.
D. V. Luu, Higher-order necessary and sufficient conditions for strict local Pareto minima in terms of Studniarskis derivatives, Optimization, 57: 593-605, 2008.
X. K. Sun, S. J. Li, Lower Studniarski derivative of the perturbation map in parametrized vector optimization, Optimization Letters, 5: 601-614, 2011.
Nguyen Le Hoang Anh, Higher-orderoptimality conditions in set-valued optimization using Studniarskiderivatives and applications to duality, Positivity, 18: 449-473, 2014.
M. Studniarski, Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim., 24:1044-1049, 1986.
Guolin Yu, Higher-order optimality conditions and duality for approximate solutions in non-convex set-valued optimization, Acta Mathematicae Applicatae Sinica, to appear.
A. Taa, Set-valued derivatives of multifunctions and optimality conditions, Numerical Functional Analysis and Optimization, 19:121-140, 1998.
F. Flores-Bazan, Radial epiderivatives and asymptotic function in nonconvex vector optimization, SIAM J. Optim. 14: 284-305, 2003.
R. Kasimbeyli, Radial epiderivatives and set-valued optimization, Optimization, 58: 521-534, 2009.
F. Flores-Bazan, B. Jimenez, Strict efficiency in set-valued optimization, SIAM J. Control Optim. 48: 881-908, 2009
Nguyen Le Hoang Anh, Phan Quoc Khanh, Le Thanh Tung, Higher-order radial derivatives and optimality conditions in nonsmooth vector optimization, Nonlinear Analysis, 74: 7365-7379, 2011.
Nguyen Le Hoang Anh, Phan Quoc Khanh, Higher-order optimality conditions in set-valued optimization using radial sets and radial derivatives, Journal of Global Optimization, 56: 519-536, 2013.
Nguyen Le Hoang Anh, Phan Quoc Khanh, Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives, Journal of Global Optimization, 58: 693-709, 2014.
H. W. Corley, Existence and Lagrangian duality for maximization of set- valued functions, Journal of Optimization Theory and Applications, 54: 489-501, 1987.
D. Bhatia, A. Mehra, Lagrangian duality for preinvex set-valued functions, Journal of Mathematical Analysis and Applications, 214: 599-612, 1997.
Z. F. Li, G. Y. Chen, Lagrangian multipliers, saddle points, and duality in vector optimization of set-valued maps, Journal of Mathematical Analysis and Applications, 215: 297-316, 1997.
Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps, Journal of Optimization Theory and Applications, 98: 623-649, 1998.
X. M. Yang, D. Li, S. Y. Wang, Near-subconvexlikeness in vector optimization with set-Valued functions, Journal of Optimization Theory and Applications, 110: 413-427, 2001.
Maria Alonso-Durán, Luis Rodriguez-Marin,
On approximate solutions in set-valued optimization problems, Journal of Computational and Applied Mathematics, 2012, 236: 4421-4427.
P. H. Sach, New generalized convexity notion for set-valued maps and application to vector optimization, Journal of Optimization Theory and Applications, 125: 157-179, 2005.
Guolin Yu, Sanyang Liu, Globally proper saddle point in ic-cone-convex like set-valued optimization problems, Acta Mathematica Sinica, English Series, 25: 1921-1928, 2009.
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