Convexifying positive polynomials and sums of squares approximation

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 $σ_0+φ(g_1) g_1+\cdots +φ(g_r) g_r$, where $σ_0\in \mathbb{R}[x_1,\ldots,x_n]$ and $φ\in\mathbb{R}[t]$ are sums of squares of polynomials. In particular, if $X$ is compact, and $h(x):=R^2-|x|^2$ is positive on $X$, then $f=σ_{0}+σ_1 h+φ(g_1) g_1+\cdots +φ(g_r) g_r$ for some sums of squares $σ_{0},σ_1\in \mathbb{R}[x_1,\ldots,x_n]$ and $φ\in\mathbb{R}[t]$, 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.

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