Matrix Cubes Parametrized by Eigenvalues

TL;DR

This paper introduces a tensor algebra approach to solve a class of semidefinite programming problems involving matrix cubes defined by eigenvalues, providing new LMI representations and geometric insights.

Contribution

It generalizes previous work on k-ellipses and k-ellipsoids by developing an algebraic geometric framework for matrix cube problems parametrized by eigenvalues.

Findings

Provides an LMI representation for the feasible set

Analyzes the boundary using algebraic geometry

Generalizes earlier geometric results

Abstract

An elimination problem in semidefinite programming is solved by means of tensor algebra. It concerns families of matrix cube problems whose constraints are the minimum and maximum eigenvalue function on an affine space of symmetric matrices. An LMI representation is given for the convex set of all feasible instances, and its boundary is studied from the perspective of algebraic geometry. This generalizes the earlier work [12] with Parrilo on k-ellipses and k-ellipsoids.