A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate

Min Sun,Jing Liu

doi:10.1155/2014/386030

Abstract

A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factorγis attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.

Highlights

Let F : Rn → Rn be a continuous nonlinear mapping and C a nonempty closed convex set of Rn
Suppose that F(⋅) is monotone and let {xk} and {yk} be the sequences generated by Algorithm 1; {xk} and {yk} are both bounded; it holds that kl→im∞αk󵄩󵄩󵄩󵄩dk󵄩󵄩󵄩󵄩2 = 0
We prove the global convergence of Algorithm 1

Summary

Introduction

Let F : Rn → Rn be a continuous nonlinear mapping and C a nonempty closed convex set of Rn. A drawback of this method is that it needs to solve a linear equation inexactly at each iteration, and its variants [8, 10] have this drawback It is well-known that the spectral gradient method and the conjugate gradient method are two efficient methods for solving large-scale unconstrained optimization problems due to their simplicity and low storage. Zhang and Zhou [12] presented a spectral gradient projection method (SGP) for solving unconstrained monotone equations, which does not utilize any merit function. 13], we propose a matrix-free method for solving nonlinear constrained equations, which can be viewed as a combination of the well-known PRP conjugate gradient method and the famous hyperplane projection method, and it possesses linear convergence rate under standard conditions.

Algorithm and Convergence Analysis

Convergence Rate

Numerical Results

Method

Conclusions