On sampling Kaczmarz-Motzkin methods for solving large-scale nonlinear systems

TL;DR

This paper introduces a novel sampling Kaczmarz-Motzkin method for efficiently solving large-scale nonlinear equations, with variants for constrained problems, demonstrating improved performance over existing methods.

Contribution

The paper proposes the NSKM, PSKM, and APSKM methods for nonlinear systems, providing convergence proofs and practical algorithms for large-scale and constrained problems.

Findings

NSKM outperforms nonlinear randomized Kaczmarz in calculation time

Convergence of the proposed methods is established under certain conditions

The methods are effective for large-scale and constrained nonlinear systems

Abstract

In this paper, for solving large-scale nonlinear equations we propose a nonlinear sampling Kaczmarz-Motzkin (NSKM) method. Based on the local tangential cone condition and the Jensen's inequality, we prove convergence of our method with two different assumptions. Then, for solving nonlinear equations with the convex constraints we present two variants of the NSKM method: the projected sampling Kaczmarz-Motzkin (PSKM) method and the accelerated projected sampling Kaczmarz-Motzkin (APSKM) method. With the use of the nonexpansive property of the projection and the convergence of the NSKM method, the convergence analysis is obtained. Numerical results show that the NSKM method with the sample of the suitable size outperforms the nonlinear randomized Kaczmarz (NRK) method in terms of calculation times. The APSKM and PSKM methods are practical and promising for the constrained nonlinear problem.