Using function-values in multi-step quasi-Newton methods

In previous work, the authors (1993, 1994) developed the concept of multi-step quasi-Newton methods, based on the use of interpolating polynomials determined by data from the m most recent steps. Different methods for parametrizing these polynomials were studied by the authors (1993), and several methods were shown (empirically) to yield substantial gains over the standard (one-step) BFGS method for unconstrained optimization. In this paper, we will consider the issue of how to incorporate function-value information within the framework of such multi-step methods. This is achieved, in the case of two-step methods, through the use of a carefully chosen rational form to interpolate the three most recent iterates. The results of numerical experiments on the new methods are reported.

[1]  Issam A. R. Moghrabi,et al.  Multi-step quasi-Newton methods for optimization , 1994 .

[2]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

[3]  Shmuel S. Oren,et al.  Optimal conditioning of self-scaling variable Metric algorithms , 1976, Math. Program..

[4]  Ya-Xiang Yuan,et al.  A modified BFGS algorithm for unconstrained optimization , 1991 .

[5]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[6]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[7]  M. C. Biggs.  Minimization Algorithms Making Use of Non-quadratic Properties of the Objective Function , 1971 .

[8]  John A. Ford,et al.  On the use of function-values in unconstrained optimisation , 1989 .

[9]  J. Nocedal,et al.  A tool for the analysis of Quasi-Newton methods with application to unconstrained minimization , 1989 .

[10]  D. F. Shanno,et al.  Matrix conditioning and nonlinear optimization , 1978, Math. Program..

[11]  John A. Ford,et al.  On the use of curvature estimates in quasi-Newtonian methods , 1991 .

[12]  Jorge Nocedal,et al.  Theory of algorithms for unconstrained optimization , 1992, Acta Numerica.

[13]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[14]  A. Iserles,et al.  Acta numerica 1992 , edited by A. Iserles. Pp. 407. £19.95 1992. ISBN 0-521-41026-6 (Cambridge University Press) , 1993, The Mathematical Gazette.

[15]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[16]  Issam A. R. Moghrabi,et al.  Alternative parameter choices for multi-step Quasi-Newton methods , 1993 .