Spectrally Constrained Optimization

TL;DR

This paper develops methods for solving smooth matrix optimization problems with eigenvalue constraints, including exact solutions for linear objectives and algorithms for non-convex cases, supported by numerical experiments.

Contribution

It introduces solution techniques for eigenvalue-constrained matrix optimization, including exact global minima for linear objectives and first-order algorithms for non-convex problems.

Findings

Exact solutions for linear objectives under eigenvalue constraints.

First-order algorithms with proven sublinear convergence for non-convex problems.

Numerical experiments demonstrating method applicability.

Abstract

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions, i.e., $F(X) = \langle C, X \rangle$, and perform exact projections onto the eigenvalue constraint set. Two first-order algorithms are developed to obtain first-order stationary points for general non-convex objective functions. Both methods are proven to converge sublinearly when the constraint set is convex. Numerical experiments demonstrate the applicability of both the model and the methods.