Univariate Polynomial Optimization with Sum-of-Squares Interpolants

One of the most common tools in polynomial optimization is the approximation of the cone of nonnegative polynomials with the cone of sum-of-squares polynomials. This leads to polynomial-time solvable approximations for many NP-hard optimization problems using semidefinite programming (SDP). While theoretically satisfactory, the translation of optimization problems involving sum-of-squares polynomials to SDPs is not always practical. First, in the common SDP formulation, the dual variables are semidefinite matrices whose condition numbers grow exponentially with the degree of the polynomials involved, which is detrimental for a floating-point implementation. Second, the SDP representation of sum-of-squares polynomials roughly squares the number of optimization variables, increasing the time and memory complexity of the solution algorithms by several orders of magnitude. In this paper we focus on the first, numerical, issue. We show that a reformulation of the sum-of-squares SDP using polynomial interpolants yields a substantial improvement over the standard formulation, and problems involving sum-of-squares interpolants of hundreds of degrees can be handled without difficulty by commonly used semidefinite programming solvers. Preliminary numerical results using semi-infinite optimization problems align with the theoretical predictions. In all problems considered, available memory is the only factor limiting the degrees of polynomials.