Abstract

This article deals with the problem of the existence of a weakly efficient solution to quasiconvex vector optimization problems in a finite dimensional setting on the real-line. This consideration is motivated by algorithmic purposes, because it is expected that, like in scalar minimization, one must solve a one-dimensional problem to find the next iterate. We start by recalling a notion of nonconvexity weaker than quasiconvexity for vector functions introduced ealier by one of the authors in a previous paper. Afterwards, we characterize the nonemptiness and/or compactness of the weakly efficient solution set. Then, this set is described as much as possible in the multiobjective case, and the bicriteria problem is carefully analysed when each component is lower semicontinuous and quasiconvex. Several examples showing the applicability of our results are presented, and various algorithms are stated to compute the overall weakly efficient solution set.