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multiobjective programming; nonsmooth constrained optimization; second-order optimality conditions; nondominated solutions; local Pareto optimal solutions
We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class $C^{1,1}$. Second-order optimality conditions for local Pareto solutions are derived as a special case.
[1] M. S. Bazaraa, C. M. Shetty: Foundations of Optimization. LN in Econom. and Math. Systems, vol. 122, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
[2] A. Ben-Tal: Second-order and related extremality conditons in nonlinear programming. J. Optim. Theory Appl. 31 (1980), 143–165.
[3] A. Cambini, L. Martein and R. Cambini: A New Approach to Second Order Optimality Conditions in Vector Optimization. Advances in Multiple Objective and Goal Programming, LN in Econom. and Math. Systems, vol. 455, Springer, Berlin, 1997.
[4] J. B. Hiriart-Urruty, J. J. Strodiot and V. H. Nguyen: Generalized Hessian matrix and second-order optimality conditions for problems with $C^{1,1}$ data. Appl. Math. Optim. 11 (1984), 43–56.
[5] S. Huang: Second-order conditions for nondominated solution in generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci. 5 (1985), 172–184. (Chinese)
[6] D. Klatte, K. Tammer: On second-order sufficient optimality conditions for $C^{1,1}$ optimization problems. Optimization 19 (1988), 169–179.
[7] M. Křížek, P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Longman, Harlow, 1990.
[8] L. Liu: The second-order conditions of nondominated solutions for $C^{1,1}$ generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci. 4 (1991), 128–138.
[9] L. Liu: The second order conditions for $C^{1,1}$ nonlinear mathematical programming. pp. 153–158.
[10] L. Liu, M. Křížek: The second order optimality conditions for nonlinear mathematical programming with $C^{1,1}$ data. Appl. Math. 42 (1997), 311–320.
[11] D. T. Luc: Taylor’s formula for $C^{k,1}$ functions. SIAM J. Optim. 5 (1995), 659–669.
[12] Ch. Malivert: First and second order optimality conditions in vector optimization. Ann. Sci. Math. Québec 14 (1990), 65–79.
[13] G. P. McCormick: Second order conditions for constrained minima. SIAM. J. Appl. Math. 15 (1967), 641–652.
[14] K. Miettinen: Nonlinear Multiobjective Optimization. Kluwer, Dordrecht, 1998.
[15] S. Sáks: Theory of the Integral. Hafner Publishing Co., New York, 1937.
[16] S. Wang: Second-order necessary and sufficient conditions in multiobjective programming. Numer. Funct. Anal. Optim. 12 (1991), 237–252.
[17] D. E. Ward: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. Optim. Theory Appl. 80 (1994), 551–571.
[18] D. E. Ward: A comparison of second-order epiderivatives: calculus and optimality conditions. J. Math. Anal. Appl. 193 (1995), 465–482.
[19] X. Q. Yang: Generalized second-order derivatives and optimality conditions. Nonlinear Anal. 23 (1994), 767–784.
[20] X. Q. Yang, V. Jeyakumar: Generalized second-order directional derivatives and optimization with $C^{1,1}$ functions. Optimization 26 (1992), 165–185.
[21] X. Q. Yang, V. Jeyakumar: First and second-order optimality conditions for convex composite multiobjective optimization. J. Optim. Theory Appl. 95 (1997), 209–224.
[22] P. L. Yu: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjective. J. Optim. Theory Appl. 17 (1974), 320–377.
[23] P. L. Yu: Multiple-Criteria Decision Making: Concepts, Techniques and Extensions. Plenum Press, New York, 1985.
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