Decomposed Structured Subsets for Semidefinite and Sum-of-Squares Optimization

TL;DR

This paper introduces decomposed structured subsets to improve the scalability of semidefinite and sum-of-squares optimization by providing tighter bounds through an equivalent conversion approach, demonstrated on control and polynomial optimization problems.

Contribution

It proposes a novel framework of decomposed structured subsets for SDP approximation, enhancing bound tightness and scalability over existing methods.

Findings

Decomposed structured subsets yield tighter bounds than direct approximations.

The method improves scalability for large SDPs in control and polynomial optimization.

An adapted basis pursuit method refines bounds iteratively.

Abstract

Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of matrix variables and the number of constraints increases. To improve scalability, SDPs can be approximated with lower and upper bounds through the use of structured subsets (e.g., diagonally-dominant and scaled-diagonally dominant matrices). Meanwhile, any underlying sparsity or symmetry structure may be leveraged to form an equivalent SDP with smaller positive semidefinite constraints. In this paper, we present a notion of decomposed structured subsets}to approximate an SDP with structured subsets after an equivalent conversion. The lower/upper bounds found by approximation after conversion become tighter than the bounds obtained by approximating the original SDP directly. We apply decomposed structured subsets to semidefinite and sum-of-squares optimization problems with examples of H-infinity norm estimation and constrained polynomial optimization. An existing basis pursuit method is adapted into this framework to iteratively refine bounds.