A gradient-related algorithm with inexact line searches. (English) Zbl 1050.65061
The authors present a gradient-related algorithm for solving large-scale unconstrained optimization problems. It is a feasible direction method. The line search direction is selected to be a combination of the current gradient and some previous search directions. The step-size is determined by using various inexact line searches.
Reviewer: Du Ding-Zhu (Minneapolis)
Keywords:
gradient-related algorithm; inexact line search; feasible direction method; unconstrained optimization; convergenceReferences:
| [1] | Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods (1982), Academic Press: Academic Press New York · Zbl 0453.65045 |
| [2] | Cohen, A. I., Stepsize analysis for descent methods, J. Optim. Theory Appl., 33, 2, 187-205 (1981) · Zbl 0421.49030 |
| [3] | Cantrell, J. W., On the relation between the memory gradient and the Fletcher-Reeves method, J. Optim. Theory Appl., 4, 1, 67-71 (1969) · Zbl 0167.08804 |
| [4] | Cragg, E. E.; Levy, A. V., Study on a supermemory gradient method for the minimization of functions, J. Optim. Theory Appl., 4, 3, 191-205 (1969) · Zbl 0172.19002 |
| [5] | Y.H. Dai, Y. Yuan, Nonlinear Conjugate Gradient Method, Shanghai Science and Technology Press, Shanghai, 1999.; Y.H. Dai, Y. Yuan, Nonlinear Conjugate Gradient Method, Shanghai Science and Technology Press, Shanghai, 1999. · Zbl 0957.65061 |
| [6] | Dai, Y. H.; Yuan, Y., A nonlinear conjugate gradient with a strong Wolfe convergence property, SIAM J. Optim., 10, 177-182 (1999) · Zbl 0957.65061 |
| [7] | Dai, Y. H.; Yuan, Y., Convergence properties of the Fletcher-Reeves method, IMA J. Numer. Anal., 16, 155-164 (1996) · Zbl 0851.65049 |
| [8] | Dixon, L. C.W., Conjugate directions without line searches, J. Institute of Math. Appl., 11, 317-328 (1973) · Zbl 0259.65060 |
| [9] | R. Fletcher, Practical Methods of Optimization, Vol. 1, Unconstrained Optimization, New York, Wiley, 1987.; R. Fletcher, Practical Methods of Optimization, Vol. 1, Unconstrained Optimization, New York, Wiley, 1987. · Zbl 0905.65002 |
| [10] | Flecther, R.; Revees, C. M., Function minimization by conjugate gradients, Comput. J., 7, 149-154 (1964) · Zbl 0132.11701 |
| [11] | Grippo, L.; Lucidi, S., A globally convergent version of the Polak-Ribière conjugate gradient method, Math. Programming, 78, 375-391 (1997) · Zbl 0887.90157 |
| [12] | Gilbert, J. C.; Nocedal, J., Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim., 2, 1, 21-42 (1992) · Zbl 0767.90082 |
| [13] | Grippo, L.; Lampariello, F.; Lucidi, S., A class of nonmonotone stability methods in unconstrained optimization, Numer. Math., 62, 779-805 (1991) · Zbl 0724.90060 |
| [14] | Horst, R.; Pardalos, P. M.; Thoai, N. V., Introduction to Global Optimization (1995), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Holland · Zbl 0836.90134 |
| [15] | Hu, Y. F.; Storey, C., Global convergence restart for conjugate gradient methods, J. Optim. Theory Appl., 72, 1, 399-405 (1991) · Zbl 0794.90063 |
| [16] | M.R. Hestenes, Conjugate Direction Methods in Optimization, Springer, New York Inc., 1980.; M.R. Hestenes, Conjugate Direction Methods in Optimization, Springer, New York Inc., 1980. · Zbl 0439.49001 |
| [17] | Huang, H. Y.; Chambliss, J. P., Quadratically convergent algorithms and one-dimensional search schemes, J. Optim. Theory Appl., 11, 2, 175-188 (1973) · Zbl 0238.65033 |
| [18] | Himmelblau, D. M., Applied Nonlinear Programming (1972), McGraw-Hill Book Company: McGraw-Hill Book Company New York · Zbl 0521.93057 |
| [19] | Hestenes, M. R.; Stiefel, E., Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bureau Stand., 29, 409-439 (1952) · Zbl 0048.09901 |
| [20] | Luenberger, D. C., Linear and Nonlinear Programming (1989), Addition-Wesley: Addition-Wesley Reading, MA |
| [21] | Miele, A.; Cantrell, J. W., Study on a memory gradient method for the minimization of functions, J. Optim. Theory Appl., 3, 6, 459-470 (1969) · Zbl 0165.22702 |
| [22] | Mangasarian, O. L., Nonlinear Programming (1969), McGraw-Hill: McGraw-Hill New York · Zbl 0194.20201 |
| [23] | Nocedal, J.; Wright, S. J., Numerical Optimization, New York (1999), Springer: Springer Berlin · Zbl 0930.65067 |
| [24] | Polak, E., Optimization: Algorithms and Consistent Approximations (1997), Springer: Springer New York · Zbl 0899.90148 |
| [25] | Polak, E.; Ribière, G., Note sur la convergence des methodes de directions conjuguees, Revue Francaise d’Inform. Rec. Oper., 16, 35-43 (1969) · Zbl 0174.48001 |
| [26] | Polyak, B. T., The conjugate gradient algorithm in extremal problems, U.S.S.R. Comput. Math. Math. Phys., 9, 94-112 (1961) · Zbl 0229.49023 |
| [27] | Shanno, D. F., Conjugate gradient methods with inexact searches, Math Oper Res., 3, 244-256 (1978) · Zbl 0399.90077 |
| [28] | Shi, Z. J., A new memory gradient method under exact line search, Asian Pacific J. Oper. Res., 20, 2 (2003) · Zbl 1165.90646 |
| [29] | Shi, Z. J., Restricted PR conjugate gradient method and its global convergence, Adv. Math., 31, 1, 47-55 (2002) · Zbl 1264.90179 |
| [30] | Shi, Z. J., A super-memory gradient method for unconstrained minimization, J. Eng. Math., 17, 2, 99-104 (2000) · Zbl 0970.90520 |
| [31] | Vrahatis, M. N.; Androulajis, G. S.; Lambrinos, J. N.; Magoulas, G. D., A class of gradient unconstrained minimization algorithms with adaptive stepsize, J. Comput. Appl. Math., 114, 2, 367-386 (2000) · Zbl 0958.65072 |
| [32] | Wolfe, M. A.; Viazminsky, C., Supermemory Descent Methods for Unconstrained Minimization, J. Optim. Theory Appl., 18, 4, 455-468 (1976) · Zbl 0304.49023 |
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