Disconnectedness and unboundedness of the solution sets of monotone vector variational inequalities

Abstract

In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities. We show that if the weak Pareto solution set of a monotone vector variational inequality is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone vector variational inequalities are raised at the end of the paper.

The 2nd and 3rd sentences in the proof of Propsition 3.3 have been replaced by "There exist an open set $U$ in ${\mathbb{R}}^n$ such that $U$ is bounded and $\A \subset U$." The author would like to thank Dr. Yu Han for pointing out the incorrect point