Cybernetics and Systems Analysis

, Volume 33, Issue 4, pp 482–497 | Cite as

Modifiedr-algorithm to find the global minimum of polynomial functions

  • N. Z. Shor
  • P. I. Stetsyuk


Global Minimum Directional Descent Dual Solution Quadratic Problem Canonical Decomposition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. M. Pardalos and J. B. Rosen, Constrained Global Optimization: Algorithms and Applications, Lect. Notes Comput. Sci., Vol. 268 (1987).Google Scholar
  2. 2.
    N. Z. Shor and S. I. Stetsenko, Quadratic Extremal Problems and Nondifferentiable Optimization [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
  3. 3.
    N. Z. Shor, “An approach to finding global extrema in polynomial problems of mathematical programming,” Kibernetika, No. 5, 102–106 (1987).Google Scholar
  4. 4.
    N. Z. Shor, “A class of bounds for global minima of polynomial functions,” Kibernetika, No. 6, 9–11 (1987).Google Scholar
  5. 5.
    P. S. Aleksandrov (ed.), Hilbert Problems [in Russian], Nauka, Moscow (1969).Google Scholar
  6. 6.
    N. Z. Shor, “Quadratic programming problems,” Izv. Akad. Nauk SSSR, Tekhn. Kibern., No. 1, 128–139 (1987).Google Scholar
  7. 7.
    D. Hilbert, “lieber die Darstellung definiter Formen als Summen von Formen quadraten,” Math. Ann. Leipzig, 22, 342–350 (1888).CrossRefMathSciNetGoogle Scholar
  8. 8.
    E. Artin, “Ueber die Zerlegung definiter Funktionen in Quadrate” Hamburg Abh., No. 5, 100–115 (1927).Google Scholar
  9. 9.
    G. Polya, “Ueber positive Darstellung von Polynomen,” Vierteljahrsschrift d. naturforschenden Gesselsch., Zurich, 73, 141–145 (1928).Google Scholar
  10. 10.
    N. Z. Shor, “Dual estimates in multiextremal problems,” J. Global Optim., No. 2, 411–418 (1992).Google Scholar
  11. 11.
    N. Z. Shor and N. G. Zhurbenko, “A minimization method with space dilation in the direction of the difference of two successive gradients,” Kibernetika, No. 3, 51–591971).Google Scholar
  12. 12.
    N. Z. Short, Minimization Methods for Nondifferentiable Functions and Their Applications [in Russian], Naukova Dumka, Kiev (1979).Google Scholar
  13. 13.
    N. G. Zhurbenko and T. V. Marchuk, An Algorithm for Minimization of Nonsmooth Functionals (r-Algorithm) [in Russian], AN UkrSSR, RFAP, No. 22 (1979).Google Scholar
  14. 14.
    N. Z. Shor, “Minimization of matrix functions and nondifferentiable optimization,” in: Surveys in Applied and Industrial Mathematics [in Russian], Vol. 2, 113–138 (1995).zbMATHGoogle Scholar
  15. 15.
    C. Lemarechal, “Numerical experiments in nonsmooth optimization,” in: E. A. Nurminski (ed.), Progress in Nondifferentiable Optimization, CP-82-S2, IIASA, Laxenburg (1982), pp. 61–84.Google Scholar
  16. 16.
    S. I. Nesterova and V. A. Skokov, “Numerical analysis of programs for nonsmooth unconstrained optimization,” Ekon. Mat. Metody, 30, No. 2, 136–145 (1994).zbMATHGoogle Scholar
  17. 17.
    I. N. Molchanov, L. D. Nikolenko, M. F. Yakovlev, et al., A Software System for Solving Systems of Linear Algebraic Equations with Automatic Selection of the Solution Method (ARAS PC) [in Russian], No. 007135, GRFAP (1984).Google Scholar
  18. 18.
    Scientific Programs in FORTRAN. A Programmer's Manual, Vol. 2: Matrix Algebra and Linear Algebra [in Russian], Statistika, Moscow (1974).Google Scholar
  19. 19.
    L. Lovasz, “On the Shannon capacity of a graph,” IEEE Trans. Inform. Theory, 25, 1–7 (1979).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    N. Z. Shor and O. A. Berezovskii, “New algorithms for solving the maximum cut problem in a graph,” Kibern. Sist. Anal., No. 2, 100–106 (1995).Google Scholar
  21. 21.
    P. I. Stetsyuk, “Orthogonalizing linear operators in convex programming. I,” Kibern. Sist. Anal., No. 3, 97–119 (1997).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • N. Z. Shor
  • P. I. Stetsyuk

There are no affiliations available

Personalised recommendations