Level set methods for finding critical points of mountain pass type

TL;DR

This paper introduces a convergent numerical method for locating mountain pass critical points, applicable to nonsmooth cases and with superlinear convergence in smooth scenarios, with applications to matrix eigenvalue problems.

Contribution

It presents a novel numerical approach for critical point computation that is robust in nonsmooth contexts and demonstrates its application to the Wilkinson problem.

Findings

Method converges in nonsmooth cases

Achieves superlinear convergence in smooth cases

Applied successfully to matrix eigenvalue distance problem

Abstract

Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case. We apply these techniques to describe a strategy for the Wilkinson problem of calculating the distance of a matrix to a closest matrix with repeated eigenvalues. Finally, we relate critical points of mountain pass type to nonsmooth and metric critical point theory.