Fast ADMM for sum-of-squares programs using partial orthogonality

TL;DR

This paper introduces a fast ADMM-based method leveraging partial orthogonality in SOS programs' SDPs, significantly improving efficiency for large-scale polynomial optimization and stability analysis.

Contribution

It develops a novel ADMM algorithm exploiting partial orthogonality in SOS SDPs, enabling faster solutions for large-scale problems.

Findings

Outperforms state-of-the-art solvers on large SOS problems

Efficient projection using diagonal plus low-rank structure

Implemented in the CDCS solver package

Abstract

When sum-of-squares (SOS) programs are recast as semidefinite programs (SDPs) using the standard monomial basis, the constraint matrices in the SDP possess a structural property that we call \emph{partial orthogonality}. In this paper, we leverage partial orthogonality to develop a fast first-order method, based on the alternating direction method of multipliers (ADMM), for the solution of the homogeneous self-dual embedding of SDPs describing SOS programs. Precisely, we show how a "diagonal plus low rank" structure implied by partial orthogonality can be exploited to project efficiently the iterates of a recent ADMM algorithm for generic conic programs onto the set defined by the affine constraints of the SDP. The resulting algorithm, implemented as a new package in the solver CDCS, is tested on a range of large-scale SOS programs arising from constrained polynomial optimization problems and from Lyapunov stability analysis of polynomial dynamical systems. These numerical experiments demonstrate the effectiveness of our approach compared to common state-of-the-art solvers.