Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

TL;DR

This paper reviews recent advances in sum of squares (SOS) optimization, introduces new methods to improve computational efficiency through sparsity exploitation and alternative convex programming, and discusses limitations in approximating SOS cones.

Contribution

It presents two novel methods for enhancing SOS optimization efficiency: one using sparsity patterns and another bypassing SDPs with linear and second order cone programs.

Findings

Sparsity-based methods significantly speed up SOS computations.

Chordal sparsity patterns further improve efficiency.

Limitations exist in approximating SOS cones with second order cones.

Abstract

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.