Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems. (English) Zbl 1154.90623

Summary: It is well known that the sufficient descent condition is very important to the global convergence of the nonlinear conjugate gradient method. In this paper, some modified conjugate gradient methods which possess this property are presented. The global convergence of these proposed methods with the weak Wolfe-Powell (WWP) line search rule is established for nonconvex function under suitable conditions. Numerical results are reported.


90C52 Methods of reduced gradient type
90C06 Large-scale problems in mathematical programming


Full Text: DOI


[1] Dai Y.: A nonmonotone conjugate gradient algorithm for unconstrained optimization. J. Syst. Sci. Complex. 15, 139–145 (2002) · Zbl 1019.90039
[2] Dai Y., Liao L.Z.: New conjugacy conditions and related nonlinear conjugate methods. Appl. Math. Optim. 43, 87–101 (2001) · Zbl 0973.65050
[3] Dai Y., Yuan Y.: A nonlinear conjugate gradient with a strong global convergence properties. SIAM J. Optim. 10, 177–182 (2000) · Zbl 0957.65061
[4] Dai, Y., Yuan, Y.: Nonlinear conjugate gradient Methods. Shanghai Scientific and Technical Publishers (1998) · Zbl 0914.90219
[5] Dolan E.D., Moré J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002) · Zbl 1049.90004
[6] Fletcher R.: Practical method of optimization, vol I: unconstrained optimization, 2nd edn. Wiley, New York (1997)
[7] Fletcher R., Reeves C.: Function minimization bu conjugate gradients. Comput. J. 7, 149–154 (1964) · Zbl 0132.11701
[8] Gibert J.C., Nocedal J.: Global convergence properties of conugate gradient methods for optimization. SIAM J. Optim. 2, 21–42 (1992) · Zbl 0767.90082
[9] Hager W.W., Zhang H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005) · Zbl 1093.90085
[10] Hager W.W., Zhang H.: Algorithm 851: CG D ESCENT, A conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32, 113–137 (2006) · Zbl 1346.90816
[11] Hager W.W., Zhang H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35–58 (2006) · Zbl 1117.90048
[12] Hestenes M.R., Stiefel E.: Method of conjugate gradient for solving linear equations. J, Res. Nat. Bur. Stand. 49, 409–436 (1952) · Zbl 0048.09901
[13] Li G., Tang C., Wei Z.: New conjugacy condition and related new conjugate gradient methods for unconstrained optimization problems. J. Comput. Appl. Math. 202, 532–539 (2007) · Zbl 1116.65069
[14] Liu Y., Storey C.: Effcient generalized conjugate gradient algorithms part 1: theory. J. Optim. Theory Appl. 69, 17–41 (1992)
[15] Polak E., Ribiere G.: Note sur la xonvergence de directions conjugees. Rev. Francaise informat Recherche Operatinelle, 3e Annee 16, 35–43 (1969)
[16] Polyak B.T.: The conjugate gradient method in extreme problems. USSR Comp. Math. Math. Phys. 9, 94–112 (1969) · Zbl 0229.49023
[17] Powell, M.J.D.: Nonconvex minimization calculations and the conjugate gradient method. Lecture Notes in Mathematics, vol. 1066, pp. 122–141. Spinger, Berlin (1984) · Zbl 0531.65035
[18] Powell M.J.D.: Convergence properties of algorithm for nonlinear optimization. SIAM Rev. 28, 487–500 (1986) · Zbl 0624.90091
[19] Wei Z., Li G., Qi L.: New nonlinear conjugate gradient formulas for large-scale unconstrained optimization problems. Appl. Math. Comput. 179, 407–430 (2006) · Zbl 1106.65055
[20] Wei Z., Yao S., Liu L.: The convergence properties of some new conjugate gradient methods. Appl. Math. Comput. 183, 1341–1350 (2006) · Zbl 1116.65073
[21] Yu, G.H.: Nonlinear self-scaling conjugate gradient methods for large-scale optimization problems. Thesis of Doctor’s Degree, Sun Yat-Sen University (2007)
[22] Yu, G., Guan, L.: Chen Spectral conjugate gradient methods with sufficient descent property for large-scale unconstrained optimization. Optimization methods and software (in press, 2007)
[23] Yuan Y., Sun W.: Theory and methods of optimization. Science Press of China, Beijing (1999)
[24] Zhang L., Zhou W., Li D.: A descent modified Polak-Ribière-Polyak conjugate method and its global convergence. IMA J. Numer. Anal. 26, 629–649 (2006) · Zbl 1106.65056
[25] Zoutendijk G.: Nonlinear programming computational methods. In: Abadie, J.(eds) Integer and nonlinear programming, pp. 37–86. North-Holland, Amsterdam (1970) · Zbl 0336.90057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.