Polynomial Optimization with Real Varieties

Abstract

We study the optimization problem of minimizing $f(x)\ {\rm subject\ to}\ h(x)=0,\,\, g(x)\geq 0$ with $f$ a polynomial and $h,g$ two tuples of polynomials in $x\in \mathbb{R}^n$. Lasserre's hierarchy is a sequence of sum of squares relaxations for finding the global minimum $f_{min}$. Let $K$ be the feasible set. We prove the following results: (i) If the real variety $V_{\mathbb{R}}(h)$ is finite, then Lasserre's hierarchy has finite convergence, no matter if the complex variety $V_{\mathbb{C}}(h)$ is finite or not. This solves an open question in the survey by M. Laurent [IMA Vol. Math. Appl. 149, Springer, New York, 2009, pp. 157--270]. (ii) If $K$ and $V_{\mathbb{R}}(h)$ have the same vanishing ideal, then the finite convergence of Lasserre's hierarchy is independent of the choice of defining polynomials for the real variety $V_{\mathbb{R}}(h)$. (iii) If only $K$ is finite, a refined version of Lasserre's hierarchy (using the preordering of $g$) has finite convergence.