Generalized Convexity and Characterization of (Weak) Pareto-Optimality in Nonsmooth Multiobjective Optimization Problems

Abstract

Efforts to characterize optimality in nonsmooth and/or nonconvex optimization problems have made rapid progress in the past four decades. Nonsmooth analysis, which refers to differential analysis in the absence of differentiability, has grown rapidly in recent years, and plays a vital role in functional analysis, information technology, optimization, mechanics, differential equations, decision making, etc. Furthermore, convexity has been increasingly important nowadays in the study of many pure and applied mathematical problems. In this paper, some new connections between three major fields, nonsmooth analysis, convex analysis, and optimization, are provided that will help to make these fields accessible to a wider audience. In this paper, at first, we address some newly reported and interesting applications of multiobjective optimization in Management Science and Biology. Afterwards, some sufficient conditions for characterizing the feasible and improving directions of nonsmooth multiobjective optimization problems are given, and using these results a necessary optimality condition is proved. The sufficient optimality conditions are given utilizing a generalized convexity notion. Establishing necessary and sufficient optimality conditions for nonsmooth fractional programming problems is the next aim of the paper. We follow the paper by studying (strictly) prequasiinvexity and pseudoinvexity. Finally, some connections between these notions as well as some applications of these concepts in optimization are given.