Empirical Properties of Optima in Free Semidefinite Programs

Abstract

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form$$L_A(X)=I-A_1{X}_1-\dots -A_g{X}_g\succcurlyeq 0.$$ 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 × n, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succcurlyeq 0$ 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 × 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, $\ker (L_A(X^{\mathcal{l}}))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^{\mathcal{l}},\dots ,X_g^{\mathcal{l}})$. 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 needed.

KEYWORDS:

• Matrix convex set
• extreme point
• linear matrix inequality
• spectrahedron
• Arveson boundary
• free semidefinite programming

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 46L07
• Secondary 90C22

Conflict of interest

No potential conflict of interest was reported by the author(s).

Notes

1 As a consequence of Helton et al. [30, Theorem 3.1], the existence of any defining tuple of diagonal matrices implies the existence of a minimal defining tuple of diagonal matrices. However, a non-minimal defining tuple for a free polytope need not be simultaneously diagonalizable.

2 If the algorithm is designed so that each ${\widehat{\gamma }}^j$ is a classical extreme point of the relevant spectrahedron, then this method always succeeds in at most μ steps.

3 Since $K_{g,d,n}$ is discrete random variable, some authors may use the term probability mass function (PMF) instead of probability density function (PDF).

4 NCSE has since been updated to use the Mathematica 12 SDP by default. As an option, a user may still use the NCAlgebra SDP.

Additional information

Funding

Research supported by the NSF grant DMS-1500835.