Numbers of the connected components of the solution sets of monotone affine vector variational inequalities

TL;DR

This paper investigates the maximum number of connected components in the solution sets of monotone affine vector variational inequalities, revealing dependence on criteria count and variables, and providing bounds that partially answer a prior open question.

Contribution

It establishes new upper and lower bounds for the number of connected components, advancing understanding of solution set topology in monotone affine vector variational inequalities.

Findings

Provides bounds on the number of connected components

Shows dependence on number of criteria and variables

Partially answers an open question in the field

Abstract

This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question~2 in [N.D. Yen and J.-C. Yao, \textit{Monotone affine vector variational inequalities}, Optimization 60 (2011), pp. 53--68] and point out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality under investigation.