A supermemory projection-based method for solving nonlinear equations with an application

Supermemory gradient (SMG) methods are generalizations of classical conjugate gradient (CG) methods and thus inherit their fundamental properties. Unlike CG methods, which utilize only one-step memory, SMG methods effectively exploit multi-step iterative information to construct new iterates, offering the potential for significantly accelerated convergence. While SMG techniques have been primarily applied to unconstrained optimization, their extension to the solution of nonlinear equations remains largely unexplored. In this paper, we develop a supermemory projection-based method for solving large-scale systems of nonlinear equations. The proposed method incorporates an inertial term (based on the acceleration techniques introduced by Polyak), along with correction terms designed to enhance stability and convergence. Under the assumption of pseudomonotonicity, we establish the global convergence of the method. Numerical experiments are presented to demonstrate the efficiency of the proposed approach in solving large-scale nonlinear systems, including its application to sparse logistic regression, which is a widely studied problem in machine learning.