An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems Over Symmetric Cones

Abstract

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let $\varepsilon$ and $\varepsilon_{+}$ be a finite dimensional real vector space and a symmetric cone embedded in $\varepsilon$; examples of $\varepsilon$ and $\varepsilon_{+}$ include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a pair of the N-dimensional Euclidean space and N-dimensional second-order cones, and a pair of the space of m × m real symmetric (or complex Hermitian) matrices and the cone of their positive semidefinite matrices. Sums of squares relaxations are further extended to a polynomial optimization problem over $\varepsilon_{+}$, i.e., a minimization of a real valued polynomial a(x) in the n-dimensional real variable vector x over a compact feasible region $\{ {\bf x} : b({\bf x}) \in \varepsilon_{+}\}$, where b(x) denotes an $\varepsilon$- valued polynomial in x. It is shown under a certain moderate assumption on the $\varepsilon$-valued polynomial b(x) that optimal values of a sequence of sums of squares relaxations of the problem, which are converted into a sequence of semidefinite programs when they are numerically solved, converge to the optimal value of the problem.