Sum-of-squares optimization without semidefinite programming

TL;DR

This paper introduces a primal-dual interior-point method for sum-of-squares optimization that bypasses semidefinite programming, leading to significant improvements in computational efficiency for high-degree polynomial problems.

Contribution

The paper presents a novel approach combining non-symmetric conic optimization and polynomial interpolation to directly optimize over the sum-of-squares cone, avoiding SDP reformulation.

Findings

Method is several orders of magnitude faster than SDP for high-degree polynomials

Reduces theoretical time and space complexity compared to traditional SDP-based methods

Allows recovery of SDP solutions with minimal additional effort

Abstract

We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining non-symmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation which requires a large number of auxiliary variables. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach. Although our approach avoids the semidefinite programming reformulation, an optimal solution to the semidefinite program can be recovered with little additional effort. Computational results confirm that for problems involving high-degree polynomials, the proposed method is several orders of magnitude faster than semidefinite programming.