Skip to main content
SpringerLink
Log in

An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

LetH 1 andH 2 denote Hilbert spaces and suppose thatD is a subset ofH 1. This paper establishes the local and linear convergence of a general iterative technique for finding the zeros ofG:D→H 2 subject to the general constraintP(x)=x, whereP:D→D. The results are then applied to several classes of problems, including those of least squares, generalized eigenvalues, and constrained optimization. Numerical results are obtained as the procedure is applied to finding the zeros of polynomials in several variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Altman, M.: Connection between gradient methods and Newton's method for functionals. Bull. Acad. Polon. Sci.9(12), 877–880 (1961)

    Google Scholar 

  2. Blum, E. K.: Numerical analysis and computation: Theory and practice. Reading, Mass.: Addison-Wesley 1972

    Google Scholar 

  3. Blum, E. K., Rodrique, G. H.: Solution of eigenvalue problems and least squares problems in Hilbert space by a gradient method. J. Comput. Sys. Sci.8, 220–238 (1974)

    Google Scholar 

  4. Fridman, V. M.: An iteration process with minimum errors for a nonlinear operator equation. Dokl. Akad. Nauk SSR139 (61), 1063–1066

  5. Goldstein, A. A.: Constructive real analysis. New York: Harper and Row 1967

    Google Scholar 

  6. McCormick, S. F.: A general approach to one-step iterative methods with application to eigenvalue problems. J. Comput. Syst. Sci.6, 354–372 (1972)

    Google Scholar 

  7. McCormick, S. F., Rodrique, G. H.: A uniform approach to gradient methods for linear operator equations. To appear in J.M.A.A.

  8. Ortega, J. M., Rheinboldt, W. C.: Iterative solution of non-linear equations in several variables. New York: Academic Press 1970

    Google Scholar 

  9. Polak, E.: Computational methods in optimization. New York: Academic Press 1971

    Google Scholar 

  10. Rheinboldt, W. C.: A unified convergence theory for a class of iterative processes. SIAM J. Numer. Anal.5, 42–63 (1968)

    Article  Google Scholar 

  11. Rodrique, G. H.: A gradient method for the matrix eigenvalue problem. This Journal22, 1–16 (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Colorado State University, 80523, Fort Collins, Colorado, USA

    Steve F. McCormick

Authors
  1. Steve F. McCormick
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by NSF grant G J 34737.

Rights and permissions

Reprints and permissions

About this article

Cite this article

McCormick, S.F. An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems. Numer. Math. 23, 371–385 (1975). https://doi.org/10.1007/BF01437037

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01437037

Keywords

Search

Navigation