Two-step inertial derivative-free projection method for solving nonlinear equations with application

Large-scale systems of nonlinear equations play a fundamental role in various applications, spanning from differential equations to economics, engineering, management science, probability theory, and various other applied sciences. This paper proposes a derivative-free iterative method to find ϵ-approximate solutions of large-scale nonlinear equations. The proposed scheme combines the hyperplane projection technique with a two-step inertial extrapolation, presenting a new approach. The proposed method does not require Jacobian information and thus can be applied to solve nonsmooth equations. Under standard assumptions, we prove the proposed method's global convergence and nonasymptotic O(1/k) convergence rate. Furthermore, we include numerical results and comparisons to demonstrate the efficiency of the proposed method. Finally, we demonstrate the applicability of the proposed method in solving regularized decentralized logistic regression, a popular problem in machine learning applications.