Complexity for exact polynomial optimization strengthened with Fritz John conditions

TL;DR

This paper establishes explicit degree bounds for polynomial representations based on Fritz John conditions and demonstrates their application in ensuring finite convergence rates of semidefinite relaxation hierarchies in polynomial optimization.

Contribution

It provides explicit degree bounds for polynomial representations using Fritz John conditions and applies these bounds to analyze convergence rates of semidefinite relaxations.

Findings

Explicit degree bounds depending on n, m, d for polynomial representations.

Finite convergence rates for hierarchies of semidefinite relaxations.

Enhanced understanding of polynomial optimization via Fritz John conditions.

Abstract

Let $f,g_1,\dots,g_m$ be polynomials of degree at most $d$ with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Assume that $f$ is non-negative on a basic semi-algebraic set $S$ defined by polynomial inequalities $g_j(x)\ge 0$, for $j=1,\dots,m$. Our previous work [arXiv:2205.04254 (2022)] has stated several representations of $f$ based on the Fritz John conditions. This paper provides some explicit degree bounds depending on $n$, $m$, and $d$ for these representations. In application to polynomial optimization, we obtain explicit rates of finite convergence of the hierarchies of semidefinite relaxations based on these representations.