Convexifying positive polynomials and sums of squares approximation

TL;DR

This paper demonstrates that nonnegative polynomials on certain sets can be uniformly approximated by sums of squares and explores conditions under which polynomial multiples become convex, with applications to semidefinite optimization.

Contribution

It introduces a new approximation scheme for nonnegative polynomials using sums of squares and provides criteria for polynomial convexification on semialgebraic sets.

Findings

Polynomials nonnegative on semialgebraic sets can be approximated by sums of squares plus boundary terms.

Conditions are established for when polynomial multiples are convex on convex semialgebraic sets.

Applications include methods for finding lower critical points in polynomial optimization.

Abstract

We show that if a polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ is nonnegative on a closed basic semialgebraic set $X=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\ldots,g_r (x)\ge 0\}$, where $g_1,\ldots,g_r\in\mathbb{R}[x_1,\ldots,x_n]$, then $f$ can be approximated uniformly on compact sets by polynomials of the form $\sigma_0+\varphi(g_1) g_1+\cdots +\varphi(g_r) g_r$, where $\sigma_0\in \mathbb{R}[x_1,\ldots,x_n]$ and $\varphi\in\mathbb{R}[t]$ are sums of squares of polynomials. In particular, if $X$ is compact, and $h(x):=R^2-|x|^2 $ is positive on $X$, then $f=\sigma_{0}+\sigma_1 h+\varphi(g_1) g_1+\cdots +\varphi(g_r) g_r$ for some sums of squares $\sigma_{0},\sigma_1\in \mathbb{R}[x_1,\ldots,x_n]$ and $\varphi\in\mathbb{R}[t]$, where $|x|^2={x_1^2+\cdots+x_n^2}$. We apply a quantitative version of those results to semidefinite optimization methods. Let $X$ be a convex closed semialgebraic subset of $\mathbb{R}^n$ and let $f$ be a polynomial which is positive on $X$. We give necessary and sufficient conditions for the existence of an exponent $N\in\mathbb{N}$ such that $(1+|x|^2)^Nf(x)$ is a convex function on $X$. We apply this result to searching for lower critical points of polynomials on convex compact semialgebraic sets.