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 (VVIs). We show that if the weak Pareto solution set of a monotone VVI 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 VVIs are raised at the end of the paper.

Keywords:

• Monotone vector variational inequality
• solution set
• disconnectedness
• unboundedness
• scalarization formula

2010 Mathematics Subject Classifications:

• 49J40
• 47H05
• 90C29
• 90C33
• 54D05

Acknowledgments

The author would like to thank Prof. Nguyen Dong Yen for encouragement and the anonymous referees for valuable remarks and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.01–2018.306].