Block SOS Decomposition

TL;DR

This paper introduces the concept of block SOS decomposable polynomials and proves that such polynomials are extremely rare among all SOS polynomials, indicating limited applicability of certain preprocessing algorithms for SOS decomposition.

Contribution

It defines block SOS decomposable polynomials and proves they form a measure-zero subset of SOS polynomials, highlighting the limited scope of existing preprocessing methods.

Findings

Block SOS decomposable polynomials are rare among SOS polynomials.

Preprocessing algorithms based on support properties apply to very few polynomials.

Most SOS polynomials cannot be simplified into smaller SDP problems using these methods.

Abstract

A widely used method for solving SOS (Sum Of Squares) decomposition problem is to reduce it to the problem of semi-definite programs (SDPs) which can be efficiently solved in theory. In practice, although many SDP solvers can work out some problems of big scale, the efficiency and reliability of such method decrease greatly while the input size increases. Recently, by exploiting the sparsity of the input SOS decomposition problem, some preprocessing algorithms were proposed [5,17], which first divide the input problem satisfying special definitions or properties into smaller SDP problems and then pass the smaller ones to SDP solvers to obtain reliable results efficiently. A natural question is that to what extent the above mentioned preprocessing algorithms work. That is, how many polynomials satisfying those definitions or properties are there in the SOS polynomials? In this paper, we define a concept of block SOS decomposable polynomials which is a generalization of those special classes in [5] and [17]. Roughly speaking, it is a class of polynomials whose SOS decomposition problem can be transformed into smaller ones (in other words, the corresponding SDP matrices can be block-diagnolized) by considering their supports only (coefficients are not considered). Then we prove that the set of block SOS decomposable polynomials has measure zero in the set of SOS polynomials. That means if we only consider supports (not with coefficients) of polynomials, such algorithms decreasing the size of SDPs for those SDP-based SOS solvers can only work on very few polynomials. As a result, this shows that the SOS decomposition problems that can be optimized by the above mentioned preprocessing algorithms are very few.