Three modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property

Min Sun,Jing Liu

doi:10.1186/s13660-015-0649-9

Abstract

In this paper, three modified Polak-Ribiere-Polyak (PRP) conjugate gradient methods for unconstrained optimization are proposed. They are based on the two-term PRP method proposed by Cheng (Numer. Funct. Anal. Optim. 28:1217-1230, 2007), the three-term PRP method proposed by Zhang et al. (IMA J. Numer. Anal. 26:629-640, 2006), and the descent PRP method proposed by Yu et al. (Optim. Methods Softw. 23:275-293, 2008). These modified methods possess the sufficient descent property without any line searches. Moreover, if the exact line search is used, they reduce to the classical PRP method. Under standard assumptions, we show that these three methods converge globally with a Wolfe line search. We also report some numerical results to show the efficiency of the proposed methods.

Highlights

Consider the unconstrained optimization problem: min f (x), x ∈ Rn, ( )where f : Rn → R is continuously differentiable, and its gradient g(x) is available
We focus our attention on the PRP method, in which the parameter βk is given by βkPRP
We take a little modification to the βkPRP and propose three modified PRP methods based on the iterate directions dkCTPRP, dkZTPRP, and dkYTPRP, which possess the sufficient descent property for any line search and global convergence with a Wolfe line search

Summary

Introduction

Consider the unconstrained optimization problem: min f (x), x ∈ Rn, ( )where f : Rn → R is continuously differentiable, and its gradient g(x) is available. The Wolfe line search is finding a step size αk satisfying f (xk + αkdk) ≤ f (xk) + ραkgk dk, ( ) Polak and Ribière [ ] proved that the PRP method with the exact line search is globally convergent under a strong convexity assumption for the objective function f . Gilbert and Nocedal [ ] conducted an elegant analysis and showed that the PRP method is globally convergent if βkPRP is restricted to be non-negative (denoted βkPRP+) and αk is determined by a line search step satisfying the sufficient descent condition gk dk ≤ –c gk , c > , ( )

Results

Conclusion