Speaker: Masakazu Muramatsu

Joined work with Masakazu Kojima

Title: An Extension of Sums of Squares Relaxations to Polynomial Optimization Problems over Symmetric Cones

Abstract:

This work is based on a recent paper by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. In this talk, we further extend his results to polynomial optimization problems over symmetric cones. Let ${\cal E}$ and ${\cal E}_+$ be a Euclidean Jordan algebra and its associated symmetric cone, respectively. We consider a minimization of a real valued polynomial $a(x)$ in the $n$-dimensional real variable vector $x$ over a compact feasible region $\{ x : b(x) \in {\cal E}_+\}$, where $b(x)$ denotes an $\cal E$-valued polynomial in $x$. It is shown under a certain moderate assumption on the $\cal E$-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 original problem. The proof is based on a generalized version of Putinar's lemma, which is newly developed for $\cal E$-valued polynomials.