Shi, Z.-J. A new class of memory gradient methods with inexact line searches. (English) Zbl 1072.65086 J. Numer. Math. 13, No. 1, 53-71 (2005). The aim of this study is to present a new class of memory gradient methods with inexact line searches for unconstrained minimization problems. In order to accelerate the convergence rate of algorithms and to avoid additional function evaluations, the proposed techniques use more previous iterative information than other methods to generate a search direction and perform inexact line searches to select an appropriate step-size at each iteration. The new method does need storing and computing matrices (associated with the Hessian of objective functions) at each iteration, as in ‘memory-less’ quasi-Newton methods, thus it proves to be suitable especially to solve large scale unconstrained minimization problems.The author shows that, under weak mild conditions, the new class of memory gradient methods has the desired property of global convergence. The convergence rate is investigated in the case of special objective functions, and the method is proved to converge more stably than similar approaches, being available to solve some ill-conditioned problems. A series of comparative numerical experiments illustrates the effectiveness of the proposed memory gradient methods in solving unconstrained minimization problems. Using principally the Armijo’s rule and Wolfe’s rule as inexact line search rules, the computational results are favourably compared to other classical methods such as Fletcher-Reeves, Polak-Ribière, Hestenes-Stiefel conjugate gradient methods and steepest descent method. The new algorithm is shown to behave effectively stable, and to provide robust and practical computational qualities when solving large scale minimization problems. Reviewer: Silvia Curteanu (Iaşi) Cited in 2 Documents MSC: 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming 90C52 Methods of reduced gradient type 90C53 Methods of quasi-Newton type 90C06 Large-scale problems in mathematical programming Keywords:unconstrained optimization; memory gradient method; inexact line search; large scale minimization; global convergence; comparison of methods; Polak-Ribière method; convergence acceleration; Fletcher-Reeves method; algorithms; numerical experiments; Armijo’s rule; Wolfe’s rule Software:minpack PDFBibTeX XMLCite \textit{Z. J. Shi}, J. Numer. Math. 13, No. 1, 53--71 (2005; Zbl 1072.65086) Full Text: DOI References: [1] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Academic Press Inc., 1982. · Zbl 0572.90067 [2] DOI: 10.1007/BF00935546 · Zbl 0421.49030 · doi:10.1007/BF00935546 [3] DOI: 10.1093/imamat/11.3.317 · Zbl 0259.65060 · doi:10.1093/imamat/11.3.317 [4] DOI: 10.1016/S0377-0427(02)00701-X · Zbl 1025.65035 · doi:10.1016/S0377-0427(02)00701-X [5] DOI: 10.1016/0377-0427(95)00178-6 · Zbl 0856.65073 · doi:10.1016/0377-0427(95)00178-6 [6] DOI: 10.1016/0898-1221(95)00230-8 · Zbl 0874.65046 · doi:10.1016/0898-1221(95)00230-8 [7] DOI: 10.1016/0377-0427(94)90309-3 · Zbl 0807.65062 · doi:10.1016/0377-0427(94)90309-3 [8] Ford I. A, Optim. Meth. Software 2 pp 357– (1993) [9] DOI: 10.1137/0802003 · Zbl 0767.90082 · doi:10.1137/0802003 [10] Grippo S., Math. Prog. 78 pp 375– (1997) [11] DOI: 10.1007/BF01385810 · Zbl 0724.90060 · doi:10.1007/BF01385810 [12] DOI: 10.1007/BF00935882 · Zbl 0238.65033 · doi:10.1007/BF00935882 [13] DOI: 10.1007/BF01585109 · Zbl 0505.90064 · doi:10.1007/BF01585109 [14] DOI: 10.1023/A:1021723128049 · Zbl 0956.90047 · doi:10.1023/A:1021723128049 [15] DOI: 10.1007/BF01589116 · Zbl 0696.90048 · doi:10.1007/BF01589116 [16] DOI: 10.1016/S0898-1221(01)00229-2 · Zbl 0979.90108 · doi:10.1016/S0898-1221(01)00229-2 [17] DOI: 10.1007/s10107-002-0367-1 · Zbl 1049.90126 · doi:10.1007/s10107-002-0367-1 [18] DOI: 10.1016/S0893-9659(01)00162-8 · Zbl 1175.90419 · doi:10.1016/S0893-9659(01)00162-8 [19] DOI: 10.1145/355934.355936 · Zbl 0454.65049 · doi:10.1145/355934.355936 [20] J. Nocedal and S. J. Wright, Numerical Optimization. Springer-Verlag, New York, 1999. · Zbl 0930.65067 [21] Acta Numerica 7 pp 287– (1998) · Zbl 0894.00025 [22] J. Appl. Sci. 21 (3) pp 241– (2003) [23] Advances in Math. 31 (1) pp 47– (2002) [24] Shi A, Asia-Pacific J. Operational Research 20 pp 275– (2003) [25] DOI: 10.1016/j.amc.2003.08.058 · Zbl 1072.65087 · doi:10.1016/j.amc.2003.08.058 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.