On duality theorems for nonsmooth Lipschitz optimization problems. (English) Zbl 0987.90072

Summary: A nonsmooth Lipschitz Vector optimization Problem (VP) is considered. Using the Fritz John type necessary optimality conditions for (VP), we formulate the Mond-Weir dual problem (VD) and establish duality theorems for VP and VD under (strict) pseudoinvexity assumptions on the functions. Our duality theorems do not require a constraint qualification.


90C29 Multi-objective and goal programming
90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
Full Text: DOI


[1] Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley–Interscience, New York, NY, 1983. · Zbl 0582.49001
[2] Giorgi, G., and Guerraggio, A., Various Types of Nonsmooth Invexity, Journal of Information and Optimization Sciences, Vol. 17, pp. 137–150, 1996. · Zbl 0859.49020
[3] Mond, B., and Weir, T., Generalized Concavity and Duality, Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W. T. Ziemba, Academic Press, New York, NY, pp. 263–279, 1981. · Zbl 0538.90081
[4] Giorgi, G., and Guerraggio, A., The Notion of Invexity in Vector Optimization: Smooth and Nonsmooth Case, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer Academic Publishers, Dordrecht, Holland, pp. 389–405, 1998. · Zbl 0953.90049
[5] Jeyakumar, V., and Mond, B., On Generalized Convex Mathematical Programming, Journal of the Australian Mathematical Society, Vol. 34B, pp. 43–53, 1992. · Zbl 0773.90061
[6] Lee, G. M., Nonsmooth Invexity in Multiobjective Programming, Journal of Information and Optimization Sciences, Vol. 15, pp. 127–136, 1994. · Zbl 0826.90100
[7] Mishra, S. K., and Mukherjee, R. N., On Generalized Convex Multiobjective Nonsmooth Programming, Journal of the Australian Mathematical Society, Vol. 38B, pp. 140–148, 1996. · Zbl 0858.90111
[8] Minami, M., Weak Pareto-Optimal Necessary Conditions in a Nondifferentiable Multiobjective Program on a Banach Space, Journal of Optimization Theory and Applications, Vol. 41, pp. 451–461, 1983. · Zbl 0502.90076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.