On the solution existence and stability of polynomial optimization problems

TL;DR

This paper establishes conditions under which solutions to polynomial optimization problems exist and are stable, providing new theorems and stability analysis for both regular and non-regular cases.

Contribution

It introduces a regularity condition for polynomial optimization, proving solution existence, stability, and extending classical theorems to this context.

Findings

Solution existence under regularity condition

Stability results including upper semicontinuity

Extension of Frank-Wolfe and Eaves theorems

Abstract

This paper introduces and investigates a regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. Under this regularity condition, the normalization argument in asymptotic analysis enables us to see the solution existence as well as the solution stability of these problems. We prove a Frank-Wolfe type theorem for regular optimization problems and an Eaves type theorem for non-regular pseudoconvex optimization problems. Moreover, we show results on the stability such as upper semicontinuity and local upper-H\"{o}lder stability of the solution map of polynomial optimization problems. At the end of the paper, we discuss the genericity of the regularity condition.