Journal of Optimization Theory and Applications

, Volume 83, Issue 3, pp 587–612 | Cite as

Quasi-Newton method by Hermite interpolation

  • T. F. Sturm
Contributed Papers

Abstract

This paper describes a new attempt to solve the problem of computing a local minimizer of a sufficiently often differentiable unconstrained objective function. In every step of the iteration, a special Hermite interpolant is constructed. Old iteration points serve as points of support with the function value and gradient information. This yields a quasi-Newton algorithm with quadratic convergence order.

Key Words

Nonlinear unconstrained optimization Hermite interpolation quasi-Newton methods quadratic convergence 

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References

  1. 1.
    Dennis, J. E., Jr., andSchnabel, R. B.,Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.Google Scholar
  2. 2.
    Dennis, J. E., Jr., andSchnabel, R. B.,A View of Unconstrained Optimization, Optimization, Edited by G. L. Nemhauser, A. H. G. Rinnooy Kan, and M. J. Todd, North-Holland, Amsterdam, Holland, Vol. 1, pp. 1–72, 1989.Google Scholar
  3. 3.
    Gill, P. E., Murray, W., andWright, M. H.,Practical Optimization, Academic Press, New York, New York, 1981.Google Scholar
  4. 4.
    Ritter, K.,Numerical Methods for Nonlinear Programming Problems, Modern Applied Mathematics: Optimization and Operations Research, Edited by B. Korte, North-Holland, Amsterdam, Holland, pp. 227–264, 1982.Google Scholar
  5. 5.
    Schnabel, R. B.,Sequential and Parallel Methods for Unconstrained Optimization, Technical Report CU-CS-414-88, University of Colorado, 1988.Google Scholar
  6. 6.
    Schnabel, R. B., andChow, T. T.,Tensor Methods for Unconstrained Optimization Using Second Derivatives, SIAM Journal on Optimization, Vol. 1, pp. 293–315, 1991.CrossRefGoogle Scholar
  7. 7.
    Peschl, E.,Analytische Geometrie, Bibliographisches Institut, Mannheim, Germany, 1961.Google Scholar
  8. 8.
    Chui, C. K., andLai, M. J.,Multivariate Vertex Splines and Finite Elements, Journal of Approximation Theory, Vol. 60, pp. 245–343, 1990.CrossRefGoogle Scholar
  9. 9.
    Ciarlet, P. G., andRaviart, P. A., General Lagrange and Hermite Interpolation in ℝn with Applications to Finite-Element Methods, Archive for Rational Mechanics and Analysis, Vol. 46, pp. 177–199, 1972.CrossRefGoogle Scholar
  10. 10.
    Engels, H.,Hermite Interpolation in N Variables and Minimal Cubature Formulae, Multivariate Approximation Theory—3, Edited by W. Schempp and K. Zeller, pp. 154–165, 1985.Google Scholar
  11. 11.
    Gregory, J. A.,Interpolation to Boundary Data on the Simplex, Computer-Aided Geometric Design, Vol. 2, pp. 43–52, 1985.CrossRefGoogle Scholar
  12. 12.
    Sturm, T. F.,Ein Quasi-Newton-Verfahren durch Hermite Interpolation, Dissertation, Technische Universität München, 1991.Google Scholar
  13. 13.
    Sturm, T. F.,A Unique Multivariate Hermite Interpolant on the Simplex, Journal of Mathematical Analysis and Applications (to appear).Google Scholar
  14. 14.
    Greiner, M., Kölbl, A., andKredler, C.,User's Guide for PADMOS: Pascal Units for Optimization and Automatic Differentiation, Report TUM-MATH-09-90-I00-300/1.-FMI, Technische Universität München, 1990.Google Scholar
  15. 15.
    Moré, J. J., Garbow, B. S., andHillstrom, K. E.,Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software, Vol. 7, No. 1, pp. 17–41, 1981.CrossRefGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • T. F. Sturm
    • 1
  1. 1.Institut für Angewandte Mathematik und StatistikTechnische Universität MünchenMünchenGermany

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