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Computational experience with improved conjugate gradient methods for unconstrained minimization. (English) Zbl 0771.90090
Summary: The paper contains a description of new restart procedures for the conjugate gradient methods and a numerical investigation of the influence of line search and scaling on their efficiency. Computational results obtained by means of 15 sufficiently difficult problems are given.
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
49M37 Numerical methods based on nonlinear programming
90-08 Computational methods for problems pertaining to operations research and mathematical programming
minpack; ve08; L-BFGS
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[1] M. Al-Bali: Descent property and global convergence of the Fletcher-Reeves method with inexact line search. IMA J. Numer. Anal. 5 (1985), 121-124. · Zbl 0578.65063
[2] P. Baptist, J. Stoer: On the relation between quadratic termination and convergence properties of minimization algorithms. Part 2. Applications. Numer. Math. 28 (1977), 367-391. · Zbl 0366.65028
[3] P. Bjorstadt, J. Nocedal: Analysis of a new algorithm for one-dimensional minimization. Computing 22 (1979), 93-100. · Zbl 0401.65041
[4] A. R. Conn N. I. M. Gould, P. L. Toint: Testing a class of methods for solving minimization problems with simple bounds on the variables. Math. Comp. 50 (1988), 399-430. · Zbl 0645.65033
[5] W.C. Davidon: Variable metric method for minimization. A.E.C. Research and Development Report ANL-5990, 1959.
[6] W.C. Davidon: Optimally conditioned optimization algorithms without line searches. Math. Pro- gramming 9 (1975), 1-30. · Zbl 0328.90055
[7] R. S. Dembo, T. Steihaug: Truncated-Newton algorithms for large-scale unconstrained minimization. Math. Programming 26 (1983), 190-212. · Zbl 0523.90078
[8] R. Fletcher: A FORTRAN subroutine for minimization by the method of conjugate gradients. Report No. AERE-R7073, Atomic Energy Research Establishment, Harwell 1972.
[9] R. Fletcher, M.J. D. Powell: A rapidly convergent descent method for minimization. Computer J. 6 (1963), 163-168. · Zbl 0132.11603
[10] R. Fletcher, CM. Reeves: Function minimization by conjugate gradients. Computer J. 7 (1964), 149-154. · Zbl 0132.11701
[11] J.C. Gilbert, and J. Nocedal: Global convergence properties of conjugate gradient methods for optimization. Report No. 1268, Institut National de Recherche en Inforrnatique et. en Automatique, 1990. · Zbl 0767.90082
[12] A. Griewank, P. L. Toint: Partitioned variable metric updates for large structured optimization problems. Numer. Math. 39 (1982), 119-137. · Zbl 0482.65035
[13] M. R. Hestenes, CM. Stiefel: Methods of conjugate gradient for solving linear systems. J. Res. Nat. Bur. Standards 49 (1964), 409-436. · Zbl 0048.09901
[14] Y. F. Hu, C. Storey: A Global Convergence Result for Conjugate Gradient Methods. Report No. A134, Loughborough University of Technology, 1990. · Zbl 0794.90063
[15] K.M. Khoda Y. Liu, C. Storey: A Generalized Polak-Ribiére Algorithm. Report No. A128, Loughborough University of Technology, 1990. · Zbl 0795.90067
[16] L. Lukšan: Variable Metric Methods. Unconstrained Minimization. Academia, Prague 1990. In Czech. · Zbl 0716.65055
[17] L. Lukšan: Computational experience with improved variable metric methods for unconstrained minimization. Kybernetika 26 (1990), 415-431. · Zbl 0716.65055
[18] J.J. Moré B.S. Garbow, K.E. Hillstrom: Testing unconstrained optimization software. ACM Trans. Math. Software 7 (1981), 17-41. · Zbl 0454.65049
[19] J. Nocedal: Updating quasi-Newton matrices with limited storage. Math. Comp. 35 (1980), 773-782. · Zbl 0464.65037
[20] E. Polak, G. Ribiére: Note sur la convergence de methodes de directions conjugees. Revue Francaise Inform. Mech. Oper. 16-R1 (1969), 35-43. · Zbl 0174.48001
[21] M.J.D. Powell: Restart procedures of the conjugate gradient method. Math. Programming 12 (1977), 241-254. · Zbl 0396.90072
[22] M.J.D. Powell: Nonconvex Minimization Calculations and the Conjugate Gradient Method. Report No. DAMTP 1983/NA14, University of Cambridge, 1983. · Zbl 0531.65035
[23] M.J.D. Powell: Convergence Properties of Algorithms for Nonlinear Optimization. Report No. DAMPT 1985/NA1, University of Cambridge, 1985.
[24] D. F. Shanno: Conditioning of quasi-Newton methods for function minimization. Math. Comp. 24 (1970), 647-656. · Zbl 0225.65073
[25] D.F. Shanno: Globally convergent conjugate gradient algorithms. Math. Programming 33 (1985), 61-67. · Zbl 0579.90079
[26] P. L. Toint: On sparse and symmetric matrix updating subject to a linear equation. Math. Comp. 31 (1987), 954-961. · Zbl 0379.65034
[27] D. Touati-Ahmed, C. Storey: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64 (1990), 379-397. · Zbl 0666.90063
[28] G. Zoutendijk: Nonlinear programming, computational methods. Integer and Nonlinear Programming (J. Abadie, North-Holland, Amsterdam 1970, pp. 93-121. · Zbl 0336.90057
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