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Second-order conditions for efficiency in nonsmooth multiobjective optimization problems. (English) Zbl 1082.90106
Summary: We are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. We introduce a second-order constraint qualification, which is a generalization of the Abadie constraint qualification and derive second-order Kuhn-Tucker type necessary conditions for efficiency under the constraint qualification. Moreover, we give conditions which ensure that the constraint qualification holds.

MSC:
90C29 Multi-objective and goal programming
49J52 Nonsmooth analysis
90C46 Optimality conditions and duality in mathematical programming
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[1] Benson, H. P., Efficiency and Proper Efficiency in Vector Minimization Problems with Respect to Cones, Journal of Mathematical Analysis and Applications, Vol. 93, pp. 273-289, 1983. · Zbl 0519.90080 · doi:10.1016/0022-247X(83)90230-5
[2] Kuhn, H. W., and Tucker, A. W., Nonlinear Programming, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, California, 1952.
[3] Maeda, T., Multiobjective Decision Making Theory and Economic Analysis, Makino-Syoten, 1996.
[4] Yu, P. L., Multicriteria Decision Making:Concepts, Techniques, and Extensions, Plenum Press, New York, NY, 1985.
[5] Kawasaki, H., Second-Order Necessary Conditions of the Kuhn-Tucker Type under New Constraint Quali cations, Journal of Optimization Theory and Applications, Vol. 57, pp. 253-264, 1988. · Zbl 0621.90074 · doi:10.1007/BF00938539
[6] Cominetti, R., and Correa, R., A Generalized Second-Order Derivative in Nonsmooth Optimization, SIAM Journal on Control and Optimization, Vol. 28, pp. 789-809, 1990. · Zbl 0714.49020 · doi:10.1137/0328045
[7] Hiriart, 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 · doi:10.1007/BF01442169
[8] Castellani, M., A Necessary Second-Order Optimality Condition in Non-smooth Mathematical Programming, Operations Research Letters, Vol. 19, pp. 79-86, 1996. · Zbl 0865.90113 · doi:10.1016/0167-6377(96)00006-5
[9] Bazaraa, B. S., Goode, J. J., and Nashed, M. Z., On the Cones of Tangents with Applications to Mathematical Programming, Journal of Optimization Theory and Applications, Vol. 13. pp. 389-426, 1974. · Zbl 0259.90037 · doi:10.1007/BF00934938
[10] Abadie, M., Generalized Kuhn-Tucker Conditions for Mathematical Programming, SIAM Journal on Control, Vol. 7, pp. 232-241, 1969. · Zbl 0182.53101 · doi:10.1137/0307016
[11] Maeda, T., On Constraint Quali cations in Multiobjective Optimization Problems: Differentiable Case, Journal of Optimization Theory and Applications, Vol. 80, pp. 483-500, 1994. · Zbl 0797.90083 · doi:10.1007/BF02207776
[12] Yang, X. Q., On Second-Order Directional Derivatives, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 26, pp. 55-66, 1996. · Zbl 0839.90138 · doi:10.1016/0362-546X(94)00209-Z
[13] Ben-Tal, A., and Zowe, J., A Unfiied Theory of First and Second-Order Necessary Conditions for Extremum Problems in Topological Vector Spaces, Mathematical Programming Study, Vol. 19, pp. 39-76, 1982. · Zbl 0494.49020 · doi:10.1007/BFb0120982
[14] Clarke, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons, NewYork, NY, 1983. · Zbl 0582.49001
[15] Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York, NY, 1969.
[16] Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401
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