Accelerated derivative‐free method for nonlinear monotone equations with an application

Abstract

In optimization theory, to speed up the convergence of iterative procedures, many mathematicians often use the inertial extrapolation method. In this article, based on the three‐term derivative‐free method for solving monotone nonlinear equations with convex constraints [Calcolo, 2016;53(2):133‐145], we design an inertial algorithm for finding the solutions of nonlinear equation with monotone and Lipschitz continuous operator. The convergence analysis is established under some mild conditions. Furthermore, numerical experiments are implemented to illustrate the behavior of the new algorithm. The numerical results have shown the effectiveness and fast convergence of the proposed inertial algorithm over the existing algorithm. Moreover, as an application, we extend this method to solve the LASSO problem to decode a sparse signal in compressive sensing. Performance comparisons illustrate the effectiveness and competitiveness of the method.

A family of inertial‐based derivative‐free projection methods with a correction step for constrained nonlinear equations and their applications

“…Therefore, this work will only focus on the DFP methods, such as spectral gradient projection methods 7,8 and conjugate gradient projection (CGP) methods. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The inertial step is a strategy that involves using information from the previous two iterations to calculate the current iteration, and it is one of the most effective methods for accelerating the convergence of iterative methods. This idea was originally proposed by Polyak, 28 who was inspired by an implicit discretization of a second-order-in-time dissipative dynamical system, represented by…”

“…Recently, scholars have applied the inertial strategy to the DFP methods for solving unconstrained/constrained nonlinear equations. For example, by building upon the three-term DFP method 17 for solving monotone nonlinear equations with convex constraints, Ibrahim et al 9 developed an inertial method to locate solutions of nonlinear equations featuring a Lipschitz continuous and monotone operator.…”

“…Building on the discussions above, we employ a novel acceleration technique to enhance the numerical performance of the DFP method. We also use the improvement strategy (9) as the correction step to endow the DFP method with nice theoretical properties. Furthermore, we develop a family of inertial-based DFP methods with a correction step (csIDFPM).…”

“…Due to their derivative‐free nature, the latter has attracted significant attention. Therefore, this work will only focus on the DFP methods, such as spectral gradient projection methods 7,8 and conjugate gradient projection (CGP) methods 9‐27 …”

“…Recently, scholars have applied the inertial strategy to the DFP methods for solving unconstrained/constrained nonlinear equations. For example, by building upon the three‐term DFP method 17 for solving monotone nonlinear equations with convex constraints, Ibrahim et al 9 developed an inertial method to locate solutions of nonlinear equations featuring a Lipschitz continuous and monotone operator. To achieve global convergence, the following condition is satisfied, $$$$ \sum \limits_{k=0}^{\infty}\kern0.20em {\alpha}_k{\left\Vert {x}_k-{x}_{k-1}\right\Vert}^2<\infty, $$$$ where $$$ \left\Vert \cdotp \right\Vert $$$ stands for the Euclidean norm on $$$ {\mathbb{R}}^n $$$.…”