Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization

TL;DR

This paper proves that the standard semidefinite extended formulation for the cone of SOS polynomials is optimal in size, and similar minimality results are shown for related cones in polynomial optimization.

Contribution

It establishes the minimal size of LMIs needed for semidefinite extended formulations of SOS cones and related cones, proving the standard formulation's optimality.

Findings

The standard LMI size for SOS cones is proven to be optimal.

No smaller finite LMI extended formulation exists for the SOS cone.

Similar minimality results are shown for other cones like copositive and completely positive cones.

Abstract

The abbreviations LMI and SOS stand for `linear matrix inequality' and `sum of squares', respectively. The cone $\Sigma_{n,2d}$ of SOS polynomials in $n$ variables of degree at most $2d$ is known to have a semidefinite extended formulation with one LMI of size $\binom{n+d}{n}$. In other words, $\Sigma_{n,2d}$ is a linear image of a set described by one LMI of size $\binom{n+d}{n}$. We show that $\Sigma_{n,2d}$ has no semidefinite extended formulation with finitely many LMIs of size less than $\binom{n+d}{n}$. Thus, the standard extended formulation of $\Sigma_{n,2d}$ is optimal in terms of the size of the LMIs. As a direct consequence, it follows that the cone of $k \times k$ symmetric positive semidefinite matrices has no extended formulation with finitely many LMIs of size less than $k$. We also derive analogous results for further cones considered in polynomial optimization such as truncated quadratic modules, the cones of copositive and completely positive matrices and the cone of sums of non-negative circuit polynomials.