An Application of the Tarski-Seidenberg Theorem with Quantifiers to Vector Variational Inequalities

TL;DR

This paper investigates the connectedness of various solution sets in polynomial vector variational inequalities and optimization problems, proving they are semi-algebraic using the Tarski-Seidenberg Theorem with quantifiers without additional constraint qualifications.

Contribution

It introduces a novel application of the Tarski-Seidenberg Theorem with quantifiers to establish semi-algebraic properties of solution sets without needing the Mangasarian-Fromovitz constraint qualification.

Findings

Solution sets are semi-algebraic.

Connectedness structures are characterized.

No need for Mangasarian-Fromovitz constraint qualification.

Abstract

We study the connectedness structure of the proper Pareto solution sets, the Pareto solution sets, the weak Pareto solution sets of polynomial vector variational inequalities, as well as the connectedness structure of the efficient solution sets and the weakly efficient solution sets of polynomial vector optimization problems. By using the Tarski-Seidenberg Theorem with quantifiers, we are able to prove that these solution sets are semi-algebraic without imposing the Mangasarian-Fromovitz constraint qualification on the system of constraints.