Existence of Solutions and Relative Regularity Conditions for Polynomial Vector Optimization Problems

TL;DR

This paper proves the existence of solutions for polynomial vector optimization problems without convexity or compactness assumptions, introduces relative regularity conditions, and explores their properties and implications.

Contribution

It introduces and characterizes relative regularity conditions for polynomial vector optimization, establishing solution existence and linking to other regularity concepts.

Findings

Existence of efficient solutions without convexity or compactness.

Characterization of relative regularity conditions.

Frank-Wolfe type theorems for non-convex polynomial problems.

Abstract

In this paper, we establish the existence of the efficient solutions for polynomial vector optimization problems on a nonempty closed constraint set without any convexity and compactness assumptions. We first introduce the relative regularity conditions for vector optimization problems whose objective functions are a vector polynomial and investigate their properties and characterizations. Moreover, we establish relationships between the relative regularity conditions, Palais-Smale condition, weak Palais-Smale condition, M-tameness and properness with respect to some index set. Under the relative regularity and non-regularity conditions, we establish nonemptiness of the efficient solution sets of the polynomial vector optimization problems respectively. As a by-product, we infer Frank-Wolfe type theorems for a non-convex polynomial vector optimization problem. Finally, we study the local properties and genericity characteristics of the relative regularity conditions.