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

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 method in terms of calculation times. The APSKM and PSKM methods are practical and promising for the constrained nonlinear problem.