Optimality conditions for \(C^{1,1}\) constrained multiobjective problems. (English) Zbl 1030.90115

Summary: The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.


90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming
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[1] Jahn, J., Mathematical Theory of Vector Optimization in Partially-Ordered Spaces, Peter Lang, Frankfurt, Germany, 1985.
[2] Yu, P.L., Multicriteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, NY, 1985.
[3] Luc, D.T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 319, 1989.
[4] Giannessi, F., Mastroeni, G., and Pellegrini, L., On the Theory of Vector Optimization and Variational Inequalities: Image Space Analysis and Separation, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, London, England, pp. 153–216, 2000. · Zbl 0985.49005
[5] Guerraggio, A., Molho, E., and Zaffaroni, A., On the Notion of Proper Efficiency in Vector Optimization, Journal of Optimization Theory and Appli-cations, Vol. 82, pp. 1–21, 1994. · Zbl 0827.90123
[6] Khanh, P.Q., Proper Solutions of Vector Optimization Problems, Journal of Optimization Theory and Applications, Vol. 74, pp. 105–130, 1992. · Zbl 0795.90057
[7] Luc, D. T., and Schaible, S., On Efficiency and Generalized Concavity, Journal of Optimization Theory and Applications, Vol. 94, pp. 147–153, 1997. · Zbl 0886.90121
[8] Luc, D. T., On the Properly Efficient Points of Nonconvex Sets, European Jour-nal of Operations Research, Vol. 86, pp. 332–336, 1995. · Zbl 0908.90227
[9] Kuhn, H. W., and Tucker, F.H., Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, California, pp. 481–492, 1951.
[10] Luc, D.T., Contingent Derivative of Set-Valued Maps and Applications to Vector Optimization, Mathematical Programming, Vol. 50, pp. 99–111, 1991. · Zbl 0718.90080
[11] Luc, D. T., and Malivert, C., Invex Optimization Problems, Bulletin of the Australian Mathematical Society, Vol. 46, pp. 47–66, 1992. · Zbl 0755.90072
[12] Craven, B.D., Nonsmooth Multiobjective Programming, Numerical Functional Analysis and Optimization, Vol. 10, pp. 49–64, 1989. · Zbl 0645.90076
[13] Luc, D.T., A Multiplier Rule for Multiobjective Programming Problems with Continuous Data, SIAM Journal on Optimization, Vol. 13, pp. 168–178, 2002. · Zbl 1055.90063
[14] Wang, S., Second-Order Necessary and Sufficient Conditions in Multiobjective Programming, Numerical Functional Analysis and Optimization, Vol. 12, pp. 237–252, 1991. · Zbl 0764.90076
[15] Bolintineau, S., and El maghri, M., Second-Order Efficiency Conditions and Sensitivity of Efficient Points, Journal of Optimization Theory and Applications, Vol. 98, pp. 569–592, 1998. · Zbl 0915.90226
[16] Aghezzaf, B., and Hachimi, M., Second-Order Optimality Conditions in Multiobjective Optimization Problems, Journal of Optimization Theory and Applications, Vol. 102, pp. 37–50, 1999. · Zbl 1039.90062
[17] Liu, L.P., The Second-Order Conditions of Nondominated Solutions for C 1,1 Generalized Multiobjective Mathematical Programming, Systems Sciences and Mathematical Sciences, Vol. 4, pp. 128–138, 1991. · Zbl 0734.90078
[18] Guerraggio, A., and Luc, D. T., Optimality Conditions for C 1,1 Vector Optim-ization Problems, Journal of Optimization Theory and Applications, Vol. 109, pp. 615–629, 2001. · Zbl 1038.49027
[19] Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley, New York, NY, 1983. · Zbl 0582.49001
[20] Hiriart-urruty, J. B., Strodiot, J. J., and Nguyen, V.H., Generalized Hessian Matrix and Second-Order Optimality Conditions for Problems with C 1,1 Data, Applied Mathematics and Optimization, Vol. 11, pp. 43–56, 1984. · Zbl 0542.49011
[21] Klatte, D., and Tammer, K., On the Second-Order Sufficient Optimality Con-ditions for C 1,1 Optimization Problems, Optimization, Vol. 19, pp. 169–180, 1988. · Zbl 0647.49014
[22] Luc, D.T., Taylor’s Formula for C k,1 Functions, SIAM Journal on Optimiz-ation, Vol. 5, pp. 659–669, 1995. · Zbl 0852.49012
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