A conjugate gradient method with quasi-Newton approximation

Jonas Koko

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 2, page 153-165
  • ISSN: 1233-7234

Abstract

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The conjugate gradient method of Liu and Storey is an efficient minimization algorithm which uses second derivatives information, without saving matrices, by finite difference approximation. It is shown that the finite difference scheme can be removed by using a quasi-Newton approximation for computing a search direction, without loss of convergence. A conjugate gradient method based on BFGS approximation is proposed and compared with existing methods of the same class.

How to cite

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Koko, Jonas. "A conjugate gradient method with quasi-Newton approximation." Applicationes Mathematicae 27.2 (2000): 153-165. <http://eudml.org/doc/219264>.

@article{Koko2000,
abstract = {The conjugate gradient method of Liu and Storey is an efficient minimization algorithm which uses second derivatives information, without saving matrices, by finite difference approximation. It is shown that the finite difference scheme can be removed by using a quasi-Newton approximation for computing a search direction, without loss of convergence. A conjugate gradient method based on BFGS approximation is proposed and compared with existing methods of the same class.},
author = {Koko, Jonas},
journal = {Applicationes Mathematicae},
keywords = {Newton and quasi-Newton methods; unconstrained high-dimensional optimization; conjugate gradient methods; numerical examples; conjugate gradient method; minimization algorithm; quasi-Newton approximation; convergence},
language = {eng},
number = {2},
pages = {153-165},
title = {A conjugate gradient method with quasi-Newton approximation},
url = {http://eudml.org/doc/219264},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Koko, Jonas
TI - A conjugate gradient method with quasi-Newton approximation
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 2
SP - 153
EP - 165
AB - The conjugate gradient method of Liu and Storey is an efficient minimization algorithm which uses second derivatives information, without saving matrices, by finite difference approximation. It is shown that the finite difference scheme can be removed by using a quasi-Newton approximation for computing a search direction, without loss of convergence. A conjugate gradient method based on BFGS approximation is proposed and compared with existing methods of the same class.
LA - eng
KW - Newton and quasi-Newton methods; unconstrained high-dimensional optimization; conjugate gradient methods; numerical examples; conjugate gradient method; minimization algorithm; quasi-Newton approximation; convergence
UR - http://eudml.org/doc/219264
ER -

References

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  1. [1] C. G. Broyden, The convergence of a class of double-rank minimization algorithms, J. Inst. Math. Appl. 6 (1970), 60-76. Zbl0223.65023
  2. [2] J. E. Dennis and J. J. Moré, Quasi-Newton methods, motivations and theory, SIAM Rev. 19 (1977), 46-89. Zbl0356.65041
  3. [3] R. Fletcher and C. M. Reeves, Function minimization by conjugate gradient, Computer J. 7 (1964), 143-154. Zbl0132.11701
  4. [4] J. C. Gilbert and C. Lemarechal, The modules M1QN2, N1QN2, M1QN3 and N1QN3, INRIA, technical report, 1989. 
  5. [5] J. C. Gilbert and J. Nocedal, Global convergence properties of conjugate gradient methods for optimization, SIAM J. Optim. 2 (1992), 21-42. Zbl0767.90082
  6. [6] M. Hestenes and E. Stiefel, Methods of conjugate gradient for solving linear systems, J. Res. Nat. Bureau Standards B48 (1952), 409-436. Zbl0048.09901
  7. [7] Y. F. Hu and C. Storey, Efficient generalized conjugate gradient algorithms, Part 2: Implementation, J. Optim. Theory Appl. 69 (1991), 139-152. 
  8. [8] Y. F. Hu and C. Storey, Preconditioned low-order Newton methods, ibid. 79 (1993), 311-331. 
  9. [9] Y. Liu and C. Storey, Efficient generalized conjugate gradient algorithms, Part 1: Theory, ibid. 69 (1991), 129-137. 
  10. [10] J. L. Nazareth, The method of successive affine reduction for nonlinear minimization, Math. Programming 35 (1985), 97-109. Zbl0621.90076
  11. [11] J. L. Nazareth, Conjugate gradient methods less dependent on conjugacy, SIAM Rev. 28 (1986), 501-511. Zbl0625.90077
  12. [12] E. Polak and G. Ribière, Note sur la convergence des méthodes de directions conjuguées, RAIRO Rech. Opér. 16 (1969), 35-43. Zbl0174.48001

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