Optimality and duality in nonsmooth multiobjective optimization involving generalized type I functions
Convexity plays a very important role in optimization, in such a way that a number of generalizations of convexity have been proposed with the aim of obtain Karush-Kuhn-Tucker type suficient optimality conditions. In this work, the authors consider the following nonsmooth multiobjective (Pareto) optimization problem: \[ \min f(x),\; \text{ subject to } g(x)\leq 0 \tag{1} \] where \(f=(f_1,f_2, \cdots, f_p)\), \(g=(g_1,g_2, \cdots, g_m)\), \(f_i, g_j:X \rightarrow \mathbb{R}\), \(i=1,2,\dots ,p\), \(j=1,2,\dots ,m\), are locally Lipschitz functions on an open subset \(X\subset \mathbb{R}^n\) and min is considered in Pareto sense. The authors propose for the objective functions and the constraints a generalized invexity, called generalized Type I, expressed in terms of Clarke subdiferentials of the involved functions. Under these generalized Type I assumptions, using a Cottle constraint qualification, Karush-Kuhn-Tucker type necessary and sufficient conditions are given for a feasible point to be an efficient or properly efficient (in Geoffrion sense) solution for problem (1). Finally, Wolfe type and Mond-Weir type duals, under these generalized Type I assumptions, for problem (1) are analized.
scientific article; zbMATH DE number 2082276
scientific article