The greedy strategy in optimizing the Perron eigenvalue

TL;DR

This paper introduces a selective greedy method for optimizing the spectral radius of non-negative matrices within product families, including sparse matrices, with proven quadratic convergence and efficient numerical performance.

Contribution

It presents a new selective greedy algorithm applicable to all non-negative product families, overcoming limitations of previous methods for sparse matrices.

Findings

The method converges quadratically.

It efficiently finds extremal spectral radii in high-dimensional matrices.

Numerical experiments confirm rapid convergence within a few iterations.

Abstract

We address the problems of minimizing and of maximizing the spectral radius overa compact family of non-negative matrices. Those problems being hard in generalcan be efficiently solved for some special families. We consider the so-called prod-uct families, where each matrix is composed of rows chosen independently from givensets. A recently introduced greedy method works very fast. However, it is applicablemostly for strictly positive matrices. For sparse matrices, it often diverges and gives awrong answer. We present the "selective greedy method" thatworks equally well forall non-negative product families, including sparse ones.For this method, we provea quadratic rate of convergence and demonstrate its efficiency in numerical examples.The numerical examples are realised for two cases: finite uncertainty sets and poly-hedral uncertainty sets given by systems of linear inequalities. In dimensions up to 2000, the matrices with minimal/maximal spectral radii in product families are foundwithin a few iterations. Applications to dynamical systemsand to the graph theoryare considered