The level set method for the two-sided eigenproblem

TL;DR

This paper introduces a method to compute the spectrum of a max-plus eigenproblem for matrix pencils using a spectral function and mean payoff games, enabling pseudo-polynomial complexity calculations.

Contribution

It presents a novel approach to determine the spectrum of max-plus matrix pencils via a spectral function and game-theoretic oracle, extending eigenproblem analysis.

Findings

Spectrum consists of a finite union of intervals.

Spectrum can be computed with pseudo-polynomial complexity.

Introduces a spectral function linked to Chebyshev distance.

Abstract

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.