Sparse sum-of-squares (SOS) optimization: A bridge between DSOS/SDSOS and SOS optimization for sparse polynomials

TL;DR

This paper explores the relationships between sparse SOS, DSOS, and SDSOS optimization methods, showing that SSOS offers less conservative solutions and can be computationally efficient for large-scale sparse polynomial problems.

Contribution

It establishes theoretical links between SSOS, DSOS, and SDSOS, demonstrating SSOS's advantages in less conservativeness and computational performance for sparse polynomial optimization.

Findings

SSOS strictly contains DSOS/SDSOS polynomials

SSOS is less conservative than DSOS/SDSOS

SSOS can be faster than DSOS/SDSOS in large-scale problems

Abstract

Optimization over non-negative polynomials is fundamental for nonlinear systems analysis and control. We investigate the relation between three tractable relaxations for optimizing over sparse non-negative polynomials: sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares (DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS) optimization. We prove that the set of SSOS polynomials, an inner approximation of the cone of SOS polynomials, strictly contains the spaces of sparse DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less conservative than its DSOS/SDSOS counterparts. Numerical results for large-scale sparse polynomial optimization problems demonstrate this fact, and also that SSOS optimization can be faster than DSOS/SDSOS methods despite requiring the solution of semidefinite programs instead of less expensive linear/second-order cone programs.