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

TL;DR

This paper explores the topological properties of solution sets in monotone vector variational inequalities, revealing that disconnected solution sets have unbounded connected components, and raises open questions on their structure.

Contribution

It establishes a novel link between disconnectedness and unboundedness of solution components in monotone vector variational inequalities.

Findings

Disconnected solution sets have unbounded connected components.

The property applies to both weak and proper Pareto solution sets.

Raises open questions on the topological structure of these solution sets.

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.