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Journal of Convex Analysis 21 (2014), No. 4, 1007--1022
Copyright Heldermann Verlag 2014

Characterization of Weakly Efficient Solutions for Nonlinear Multiobjective Programming Problems. Duality

Beatriz Hernández-Jiménez
Depto Economía, Métodos Cuantitativos e Historia Económica, Area de Estadística e Investigación Operativa, Universidad Pablo de Olavide, Edificio 3 - Ctra Utrera - Km 1, 41013 Sevilla, Spain

Rafaela Osuna-Gómez
Depto de Estadística e Investigación Operativa, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, 41012 Sevilla, Spain

Marko A. Rojas-Medar
Depto de Ciencias Básicas, Facultad de Ciencias, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile

Convexity and generalized convexity play a central role in mathematical programming for duality results and in order to characterize the solutions set. In this paper, taking in mind Craven's notion of K-invexity function (when K is a cone in Rn) and Martin's notion of Karush-Kuhn-Tucker invexity (hereafter KKT-invexity), we define new notions of generalized convexity for a multiobjective problem with conic constraints. These new notions are both necessary and sufficient to ensure every Karush-Kuhn-Tucker point is a solution. The study of the solutions is also done through the solutions of an associated scalar problem. A Mond-Weir type dual problem is formulated and weak and strong duality results are provided. The notions and results that exist in the literature up to now are particular instances of the ones presented here.

Keywords: Generalized convexity, KKT-invexity, optimality conditions, duality.

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