Spectrahedral cones generated by rank 1 matrices

TL;DR

This paper studies spectrahedral cones generated by rank 1 matrices, classifies small cases, explores their structure, and discusses their relevance to semi-definite relaxations in optimization.

Contribution

It introduces the concept of rank one generated (ROG) spectrahedral cones, classifies all such cones for matrices up to size 4, and analyzes their structural properties.

Findings

ROG cones are linearly isomorphic iff they are convex cone isomorphic.

Many examples and construction methods for ROG cones are provided.

ROG cones relate to the exactness of semi-definite relaxations in optimization.

Abstract

Let ${\cal S}_+^n \subset {\cal S}^n$ be the cone of positive semi-definite matrices as a subset of the vector space of real symmetric $n \times n$ matrices. The intersection of ${\cal S}_+^n$ with a linear subspace of ${\cal S}^n$ is called a spectrahedral cone. We consider spectrahedral cones $K$ such that every element of $K$ can be represented as a sum of rank 1 matrices in $K$. We shall call such spectrahedral cones rank one generated (ROG). We show that ROG cones which are linearly isomorphic as convex cones are also isomorphic as linear sections of the positive semi-definite matrix cone, which is not the case for general spectrahedral cones. We give many examples of ROG cones and show how to construct new ROG cones from given ones by different procedures. We provide classifications of some subclasses of ROG cones, in particular, we classify all ROG cones for matrix sizes not exceeding 4. Further we prove some results on the structure of ROG cones. We also briefly consider the case of complex or quaternionic matrices. ROG cones are in close relation with the exactness of semi-definite relaxations of quadratically constrained quadratic optimization problems or of relaxations approximating the cone of nonnegative functions in squared functional systems.