Notes on the Dai-Yuan-Yuan modified spectral gradient method. (English) Zbl 1195.65081

Summary: We give some notes on the two modified spectral gradient methods which were developed by Y. Dai, J. Yuan and Y.-X. Jinyun [Comput. Optim. Appl. 22, No. 1, 103–109 (2002; Zbl 1008.90056)]. These notes present the relationship between their stepsize formulae and some new secant equations in the quasi-Newton method. In particular, we also introduce another two new choices of stepsize. By using an efficient nonmonotone line search technique, we propose some new spectral gradient methods. Under some mild conditions, we show that these proposed methods are globally convergent. Numerical experiments on a large number of test problems from the CUTEr library are also reported, which show that the efficiency of these proposed methods.


65K05 Numerical mathematical programming methods
90C26 Nonconvex programming, global optimization
90C53 Methods of quasi-Newton type


Zbl 1008.90056


Full Text: DOI


[1] Barzilai, J.; Borwein, J. M., Two point step size gradient method, IMA J. Numer. Anal., 8, 141-148 (1988) · Zbl 0638.65055
[2] Grippo, L.; Sciandrone, M., Nonmonotone globalization techniques for the Barzilai-Borwein gradient method, Comput. Optim. Appl., 23, 134-169 (2002) · Zbl 1028.90061
[3] Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal., 23, 707-716 (1986) · Zbl 0616.65067
[4] Raydan, M., The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7, 26-33 (1997) · Zbl 0898.90119
[5] Birgin, E. G.; Martínez, J. M.; Raydan, M., Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optim., 10, 1196-1211 (2000) · Zbl 1047.90077
[6] Xiao, Y.; Hu, Q., Subspace Barzilai-Borwein gradient method for large-scale bound constrained optimization, Appl. Math. Optim., 58, 275-290 (2008) · Zbl 1173.90584
[8] Friedlander, A.; Martínez, J. M.; Molina, B., Gradient method with restarts and generalizations, SIAM J. Numer. Anal., 36, 275-289 (1999) · Zbl 0940.65032
[9] Dai, Y. H.; Hager, W. W.; Schittkowski, K.; Zhang, H., The cyclic Barzilai-Borwein method for unconstrained optimization, IMA J. Numer. Anal., 26, 604-627 (2006) · Zbl 1147.65315
[10] Dai, Y. H.; Yuan, J.; Yuan, Y. X., Modified two-point stepsize gradient method for unconstrained optimization, Comput. Optim. Appl., 22, 103-109 (2002) · Zbl 1008.90056
[11] Wei, Z.; Li, G.; Qi, L., New quasi-Newton methods for unconstrained optimization problems, Appl. Math. Comput., 175, 1156-1188 (2006) · Zbl 1100.65054
[12] Wei, Z.; Yu, G.; Yuan, G.; Lian, Z., The superlinear convergence of a modified BFGS-type method for unconstrained optimization, Comput. Optim. Appl., 29, 315-332 (2004) · Zbl 1070.90089
[13] Zhang, J. Z.; Deng, N. Y.; Chen, L. H., New quasi-Newton equation and related methods for unconstrained optimization, J. Optim. Theory Appl., 102, 147-167 (1999) · Zbl 0991.90135
[14] Xiao, Y.; Wei, Z.; Wang, Z., A limited memory BFGS-type method for large-scale unconstrained optimization, Comput. Math. Appl., 56, 1001-1009 (2008) · Zbl 1155.90441
[15] Zhang, H.; Hager, W. W., A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim., 14, 1043-1056 (2004) · Zbl 1073.90024
[16] Dai, Y. H., On the nonmonotone line search, J. Optim. Theory Appl., 112, 315-330 (2002) · Zbl 1049.90087
[17] Conn, A. R.; Gould, N. I.M.; Toint, Ph. L., CUTE: constrained and unconstrained testing environment, ACM Trans. Math. Softw., 21, 123-160 (1995) · Zbl 0886.65058
[18] Dolan, E. D.; Moré, J. J., Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213 (2002) · Zbl 1049.90004
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