On generalized-convex constrained multi-objective optimization / Chr. Günter and Chr. Tammer

Abstract

Abstract: In this paper, we consider multi-objective optimization problems involving convex and nonconvex constraints, where the objective function is acting between a real linear topological pre-image space and a finite dimensional image space. The vector-valued objective function of the considered multi-objective optimization problem is assumed to be componentwise generalized-convex (e.g., semi-strictly quasi-convex or quasi-convex). For these problems with a not necessarily convex feasible set, we show that the set of efficient solutions can be computed completely using two corresponding multi-objective optimization problems with a new feasible set that is a convex upper set of the original feasible set in both problems. This means that it is possible to solve a problem with a nonconvex feasible set by solving two problems with convex feasible sets and to apply corresponding methods. Our approach relies on the fact that the original feasible set can be described using level sets of a certain scalar function (a kind of penalization function). At the end of the paper, we apply our approach to problems where the constraints are given by a system of inequalities with a finite number of constraint functions.