A Modified Spectral Gradient Projection Method for Solving Non-linear Monotone Equations with Convex Constraints and Its Application

Li Zheng,Yong Liang,Lei Yang

doi:10.1109/access.2020.2990298

Li Zheng, Yong Liang + Show 1 more

Open Access

https://doi.org/10.1109/access.2020.2990298

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Journal: IEEE Access	Publication Date: Jan 1, 2020
Citations: 17	License type: CC BY 4.0

Affiliation: Macau University of Science and Technology

Abstract

In this paper, we propose a derivative free algorithm for solving non-linear monotone equations with convex constraints. The proposed algorithm combines the method of spectral gradient and the projection method. We also modify the backtracking line search technique. The global convergence of the proposed method is guaranteed, under the mild conditions. Further, the numerical experiments show that the large-scale non-linear equations with convex constraints can be effectively solved with our method. The $L_{1}$ -norm regularized problems in signal reconstruction are studied by using our method.

Highlights

In this paper, we focus on finding the solution of the following non-linear systematic equations F(x) = 0, x ∈, (1)where F: Rn → Rn is a given functions and is a non-empty convex set.If = Rn, Eq(1) is an unconstrained problem, which could be solved by many methods, such as Newton method, Quasi-Newton method, Conjugate Gradient method, Fixed Point method and their variants
The proposed derivative free method is suitable to solve large-scale non-linear equations as no sub-problem needs to be solved, which is different from the classical projection method
In this paper, a new derivative free algorithm MSGP for solving non-linear equations with convex constraints is proposed by combining the method of spectral gradient and the projection method

Summary

INTRODUCTION

We focus on finding the solution of the following non-linear systematic equations. L. Zheng et al.: Modified Spectral Gradient Projection Method for Solving Non-Linear Monotone Equations. Due to the non-expansion of the projection operator, Wang et al [23] proposed a method to solve the convex constrained problem of (1) by projecting the predictor-corrector point in the constraint sets. Awwal et al [25] proposed a modified Hestenes-Stiefel (HS) spectral conjugate gradient (CG) method to solve monotone non-linear equations with convex constraints efficiently. In this paper, inspired by the works of [24] and [25], we further investigate the projection method with a modified SG parameter and propose a derivative free algorithm for solving non-linear monotone equations with convex constraints.

NUMERICAL EXPERIMENTS AND APPLICATION

NUMERICAL EXPERIMENTS

APPLICATION IN SIGNAL RECONSTRUCTION

CONCLUSION

A PROOF OF LEMMA 7

A PROOF OF THEOREM 8