Sums of Squares and Sparse Semidefinite Programming

TL;DR

This paper explores the connection between sums of squares, nonnegative polynomials on real varieties, and sparse semidefinite programming, providing quantitative approximation results with applications in optimization.

Contribution

It establishes a link between sums of squares and sparse semidefinite programming on quadratic monomial-defined varieties, with new approximation bounds.

Findings

Quantitative bounds on approximating nonnegative polynomials by sums of squares.

Application of these bounds to improve sparse semidefinite programming techniques.

Insight into the structure of PSD matrix completions for quadratic monomial ideals.

Abstract

We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a quadratic square-free monomial ideal. In this case nonnegative polynomials and sums of squares on $X$ are also natural objects in positive semidefinite matrix completion. Nonnegative quadratic forms over $X$ naturally correspond to partially specified matrices where all of the fully specified square blocks are PSD, and sums of squares quadratic forms naturally correspond to partially specified matrices which can be completed to a PSD matrix. We show quantitative results on approximation of nonnegative polynomials by sums of squares, which leads to applications in sparse semidefinite programming.