Abstract
Recent attempts to assess the performance of SSVM algorithms for unconstrained minimization problems differ in their evaluations from earlier assessments. Nevertheless, the new experiments confirm earlier observations that, on certain types of problems, the SSVM algorithms are far superior to other variable metric methods. This paper presents a critical review of these recent assessments and discusses some current interpretations advanced to explain the behavior of SSVM methods. The paper examines the new empirical results, in light of the original self-scaling theory, and introduces a new interpretation of these methods based on anL-function model of the objective function. This interpretation sheds new light on the performance characteristics of the SSVM methods, which contributes to the understanding of their behavior and helps in characterizing classes of problems which can benefit from the self-scaling approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Oren, S. S.,Self-Scaling Variable Metric Algorithms without Line Search for Unconstrained Minimization, Mathematics of Computation, Vol. 27, pp. 873–885, 1973.
Oren, S. S.,Self-Scaling Variable Metric (SSVM) Algorithms, II, Implementation and Experiments, Management Science, Vol. 20, pp. 863–874, 1974.
Oren, S. S.,On the Selection of Parameters in Self-Scaling Variable Metric Algorithms, Mathematical Programming, Vol. 7, pp. 351–367, 1974.
Oren, S. S., andLuenberger, D. G.,Self-Scaling Variable Metric (SSVM) Algorithms, I, Criteria and Sufficient Conditions for Scaling a Class of Algorithms, Management Science, Vol. 20, pp. 845–862, 1974.
Oren, S. S., andSpedicato, E.,Optimal Conditioning of Self-Scaling Variable Metric Algorithms, Mathematical Programming, Vol. 10, pp. 70–90, 1976.
Fletcher, R.,A New Approach to Variable Metric Algorithms, Computer Journal, Vol. 13, pp. 317–322, 1970.
Luenberger, D. G.,Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.
Spedicato, E.,Computational Experience with Quasi-Newton Algorithms for Minimization Problems of Moderately Large Size, Report No. CISE-N-175, CISE Documentation Service, Segrate, Milano, Italy, 1975.
Spedicato, E.,A Variable Metric Method for Function Minimization Derived from Invariancy to Nonlinear Scaling, Journal of Optimization Theory and Applications, Vol. 20, pp. 315–328, 1976.
Brodlie, K. W.,An Assessment of Two Approaches to Variable Metric Methods, Mathematical Programming, Vol. 12, pp. 344–355, 1977.
Shanno, D. F., andPhua, K. H.,Matrix Conditioning and Nonlinear Optimization, Mathematical Programming, Vol. 14, pp. 149–160, 1978.
Biggs, M. C.,Minimization Algorithms Making Use of Nonquadratic Properties of the Objective Function, Journal of the Institute of Mathematics and Its Applications, Vol. 8, pp. 315–327, 1971.
Broyden, C. G.,The Convergence of a Class of Double Rank Minimization Algorithms, 2, The New Algorithm, Journal of the Institute of Mathematics and Its Applications, Vol. 6, pp. 222–231, 1970.
Jacobson, D. H., andOksman, W.,An Algorithm that Minimizes Homogeneous Function of N Variables in N + 2Iterations and Rapidly Minimizes General Functions, Journal of Mathematical Analysis and Applications, Vol. 38, pp. 535–552, 1970.
Dennis, J. E., Gay, D. M., andWelsch, R. E. An Adaptive Nonlinear Least-Squares Algorithm, Cornell University, Department of Computer Science, Technical Report No. 77–321, 1977.
Author information
Authors and Affiliations
Additional information
Communicated by H. Y. Huang
This work was done while the author was with the Analysis Research Group, Xerox Palo Alto Research Center, Palo Alto, California.
Rights and permissions
About this article
Cite this article
Oren, S.S. Perspectives on self-scaling variable metric algorithms. J Optim Theory Appl 37, 137–147 (1982). https://doi.org/10.1007/BF00934764
Issue Date:
DOI: https://doi.org/10.1007/BF00934764