On the convergence properties of the modified Polak--Ribiere--Polyak method with the standard Armijo line search

Abstract

Zhang et al. [IMA J. Numer. Anal., 26 (2006) 629--640] proposed a modified Polak--Ribiere--Polyak method for non-convex optimization and proved its global convergence with some backtracking type line search. We further study its convergence properties. Under the standard Armijo line search condition, we show that the modified Polak--Ribiere--Polyak method has better global convergence property and locally \(R\)-linear convergence rate for non-convex minimization. Some preliminary numerical results are also reported to show its efficiency. References C. Li, A conjugate gradient type method for the nonnegative constraints optimization problems, J. Appl. Math. , Volume 2013, Article ID 986317, 6 pages. D. Li and B. Tian, $n$-step quadratic convergence of the mprp method with a restart strategy, J. Comput. Appl. Math. , 235 (2011) 4978--4990. J. J. More, B. S. Garbow and K. H. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Softw. , 7 (1981) 17--41. E. Polak and G. Ribiere, Note sur la convergence de methodes de directions conjuguees, Rev. Fr. Inform. Rech. Oper. , 16 (1969) 35--43. B. T. Polyak, The conjugate gradient method in extreme problems, USSR Comput. Math. Math. Phys. , 9 (1969) 94--112. L. Zhang, W. Zhou and D. Li, A descent modified Polak--Ribiere--Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal. , 26 (2006) 629--640. H. Zhu, Y. Xiao and S. Wu, Large sparse signal recovery by conjugate gradient algorithm based on smoothing technique, Comput. Math. Appl. , 66 (2013) 24--32.