Minimum norm symmetric quasi-Newton updates restricted to subspaces
HTML articles powered by AMS MathViewer
- by Robert B. Schnabel PDF
- Math. Comp. 32 (1978), 829-837 Request permission
Abstract:
The Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno updates have been the two most successful quasi-Newton updates for a variety of applications. One reason offered in explanation is that they constitute, in an appropriate norm and metric, the minimum norm change to the matrix, or its inverse, being approximated which preserves symmetry and obeys the quasi-Newton equation. Recent methods have reason to consider updates restricted to certain subspaces. In this paper we derive the general minimum norm symmetric quasi-Newton updates restricted to such subspaces. In the same appropriate norm and metric, the minimum norm change update to the matrix or its inverse is shown to be, respectively, the rank-two update which is a particular projection of the DFP or BFGS onto this subspace.References
- C. G. Broyden, The convergence of a class of double-rank minimization algorithms. II. The new algorithm, J. Inst. Math. Appl. 6 (1970), 222–231. MR 433870 C. DAVIDON (1959), Variable Metric Algorithm for Minimization, Argonne National Laboratory Report ANL-5990 (Rev.).
- William C. Davidon, Optimally conditioned optimization algorithms without line searches, Math. Programming 9 (1975), no. 1, 1–30. MR 383741, DOI 10.1007/BF01681328
- J. E. Dennis Jr. and Jorge J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev. 19 (1977), no. 1, 46–89. MR 445812, DOI 10.1137/1019005 FLETCHER (1970), "A new approach to variable metric methods," Comput. J., v. 13, pp. 317-322.
- R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization, Comput. J. 6 (1963/64), 163–168. MR 152116, DOI 10.1093/comjnl/6.2.163
- Donald Goldfarb, A family of variable-metric methods derived by variational means, Math. Comp. 24 (1970), 23–26. MR 258249, DOI 10.1090/S0025-5718-1970-0258249-6
- J. Greenstadt, Variations on variable-metric methods. (With discussion), Math. Comp. 24 (1970), 1–22. MR 258248, DOI 10.1090/S0025-5718-1970-0258248-4 H.-W. MEI (1977), "An analysis and implementation of Davidon’s techniques for unconstrained optimization," Ph.D. thesis, Cornell University.
- Robert B. Schnabel, Optimal conditioning in the convex class of rank two updates, Math. Programming 15 (1978), no. 3, 247–260. MR 514611, DOI 10.1007/BF01609030 B. SCHNABEL (1977), "Analyzing and improving quasi-Newton methods for unconstrained optimization," Ph.D. thesis, Cornell University.
- D. F. Shanno, Conditioning of quasi-Newton methods for function minimization, Math. Comp. 24 (1970), 647–656. MR 274029, DOI 10.1090/S0025-5718-1970-0274029-X
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp. 32 (1978), 829-837
- MSC: Primary 65K10; Secondary 65F30
- DOI: https://doi.org/10.1090/S0025-5718-1978-0492041-X
- MathSciNet review: 492041