Abstract
In this note, a general optimal conditioning problem for updates which satisfy the quasi-Newton equation is solved. The new solution is a family of updates which contains other known optimally conditioned updates but also includes new formulas of increased rank. A new factorization formula for the Broyden family and some preliminary numerical results are also given.
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References
Fletcher, R.,Practical Methods of Optimization, 2nd Edition, John Wiley and Sons, Chichester, England, 1987.
Fletcher, R.,A New Variational Result for Quasi-Newton Formula, SIAM Journal on Optimization, Vol. 1, pp. 18–21, 1991.
Powell, M. J. D.,Some Global Convergence Properties of a Variable Metric Algorithm for Minimization without Exact Line Searches, Nonlinear Programming, SIAM-AMS Proceedings, Edited by R. W. Cottle and C. E. Lemke, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, Vol. 9, pp. 53–72, 1976.
Byrd, R. H., Nocedal, J., andYuan, Y.,Global Convergence of a Class of Quasi-Newton Methods on Convex Problems, SIAM Journal on Numerical Analysis, Vol. 24, pp. 1171–1190, 1987.
Oren, S. S., andLuenberger, D. G.,Self-Scaling Variable Metric (SSVM) Algorithms, Part 1: Criteria and Sufficient Conditions for Scaling a Class of Algorithms, Management Sciences Vol. 20, pp. 845–862, 1974.
Davidon, W. C.,Optimally Conditioned Optimization Algorithms without Line Searches, Mathematical Programming, Vol. 9, pp. 1–30, 1975.
Oren, S. S., andSpedicato, E.,Optimal Conditioning of Self-Scaling Variable Metric Algorithms, Mathematical Programming, Vol. 10, pp. 70–90, 1976.
Al-Baali, M.,On Self-Scaling Updating Formulas for Quasi-Newton Methods, Report No. 87, Department of Mathematics, Faculty of Science, University of Damascus, Damascus, Syria, 1988.
Lukšan, L.,Computational Experience with Improved Variable Metric Methods for Unconstrained Minimization, Kybernetika, Vol. 5, pp. 415–431, 1990.
Hu, Y. F., andStorey C.,On Optimally and Near-Optimally Conditioned Quasi-Newton Updates, Mathematics Report A141, Loughborough University of Technology, Loughborough, England, 1991.
Hu, Y. F.,New Conjugate Gradient Methods and Their Connections with Quasi-Newton and Lower-Dimensional Newton Methods, PhD Thesis, Department of Mathematical Sciences, Loughborough University of Technology, Loughborough, England, 1992.
Ip, C. M.,On Least-Change Secant Updates in Factored Form, SIAM Journal on Numerical Analysis, Vol. 24, pp. 1126–1132, 1987.
Powell, M. J. D.,Updating Conjugate Directions by the BFGS Formula, Mathematical Programming, Vol. 38, pp. 29–46, 1987.
Siegel, D.,Updating of Conjugate Direction Matrices Using Members of Broyden's Family, Report DAMTP 1991/NA4, University of Cambridge, Cambridge, England, 1991.
Siegel, D.,Modifying the BFGS Update by a New Column Scaling Technique, Report DAMTP 1991/NA5, University of Cambridge, Cambridge, England, 1991.
Shanno, D. F., andPhua, K. H.,Matrix Conditioning and Nonlinear Optimization, Mathematical Programming, Vol. 14, pp. 149–160, 1978.
Moré, J. J., Garbow, B. S., andHillstrom, K. E.,Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software, Vol. 7, pp. 17–41, 1981.
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Communicated by L. C. W. Dixon
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Hu, Y.F., Storey, C. Family of optimally conditioned quasi-Newton updates for unconstrained optimization. J Optim Theory Appl 83, 421–431 (1994). https://doi.org/10.1007/BF02190066
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DOI: https://doi.org/10.1007/BF02190066