Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

TL;DR

This paper introduces dual certificates for sum-of-squares polynomials, enabling efficient rational decompositions and a hybrid algorithm for polynomial optimization over compact sets.

Contribution

It develops the concept of dual cone certificates, allowing rational WSOS certificates to be derived from numerical dual certificates without rounding.

Findings

Dual certificates interpret vectors as nonnegativity certificates.

Rational WSOS certificates can be constructed efficiently from dual certificates.

An almost entirely numerical hybrid algorithm computes optimal WSOS bounds with linear convergence.

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 cone 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. As a result, rational 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. As an additional algorithmic application, we present an almost entirely numerical hybrid 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.