A Modified Sufficient Descent Polak-Ribiére-Polyak Type Conjugate Gradient Method for Unconstrained Optimization Problems

In this paper, a modification to the Polak–Ribiere–Polyak (PRP) nonlinear conjugate gradient method is presented. The proposed method always generates a sufficient descent direction independent of the accuracy of the line search and the convexity of the objective function. Under appropriate conditions, the modified method is proved to possess global convergence under the Wolfe or Armijo-type line search. Moreover, the proposed methodology is adopted in the Hestenes–Stiefel (HS) and Liu–Storey (LS) methods. Extensive preliminary numerical experiments are used to illustrate the efficiency of the proposed method.

[1]  Boris Polyak.  The conjugate gradient method in extreme problems , 2015 .

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

[3]  Nicholas I. M. Gould,et al.  CUTE: constrained and unconstrained testing environment , 1995, TOMS.

[4]  Gaohang Yu,et al.  Global convergence of modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property , 2008 .

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

[6]  Liying Liu,et al.  The convergence properties of some new conjugate gradient methods , 2006, Appl. Math. Comput..

[7]  Yuhong Dai.  Nonlinear Conjugate Gradient Methods , 2011 .

[8]  Panayiotis E. Pintelas,et al.  A new class of spectral conjugate gradient methods based on a modified secant equation for unconstrained optimization , 2013, J. Comput. Appl. Math..

[9]  C. Storey,et al.  Efficient generalized conjugate gradient algorithms, part 1: Theory , 1991 .

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

[11]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[12]  E. Polak,et al.  Note sur la convergence de méthodes de directions conjuguées , 1969 .

[13]  Hongwei Liu,et al.  A New Three–Term Conjugate Gradient Method with Descent Direction for Unconstrained Optimization , 2016 .

[14]  Li Zhang,et al.  An improved Wei-Yao-Liu nonlinear conjugate gradient method for optimization computation , 2009, Appl. Math. Comput..

[15]  Weijun Zhou,et al.  A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence , 2006 .

[16]  W. Cheng.  A Two-Term PRP-Based Descent Method , 2007 .

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

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

[19]  Ya-Xiang Yuan.  Analysis on the conjugate gradient method , 1993 .

[20]  W. Hager,et al.  A SURVEY OF NONLINEAR CONJUGATE GRADIENT METHODS , 2005 .