Globally convergent Polak-Ribière-Polyak conjugate gradient methods under a modified Wolfe line search

It is well known that global convergence has not been established for the Polak-Ribiere-Polyak (PRP) conjugate gradient method using the standard Wolfe conditions. In the convergence analysis of PRP method with Wolfe line search, the (sufficient) descent condition and the restriction @b"k>=0 are indispensable (see [4,7]). This paper shows that these restrictions could be relaxed. Under some suitable conditions, by using a modified Wolfe line search, global convergence results were established for the PRP method. Some special choices for @b"k which can ensure the search direction's descent property were also discussed in this paper. Preliminary numerical results on a set of large-scale problems were reported to show that the PRP method's computational efficiency is encouraging.

[1]  C. M. Reeves,et al.  Function minimization by conjugate gradients , 1964, Comput. J..

[2]  B. V. Shah,et al.  Integer and Nonlinear Programming , 1971 .

[3]  Guan,et al.  A Globally Convergent Polak-Ribière-Polyak Conjugate Gradient Method with Armijo-Type Line Search , 2006 .

[4]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[5]  Gonglin Yuan,et al.  Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems , 2009, Optim. Lett..

[6]  M. Al-Baali.  Descent Property and Global Convergence of the Fletcher—Reeves Method with Inexact Line Search , 1985 .

[7]  Ya-Xiang Yuan,et al.  Convergence Properties of Nonlinear Conjugate Gradient Methods , 1999, SIAM J. Optim..

[8]  C. Storey,et al.  Global convergence result for conjugate gradient methods , 1991 .

[9]  D. Touati-Ahmed,et al.  Efficient hybrid conjugate gradient techniques , 1990 .

[10]  Yu-Hong Dai,et al.  Conjugate Gradient Methods with Armijo-type Line Searches , 2002 .

[11]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

[12]  M. Powell.  Nonconvex minimization calculations and the conjugate gradient method , 1984 .

[13]  Jorge Nocedal,et al.  Global Convergence Properties of Conjugate Gradient Methods for Optimization , 1992, SIAM J. Optim..

[14]  Zengxin Wei,et al.  A descent nonlinear conjugate gradient method for large-scale unconstrained optimization , 2007, Appl. Math. Comput..

[15]  Guoyin Li,et al.  Global convergence of the Polak-Ribière-Polyak conjugate gradient method with an Armijo-type inexact line search for nonconvex unconstrained optimization problems , 2008, Math. Comput..

[16]  Boris Polyak.  The conjugate gradient method in extremal problems , 1969 .

[17]  William W. Hager,et al.  A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search , 2005, SIAM J. Optim..

[18]  Luigi Grippo,et al.  Convergence conditions, line search algorithms and trust region implementations for the Polak–Ribière conjugate gradient method , 2005, Optim. Methods Softw..

[19]  Luigi Grippo,et al.  A globally convergent version of the Polak-Ribière conjugate gradient method , 1997, Math. Program..

[20]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[21]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[22]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..