Sellami, Badreddine; Chiheb Eddine Sellami, Mohamed Global convergence of a modified Fletcher-Reeves conjugate gradient method with Wolfe line search. (English) Zbl 1464.65058 Asian-Eur. J. Math. 13, No. 4, Article ID 2050081, 10 p. (2020). Summary: In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. We propose a modified Fletcher-Reeves (abbreviated FR) [R. Fletcher and C. M. Reeves, Comput. J. 7, 149–154 (1964; Zbl 0132.11701)] conjugate gradient algorithm satisfying a parametrized sufficient descent condition with a parameter \(\delta_k\) is proposed. The parameter \(\delta_k\) is computed by means of the conjugacy condition, thus an algorithm which is a positive multiplicative modification of the Hestenes and Stiefel (abbreviated HS) [M. R. Hestenes and E. Stiefel, J. Res. Natl. Bur. Stand. 49, 409–436 (1952; Zbl 0048.09901)] algorithm is obtained, which produces a descent search direction at every iteration that the line search satisfies the Wolfe conditions. Under appropriate conditions, we show that the modified FR method with the strong Wolfe line search is globally convergent of uniformly convex functions. We also present extensive preliminary numerical experiments to show the efficiency of the proposed method. MSC: 65K05 Numerical mathematical programming methods 90C25 Convex programming 90C26 Nonconvex programming, global optimization 90C27 Combinatorial optimization 90C30 Nonlinear programming Keywords:unconstrained optimization; conjugate gradient method; sufficient descent; condition; line search; global convergence Citations:Zbl 0132.11701; Zbl 0048.09901 Software:CUTE; SCALCG; CUTEr PDFBibTeX XMLCite \textit{B. Sellami} and \textit{M. Chiheb Eddine Sellami}, Asian-Eur. J. Math. 13, No. 4, Article ID 2050081, 10 p. (2020; Zbl 1464.65058) Full Text: DOI References: [1] Andrei, N., A scaled BFGs preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett.20(6) (2007) 645-650. · Zbl 1116.90114 [2] Andrei, N., Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl.38(3) (2007) 401-416. · Zbl 1168.90608 [3] Andrei, N., Scaled memoryless bfgs preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw.22(4) (2007) 561-571. · Zbl 1270.90068 [4] Birgin, E. G. and Martínez, J. M., A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim.43(2) (2001) 117-128. · Zbl 0990.90134 [5] Bongartz, I., Conn, A. R., Gould, N. and Toint, P. L., Cute: Constrained and unconstrained testing environment, ACM Trans. Math. Softw. (TOMS)21(1) (1995) 123-160. · Zbl 0886.65058 [6] Dai, Y. and Yuan, Y., An efficient hybrid conjugate gradient method for unconstrained optimization, Ann. Operat. Res.103(1-4) (2001) 33-47. · Zbl 1007.90065 [7] Dai, Y.-H. and Liao, L.-Z., New conjugacy conditions and related nonlinear conjugate gradient methods, Appl. Math. Optim.43(1) (2001) 87-101. · Zbl 0973.65050 [8] Dai, Y.-H. and Ni, Q., Testing diffierent conjugate gradient methods for large-scale unconstrained optimization, J. Comput. Math. (2003) 311-320. · Zbl 1041.65048 [9] Dai, Y.-H. and Yuan, Y., A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim.10(1) (1999) 177-182. · Zbl 0957.65061 [10] Fletcher, R. and Reeves, C. M., Function minimization by conjugate gradients, Comput. J.7(2) (1964) 149-154. · Zbl 0132.11701 [11] Hestenes, M. R. and Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Standards49 (1952) 409-436. · Zbl 0048.09901 [12] Hu, Y. and Storey, C., Global convergence result for conjugate gradient methods, J. Optim. Theory Appl.71(2) (1991) 399-405. · Zbl 0794.90063 [13] Liu, Y. and Storey, C., Efficient generalized conjugate gradient algorithms, part 1: Theory, J. Optim. Theory Appl.69(1) (1991) 129-137. · Zbl 0702.90077 [14] Polyak, B. T., The conjugate gradient method in extremal problems, USSR Comput. Math. Math. Phys.9(4) (1969) 94-112. · Zbl 0229.49023 [15] Rivaie, M., Mamat, M., Mohd, I. and Fauzi, M., Modified Hestenes-Steifel conjugate gradient coefficient for unconstrained optimization, J. Interdiscipl. Math.13(3) (2010) 241-251. · Zbl 1225.90155 [16] Shanno, D. F., Conjugate gradient methods with inexact searches, Math. Oper. Res.3(3) (1978) 244-256. · Zbl 0399.90077 [17] Touati-Ahmed, D. and Storey, C., Efficient hybrid conjugate gradient techniques, J. Optim. Theory Appl.64(2) (1990) 379-397. · Zbl 0666.90063 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.