Skip to main content
SpringerLink
Log in

Locating, characterizing and computing the stationary points of a function

Поиск, классификация и вычисление стационарных точек функции

  • Published:
Reliable Computing

Abstract

A method for the localization, characterization and computation of the stationary points of a continuously differentiable real-valued function ofn variables is presented. It is based on the combinatorial topology concept of the degree of a mapping associated with an oriented polyhedron. The method consists of two principal steps: (i) localization (and computation if required) of a stationary point in ann-dimensional polyhedron; (ii) characterization of a stationary point as a minimum, maximum or saddle point. The method requires only the signs of gradient values to be correct and it can be successfully applied to problems with imprecise values.

Abstract

Предложен метод ноиска, классификации и вычисления стационарных точек непрерывно дифференцируемой вещественной функцииn переменных. Метод основан на заимствованном из комбинаторной топологии цонятии стецени отображения, связанного с ориентированным многогранником. Метод состоит из двух основных щагов: 1) локализация (и, если необходимо, вычисление) стационарной точки вn-мерном многограннике; 2) классификация стационарной точки как точки минимума, максимума или седловой точки. Применение метода требует знания только знаков градиентов, поэтому данный метод может успещно исцользоваться для рещения задач с погрещностями в условиях.

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. Dennis, Jr., J. E. and Schnabel, R. B.Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, 1983.

  2. Grapsa, T. N. and Vrahatis, M. N.A dimension-reducing method for unconstrained optimization. J. Comput. Appl. Math., to appear.

  3. Himmelblau, D. M.Applied nonlinear programming. McGraw-Hill, NY, 1972.

    Google Scholar 

  4. Kavvadias, D. J. and Vrahatis, M. N.Locating and computing all the simple roots and extrema of a function. SIAM J. Sci. Comput., to appear.

  5. Kearfott, R. B.An efficient degree-computation method for a generalized method of bisection. Numer. Math.32 (1979), pp. 109–127.

    Article  MATH  MathSciNet  Google Scholar 

  6. Kupferschmid, M. and Ecker, J. G.A note on solution of nonlinear programming problems with imprecise function and gradient values. Math. Program. Study31 (1987), pp. 129–138.

    MathSciNet  Google Scholar 

  7. Magoulas, G. D., Vrahatis, M. N., Grapsa, T. N., and Androulakis, G. S.An efficient training method for discrete multilayer neural networks. Ann. Math. Artif. Intel., to appear.

  8. Magoulas, G. D., Vrahatis, M. N., Grapsa, T. N., and Androulakis, G. S.Neural network supervised training based on a dimension reducing method. Ann. Math. Artif. Intel., to appear.

  9. Vrahatis, M. N.Solving systems of nonlinear equations using the nonzero value of the topological degree. ACM Trans. Math. Software14 (1988), pp. 312–329.

    Article  MATH  MathSciNet  Google Scholar 

  10. Vrahatis, M. N.CHABIS: A mathematical software package for locating and evaluating roots of systems of non-linear equations. ACM Trans. Math. Software14 (1988), pp. 330–336.

    Article  MATH  MathSciNet  Google Scholar 

  11. Vrahatis, M. N.An efficient method for locating and computing periodic orbits of nonlinear mappings. J. Comp. Phys.119 (1995), pp. 105–119.

    Article  MATH  MathSciNet  Google Scholar 

  12. Vrahatis, M. N., Androulakis, G. S., and Manoussakis, G. E.A new unconstrained optimization method for imprecise function and gradient values. J. Math. Anal. Appl., to appear.

  13. Vrahatis, M. N. and Iordanidis, K. I.A rapid generalized method of bisection for solving systems of nonlinear equations. Numer. Math.49 (1986), pp. 123–138.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Patras, GR-261.10, Patras, Greece

    Michael N. Vrahatis & Evangelia C. Triantafyllou

Authors
  1. Michael N. Vrahatis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evangelia C. Triantafyllou
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

© M. N. Vrahatis, E. C. Triantafyllon, 1996

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vrahatis, M.N., Triantafyllou, E.C. Locating, characterizing and computing the stationary points of a function. Reliable Comput 2, 187–193 (1996). https://doi.org/10.1007/BF02425923

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02425923

Keywords

Search

Navigation