Some global convergence properties of the Wei-Yao-Liu conjugate gradient method with inexact line search

Abstract In [Z.Wei, S. Yao, L. Liu, The convergence properties of some new conjugate gradient methods, Applied Mathematics and Computation 183 (2006) 1341–1350], Wei et al. proposed a new conjugate gradient method called WYL method which has good numerical experiments and some excellent properties such as β k WYL ⩾ 0 . In this paper, we prove that while t k ⩽ 1 - c 2 L ‖ g k ‖ 2 ‖ d k ‖ 2 , the sufficient descent condition can be satisfied at each iteration of the WYL method. Based on this, the global convergence of the WYL method with the standard Armijo line search and the generalized Wolfe–Powell line search is established.

[1]  Guoyin Li,et al.  New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems , 2006, Appl. Math. Comput..

[2]  Zengxin Wei,et al.  A note about WYL's conjugate gradient method and its applications , 2007, Appl. Math. Comput..

[3]  Ya-Xiang Yuan,et al.  Convergence properties of the Fletcher-Reeves method , 1996 .

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

[5]  Guoyin Li,et al.  A new version of the Liu-Storey conjugate gradient method , 2007, Appl. Math. Comput..

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

[7]  Zengxin Wei,et al.  New line search methods for unconstrained optimization , 2009 .

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

[9]  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..

[10]  Jinhua Guo,et al.  A new family of conjugate gradient methods , 2009 .

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

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

[13]  Yu-Hong Dai.  New properties of a nonlinear conjugate gradient method , 2001, Numerische Mathematik.

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

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

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

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

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

[19]  Zengxin Wei,et al.  The proof of the sufficient descent condition of the Wei-Yao-Liu conjugate gradient method under the strong Wolfe-Powell line search , 2007, Appl. Math. Comput..