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Direct-prediction quasi-Newton methods in Hilbert space with applications to control problems

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Abstract

In this paper, the Hilbert-space analogue of a result of Huang, that all the methods in the Huang class generate the same sequence of points when applied to a quadratic functional with exact linear searches, is established. The convergence of a class of direct prediction methods based on some work of Dixon is then proved, and these methods are then applied to some control problems. Their performance is found to be comparable with methods involving exact linear searches.

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References

  1. Fletcher, R.,A New Approach to Variable Metric Algorithms, Computer Journal, Vol. 13, No. 3, 1970.

  2. Dixon, L. C. W.,Conjugate Directions Without Linear Searches, Journal of the Institute of Mathematics and Its Applications, Vol. 11, No. 3, 1973.

  3. Horwitz, L. B., andSarachik, P. E.,Davidon's Method in Hilbert Space, SIAM Journal on Applied Mathematics, Vol. 16, No. 4, 1968.

  4. Huang, H. Y.,Unified Approach to Quadratically Convergent Algorithms for Function Minimization, Journal of Optimization Theory and Applications, Vol. 5, No. 6, 1970.

  5. Dixon, L. C. W.,Variable Metric Algorithms: Necessary and Sufficient Conditions for Identical Behavior of Nonquadratic Functions, Journal of Optimization Theory and Applications, Vol. 10, No. 1, 1972.

  6. Turner, P. R., andHuntley, E.,Variable Metric Methods in Hilbert Space with Applications to Control Problems, Journal of Optimization Theory and Applications, Vol. 19, No. 3, 1976.

  7. Lasdon, L. S.,Conjugate Direction Methods for Optimal Control, IEEE Transactions on Automatic Control, Vol. AC-15, No. 4, 1970.

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Authors and Affiliations

  1. Department of Mathematics, University of Lancaster, Bailrigg, Lancaster, England

    P. R. Turner (Lecturer in Numerical Analysis)

  2. Department of Applied Mathematics and Computing Science, University of Sheffield, Sheffield, England

    E. Huntley (Senior Lecturer)

Authors
  1. P. R. Turner
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  2. E. Huntley
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Communicated by D. F. Lawden

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Turner, P.R., Huntley, E. Direct-prediction quasi-Newton methods in Hilbert space with applications to control problems. J Optim Theory Appl 21, 199–211 (1977). https://doi.org/10.1007/BF00932520

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  • DOI: https://doi.org/10.1007/BF00932520

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