A Family of Derivative-Free Conjugate Gradient Methods for Constrained Nonlinear Equations and Image Restoration

Abdulkarim Hassan Ibrahim,Wiyada Kumam,Poom Kumam

doi:10.1109/access.2020.3020969

Abstract

In this paper, a derivative-free conjugate gradient method for solving nonlinear equations with convex constraints is proposed. The proposed method can be viewed as an extension of the three-term modified Polak-Ribiere-Polyak method (TTPRP) and the three-term Hestenes-Stiefel conjugate gradient method (TTHS) using the projection technique of Solodov and Svaiter [Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, 1998, 355-369]. The proposed method adopts the adaptive line search scheme proposed by Ou and Li [Journal of Applied Mathematics and Computing 56.1-2 (2018): 195-216] which reduces the computational cost of the method. Under the assumption that the underlying operator is Lipschitz continuous and satisfies a weaker condition of monotonicity, the global convergence of the proposed method is established. Furthermore, the proposed method is extended to solve image restoration problem arising in compressive sensing. Numerical results are presented to demonstrate the effectiveness of the proposed method.

Highlights

Let φ : Rn → Rn be a map, and be a nonempty closed, convex set of Rn
Inspired by [32], as an attempt, using the projection method [12] and the modified line search scheme proposed in [34], we effectively extend the threeterm modified Polak-Ribiére-Polyak method (TTPRP) method and the three-term Hestenes-Stiefel conjugate gradient method (TTHS) to solve the nonlinear equation with convex constraint (1)
Based on the performance profile of the number of iterations, it is clear that DF-PRPMHS outperforms comparative methods as we can see from Figure 1 that DF-PRPMHS is a better algorithm for solving (1) since it solved about 62% of test problems with fewer iterations compared to New hybrid conjugate gradient projection method (NHCGPM), modified Hestenes-Stiefel projection method (MHSPM) and selfadaptive three-term conjugate gradient method (STTCGM) which solved 15%, 10% and 18% of test problems with less number of iterations

Summary

INTRODUCTION

Let φ : Rn → Rn be a map, and be a nonempty closed, convex set of Rn. We consider the following problem: finding a vector v such that φ(v) = 0, v ∈. Inspired by [32], as an attempt, using the projection method [12] and the modified line search scheme proposed in [34], we effectively extend the TTPRP method and the TTHS to solve the nonlinear equation with convex constraint (1). Our proposed search direction can be viewed as an affine combination of the derivative-free version of TTPRP and TTHS method, which satisfies the descent condition. ALGORITHM we begin by focusing on the conjugate gradient method designed to solve the following unconstrained optimization problem: min{φ(v) : v ∈ Rn},. Zhang, Zhou and Li [29] proposed a three term Polak-Ribiére-Polyak (TTPRP) conjugate gradient method with its search direction defined as follows.

CONVERGENCE ANALYSIS

NUMERICAL EXPERIMENT

APPLICATION TO IMAGE RESTORATION PROBLEMS

Findings

CONCLUSION