Level set methods for finding saddle points of general Morse index

TL;DR

This paper develops algorithms inspired by the mountain pass theorem to find saddle points of functions with arbitrary Morse index, proving convergence in nonsmooth cases and superlinear convergence in smooth finite-dimensional cases.

Contribution

It extends previous level set methods to general Morse index saddle points and proves convergence properties for these algorithms.

Findings

Algorithms successfully find saddle points of general Morse index.

Proven convergence in nonsmooth cases.

Achieved local superlinear convergence in smooth finite-dimensional cases.

Abstract

For a real valued function, a point is critical if its derivatives are zero, and a critical point is a saddle point if it is not a local extrema. In this paper, we study algorithms to find saddle points of general Morse index. Our approach is motivated by the multidimensional mountain pass theorem, and extends our earlier work on methods (based on studying the level sets) to find saddle points of mountain pass type. We prove the convergence of our algorithms in the nonsmooth case, and the local superlinear convergence of another algorithm in the smooth finite dimensional case.