An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones

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 epsilon and epsilon(+) be a finite dimensional real vector space and a symmetric cone embedded in epsilon; examples of epsilon and epsilon(+) 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 x 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 epsilon(+), i.e., 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) is an element of epsilon(+)}, where b( x) denotes an epsilon-valued polynomial in x. It is shown under a certain moderate assumption on the epsilon-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.