A Nonmonotone Matrix-Free Algorithm for Nonlinear Equality-Constrained Least-Squares Problems

TL;DR

This paper introduces a nonmonotone, matrix-free Levenberg-Marquardt algorithm tailored for large-scale nonlinear least squares problems with equality constraints, demonstrating superior performance in data assimilation and inverse problems.

Contribution

It presents a novel Levenberg-Marquardt method that uses inexact Jacobian-vector products and nonmonotone steps, ensuring global convergence for large-scale constrained least squares.

Findings

Algorithm reaches solution vicinity from arbitrary starting points.

Outperforms natural alternatives in test cases.

Effective for data assimilation and inverse problems.

Abstract

Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty of the problem, and motivates the need for efficient algorithms amenable to large-scale implementations. In this paper, we propose and analyze a Levenberg-Marquardt algorithm for nonlinear least squares subject to nonlinear equality constraints. Our algorithm is based on inexact solves of linear least-squares problems, that only require Jacobian-vector products. Global convergence is guaranteed by the combination of a composite step approach and a nonmonotone step acceptance rule. We illustrate the performance of our method on several test cases from data assimilation and inverse problems: our algorithm is able to reach the vicinity of a solution from an arbitrary starting point, and can outperform the most natural alternatives for these classes of problems.