Empirical properties of optima in free semidefinite programs

TL;DR

This paper empirically investigates the properties of optimal solutions in free semidefinite programs, revealing that most optimizers are classical and free extreme points, with unexpected constraint dimensions and reducibility patterns.

Contribution

It provides the first empirical analysis of optimizer properties in free spectrahedra, including the prevalence of free extreme points and new insights into their structure.

Findings

Over 99.9% of optimizers are free extreme points.

The dimension of the active constraint's kernel is about twice expected.

Most optimizers are classical extreme points.

Abstract

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede