On the existence of Pareto solutions for polynomial vector optimization problems

TL;DR

This paper investigates conditions ensuring the existence of Pareto solutions in polynomial vector optimization problems, using geometric and topological tools like the tangency variety and Palais–Smale conditions.

Contribution

It introduces a semi-algebraic set containing Pareto values, links topological conditions to solution existence, and identifies a generic class with guaranteed Pareto solutions.

Findings

Constructs a semi-algebraic set of Pareto values

Establishes links between Palais–Smale conditions and properness

Provides sufficient conditions for Pareto solution existence

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 $f\colon\mathbb{R}^n\to \mathbb{R}^m$ 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.