A New Sparse SOS Decomposition Algorithm Based on Term Sparsity

TL;DR

This paper introduces a novel sparse SOS decomposition algorithm utilizing cross sparsity patterns that focus on term sparsity, leading to more efficient computations and handling of larger polynomials.

Contribution

It presents a new sparsity pattern for SOS decomposition that refines existing methods and improves computational efficiency for sparse polynomials.

Findings

Significantly reduces computational cost.

Handles larger and more complex polynomials.

Provides a refinement over sign-symmetry methods.

Abstract

A new sparse SOS decomposition algorithm is proposed based on a new sparsity pattern, called cross sparsity patterns. The new sparsity pattern focuses on the sparsity of terms and thus is different from the well-known correlative sparsity pattern which focuses on the sparsity of variables though the sparse SOS decomposition algorithms based on these two sparsity patterns both take use of chordal extensions/chordal decompositions. Moreover, it is proved that the SOS decomposition obtained by the new sparsity pattern is always a refinement of the block-diagonalization obtained by the sign-symmetry method. %Because the new sparsity pattern covers more sparse polynomials than correlative sparsity pattern, Various experiments show that the new algorithm dramatically saves the computational cost compared to existing tools and can handle some really huge polynomials.