Nonsmooth Optimization Using Mordukhovich's Subdifferential

Abstract

In this paper, new results, which exhibit some new applications for Mordukhovich's subdifferential in nonsmooth optimization and variational problems, are established. Nonsmooth (fractional) multiobjective optimization problems in special Banach spaces are studied, and some necessary and sufficient conditions for weak Pareto-optimality for these problems are introduced. Through this work, we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-$(p,r)$-invexity. Some optimality conditions regarding the generalized KT-$(p,r)$-invexity notion and Kuhn-Tucker points are provided. Also, we seek a connection between linear (semi-) infinite programming and nonlinear programming. Some sufficient conditions for (proper) optimality under invexity are provided. A nonsmooth variational problem corresponding to a considered multiobjective problem is defined and the relations between the provided variational problem and the considered optimization problem are studied. The final part of the paper is devoted to illustrating a penalization mechanism, using the distance function as a tool, to provide some conditions to the solutions of the nonsmooth variational inequality problems. All results of the paper have been established in the absence of gradient vectors, using the properties of Mordukhovich's subdifferential in Asplund spaces.