Deflation Techniques for Finding Multiple Local Minima of a Nonlinear Least Squares Problem

TL;DR

This paper introduces two new deflation-based methods, using Gauss-Newton, to efficiently find multiple local minima in nonlinear least squares problems, with applications to inverse eigenvalue problems.

Contribution

The paper generalizes deflation techniques to develop two Gauss-Newton based methods that find multiple local minima without Hessian calculations, reducing the number of deflations needed.

Findings

Methods successfully find multiple local minima

Fewer deflations needed compared to deflated Newton method

Open source MATLAB code provided

Abstract

In this paper we generalize the technique of deflation to define two new methods to systematically find many local minima of a nonlinear least squares problem. The methods are based on the Gauss-Newton algorithm, and as such do not require the calculation of a Hessian matrix. They also require fewer deflations than for applying the deflated Newton method on the first order optimality conditions, as the latter finds all stationary points, not just local minima. One application of interest covered in this paper is the inverse eigenvalue problem (IEP) associated with the modelling of spectroscopic data of relevance to the physical and chemical sciences. Open source MATLAB code is provided at https://github.com/AlbanBloorRiley/DeflatedGaussNewton.