A class of conjugate gradient methods for convex constrained monotone equations

Abstract

The recent designed non-linear conjugate gradient method of Dai and Kou [SIAM J Optim. 2013;23:296–320] is very efficient currently in solving large-scale unconstrained minimization problems due to its simpler iterative form, lower storage requirement and its closeness to the scaled memoryless BFGS method. Just because of these attractive properties, this method was extended successfully to solve higher dimensional symmetric non-linear equations in recent years. Nevertheless, its numerical performance in solving convex constrained monotone equations has never been explored. In this paper, combining with the projection method of Solodov and Svaiter, we develop a family of non-linear conjugate gradient methods for convex constrained monotone equations. The proposed methods do not require the Jacobian information of equations, and even they do not store any matrix in each iteration. They are potential to solve non-smooth problems with higher dimensions. We prove the global convergence of the class of the proposed methods and establish its R-linear convergence rate under some reasonable conditions. Finally, we also do some numerical experiments to show that the proposed methods are efficient and promising.

Keywords:

• Monotone equations
• conjugate gradient method
• projection method
• global convergence
• sufficient descent condition

Acknowledgements

We would like to thank three anonymous referees and the associate editor for their useful comments and suggestions which improved this paper greatly.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Y. Xiao was supported by the Major State Basic Research Development Program of China (973 Program) [grant number 2015CB856003]; the National Natural Science Foundation of China [grant number 11471101]; Program for Science and Technology Innovation Talents in Universities of Henan Province [grant number 13HASTIT050].