Solution maps of polynomial variational inequalities

TL;DR

This paper studies the properties of solution maps in polynomial variational inequalities, focusing on stability, existence, and genericity, with implications for understanding solution behavior under various conditions.

Contribution

It provides new insights into the $R_0$-property, stability, and solution existence for polynomial variational inequalities, especially under semi-algebraic constraints.

Findings

Solution maps exhibit local boundedness and upper semicontinuity.

Existence and stability of solutions are established under copositivity.

Genericity results show the $R_0$-property and finite-valuedness are typical in semi-algebraic cases.

Abstract

In this paper, we investigate several properties of the solution maps of variational inequalities with polynomial data. First, we prove some facts on the $R_0$-property, the local boundedness, and the upper semicontinuity of the solution maps. Second, we establish results on the solution existence and the local upper-H\"{o}lder stability under the copositivity condition. Third, when the constraint set is semi-algebraic, we discuss the genericity of the $R_0$-property and the finite-valuedness of the solution maps.