Convergence of the Polak–Ribiére–Polyak conjugate gradient method

• Published in: Nonlinear analysis Vol. 66; no. 6; pp. 1428 - 1441
• Main Authors: Shi, Zhen-Jun, Shen, Jie
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 15.03.2007; Elsevier
• Subjects: Applied sciences; Calculus of variations and optimal control; Exact sciences and technology; Global convergence; Mathematical analysis; Mathematical programming; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming; Numerical methods in mathematical programming, optimization and calculus of variations; Operational research and scientific management; Operational research. Management science; PRP conjugate gradient method; Sciences and techniques of general use; Unconstrained optimization; PRP conjugate gradient method; Global convergence; 65K05; 90C30; Unconstrained optimization; 49M37; Conjugate gradient method; 49M37; 65K05; 90C30; Unconstrained optimization; PRP conjugate gradient method; Global convergence; Nonlinear analysis
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2006.02.001

In this paper, we consider the global convergence of the Polak–Ribiére–Polyak (abbreviated PRP) conjugate gradient method for unconstrained optimization problems. A new Armijo-type line search is proposed for the original PRP method and some convergence properties are given under some mild conditions. The new Armijo-type line search can make the PRP method choose a suitable initial step size so as to decrease the function evaluations at each iteration and improve the performance of the PRP method. Numerical results show that the PRP method with the new Armijo-type line search is more efficient than other similar methods in practical computation.