A Scaled Conjugate Gradient Method for Solving Monotone Nonlinear Equations with Convex Constraints

Sheng Wang,Hongbo Guan

doi:10.1155/2013/286486

Abstract

Based on the Scaled conjugate gradient (SCALCG) method presented by Andrei (2007) and the projection method presented by Solodov and Svaiter, we propose a SCALCG method for solving monotone nonlinear equations with convex constraints. SCALCG method can be regarded as a combination of conjugate gradient method and Newton-type method for solving unconstrained optimization problems. So, it has the advantages of the both methods. It is suitable for solving large-scale problems. So, it can be applied to solving large-scale monotone nonlinear equations with convex constraints. Under reasonable conditions, we prove its global convergence. We also do some numerical experiments show that the proposed method is efficient and promising.

Highlights

In this paper, we consider the following convex constrained monotone equations:F (x) = 0, x ∈ Ω, (1)where F : Rn → Rn is a continuous and monotone function
Based on the Scaled conjugate gradient (SCALCG) method presented by Andrei (2007) and the projection method presented by Solodov and Svaiter, we propose a SCALCG method for solving monotone nonlinear equations with convex constraints
The strictly convex function must exists unique minimum point, so the minimum point is a stable point of the convex functions, namely, the point which the gradient vector ∇f(x) = 0

Summary

Introduction

We consider the following convex constrained monotone equations:. where F : Rn → Rn is a continuous and monotone function. We consider the following convex constrained monotone equations:. Zhang and Zhou [3] combined the spectral gradient method and the projection method of Solodov and Svaiter, proposed a spectral gradient projection method. Wang et al [7] extended Solodov and Svaiter’s projection method to solve monotone equations with convex constraints. Yu et al [8] proposed a spectral gradient projection algorithm for monotone nonlinear equations with convex constraints by combining a modified spectral gradient method and the projection method. Xiao and Zhu [9] extended CG DESCENT to solve large-scale nonlinear convex constrained monotone equations in compressive sensing by combining with the projection method of Solodov and Svaiter.

The Method

Convergence Analysis

Numerical Experiments

Conclusions