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Fast solution of general nonlinear fixed point problems

  • Roberto L. V. González
  • Mabel M. Tidball
I Optimality And Duality
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)

Abstract

In this paper, we develope a general procedure to stabilize the usual Newton method in such a way that algorithms obtained always converge to the unique solution of the problem.

We show that in the case where the operator T∈C1∩H2,∞, quadratic convergence holds and when T is polyhedric, convergence in a finite number of steps is obtained. Numerical results are shown for an example issued from the field of differential games.

Keywords

Variational Inequality Optimal Control Problem Differential Game Descent Direction Quadratic Convergence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Belbas, S. A., Mayergoyz, I. D.: Applications of fixed-point methods to discrete variational and quasivariational inequalities. Numerische Mathematik, Vol. 51, pp. 631–654, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Clarke, F. H.: Optimization and Nonsmooth Analysis (Wiley, New York, 1983).zbMATHGoogle Scholar
  3. [3]
    El Tarazi, M. N.: On a monotony-preserving accelerator process for the successive approximations method. IMA Journal of Numerical Analysis. Vol. 6, pp. 439–446, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    González R., Rofman E.: On deterministic control problems: an approximation procedure for the optimal cost. Part I and II. SIAM Journal on Control and Optimization, 23, pp. 242–285, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    González, R., Sagastizabál, C.: Un algorithme pour la résolution rapide d’équations discrètes de Hamilton-Jacobi-Bellman. Comptes Rendus Acad. Sc. Paris, Serie I, Tome 311, pag. 45–50, 1990.zbMATHGoogle Scholar
  6. [6]
    González R., Tidball M.: Fast solution of general nonlinear fixed point problems, Rapport de Recherche No 1339, INRIA, 1990.Google Scholar
  7. [7]
    Marcotte P.: A new algorithm for solving variational inequalities with application to the traffic assignment problem. Mathematical Programming, Vol. 33, pp. 339–351, 1985.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables (Academic Press, New York, 1970).zbMATHGoogle Scholar
  9. [9]
    Polak E.: Computational Methods in Optimization. A Unified Approach (Academic Press, New York, 1970).Google Scholar
  10. [10]
    Ross S. M.: Applied Probability Models with Optimization Applications (Holden-Day, San Francisco, 1970).zbMATHGoogle Scholar
  11. [11]
    Tolwinski B.: Newton-type methods for stochastics games. In Differential Games and Applications, T. S. Basar, P. Bernhard (Eds.), pp. 128–144, (Springer-Verlag, Heidelberg, 1989).CrossRefGoogle Scholar

Copyright information

© International Federation for Information Processing 1992

Authors and Affiliations

  • Roberto L. V. González
    • 1
  • Mabel M. Tidball
    • 1
  1. 1.Departamento de Matemática. Facultad de Ciencias Exactas, Ingeniería y AgrimensuraUniversidad Nacional de RosarioRosarioArgentina

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