On the existence of Pareto solutions for polynomial vector optimization problems

Abstract

We are interested in the existence of Pareto solutions to the vector optimization problem $${\text{Min}}_{{\mathbb{R}}_{+}^m}\{f(x)|x\in {\mathbb{R}}^n\},$$ where $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m-1$ containing the set of Pareto values of the problem. Then we establish connections between the Palais--Smale conditions, $M$-tameness, and properness for the map $f$. Based on these results, we provide some sufficient conditions for the existence of Pareto solutions of the problem. We also introduce a generic class of polynomial vector optimization problems having at least one Pareto solution.

21 pages

References in corpus (2)

• Detecting bifurcation values at infinity of real polynomials
• Towards effective detection of the bifurcation locus of real polynomial maps

Cited by in corpus (2)

• Atypical points at infinity and algorithmic detection of the bifurcation locus of real polynomials
• On the existence and the stability of solutions in nonconvex vector optimization