A multi-step derivative-free projection method for nonlinear equations with application to sparse signal recovery

Abstract

In this paper, we propose a multi-step derivative-free projection method for solving large-scale nonlinear equations with convex constraints. The first direction is constructed as a modified steepest descent direction, while the second direction is derived by equating the spectral gradient direction with the classical conjugate gradient direction, incorporating a convex combination of the HS and DY parameters. These dual-direction mechanisms synergistically enhance the efficiency of the search process. Notably, the global convergence of the proposed method does not rely on the Lipschitz continuity assumption. Furthermore, both asymptotic and non-asymptotic convergence rates are established in terms of iteration complexity. Numerical experiments validate its efficacy in addressing large-scale nonlinear equations and sparse signal recovery problems.