Linear optimization over homogeneous matrix cones

TL;DR

This paper explores optimization over homogeneous, non-self-dual cones, focusing on cones of sparse positive semidefinite matrices, their automorphisms, and implications for interior-point methods.

Contribution

It characterizes automorphisms of sparse positive semidefinite cones, extends to linear slices, and discusses the role of homogeneous cones in optimization algorithms.

Findings

Characterization of automorphisms for sparse positive semidefinite cones

Conditions for homogeneity in linear slices of PSD cones

Homogeneous cones admit spectrahedral representations

Abstract

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the positive semidefinite matrix cone and the second order cone as important practical examples. In this paper, we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual. We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. We describe transitive subsets of the automorphism groups of the cones and their duals, and important properties of the composition of log-det barrier functions with the automorphisms in this set. Next, we consider extensions to linear slices of the positive semidefinite cone, i.e., intersection of the positive semidefinite cone with a linear subspace, and review conditions that make the cone homogeneous. In the third part of the paper we give a high-level overview of the classical algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation. We conclude by discussing the role of homogeneous cone structure in primal-dual symmetric interior-point methods.