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Abstract and Applied Analysis
Volume 2014, Article ID 386030, 6 pages
Research Article

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

1School of Mathematics and Statistics, Zaozhuang University, Shandong 277160, China
2School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, China

Received 24 January 2014; Accepted 8 May 2014; Published 19 May 2014

Academic Editor: Vladimir Danilov

Copyright © 2014 Min Sun and Jing Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 factor is attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.