Dual Certificates and Efficient Rational Sum-of-Squares Decompositions for Polynomial Optimization over Compact Sets

Abstract

We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of dual certificates, which allows us to interpret vectors from the dual of the sum-of-squares cone as rigorous nonnegativity certificates of a WSOS polynomial. Whereas conventional WSOS certificates are alternative representations of the polynomials they certify, dual certificates are distinct from the certified polynomials; moreover, each dual certificate certifies a full-dimensional convex cone of WSOS polynomials. For a theoretical application, we give a short new proof of Powers's theorems on the existence of rational WSOS certificates of positive polynomials. For a computational application, we show that exact WSOS certificates can be constructed from numerically computed dual certificates at little additional cost, without any rounding or projection steps applied to the numerical certificates. We also present an algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence.