Cybernetics and Systems Analysis

, Volume 55, Issue 6, pp 1052–1058 | Cite as

A Method for Global Minimization of Functions Using the Krawczyk Operator

  • V. Yu. SemenovEmail author
  • Ye. V. Semenova


A method is proposed for global minimization of twice continuously differentiable functions of several variables on a given interval. The method is based on the solution of a system of nonlinear equations formed by partial derivatives of an objective function using the Krawczyk operator. The application of the method is illustrated by numerical examples.


global minimization Krawczyk operator rootfinding Hessian 


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  1. 1.
    J. Dennis and R. Schnabel, Numerical Methods for Unconditional Optimization and Nonlinear Equations [Russian translation], Mir, Moscow (1988).Google Scholar
  2. 2.
    R. M. Lewis, V. Torczon, M. V. Trosset, “Direct search methods: Then and now,” J. Comp. Appl. Math., Vol. 124, Nos. 1, 2, 191–207 (2000).MathSciNetCrossRefGoogle Scholar
  3. 3.
    A. Neumaier, “Complete search in continuous global optimization and constraint satisfaction,” Acta Numerica, Vol. 13, 271–369 (2004).MathSciNetCrossRefGoogle Scholar
  4. 4.
    R. E. Moore, Interval Arithmetic and Automatic Error Analysis in Digital Computing, Ph.D. Thesis, Stanford University (1962).Google Scholar
  5. 5.
    R. B. Kearfott, “Empirical evaluation of innovations in interval branch and bound algorithms for nonlinear algebraic systems,” SIAM J. Sci. Comput., Vol. 18, No. 2, 574–594 (1997).MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. E. Moore, “A test for existence of solutions to nonlinear systems,” SIAM J. Numer. Anal., Vol. 14, No. 4, 611–615 (1977).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Neumaier and S. Zuhe, “The Krawczyk operator and Kantorovich theorem,” J. Math. Anal. Applications, Vol. 149, No. 2, 437–443 (1990).Google Scholar
  8. 8.
    V. Yu. Semenov, “A method to find all the roots of the system of nonlinear algebraic equations based on the Krawczyk operator,” Cybernetics and Systems Analysis, Vol. 51, No 5, 819–825 (2015).MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. Yu. Semenov and E. V. Semenova, “Method for localizing the zeros of analytic functions based on the Krawczyk operator,” Cybernetics and Systems Analysis, Vol. 55, No. 3, 514–520 (2019).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. Yu. Semenov, “The method of determining all real nonmultiple roots of systems of nonlinear equations,” Computational Mathematics and Mathematical Physics, Vol. 47, No. 9, 1428–1434 (2007).MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Makhoul, S. Roucos, and H Gish, “Vector quantization in speech coding,” Proc. IEEE, Vol. 73, No. 11, 19–61 (1985).CrossRefGoogle Scholar
  12. 12.
    A. Neculai, “An unconstrained optimization test functions collection,” Advanced Modeling and Optimization, Vol. 10, No. 1, 147–161 (2008).Google Scholar
  13. 13.
    V. Semenov and A. Neumaier, “Calculation of local maxima for the likelihood function of speech autoregressive parameters,” in: Proc. Int. Conf. Computational Management Science, Vienna (2010).Google Scholar
  14. 14.
    R. Byrd, J. Nocedal, and R. Waltz, “KNITRO: An integrated package for nonlinear optimization,” in: G. Di Pillo and M. Roma (eds), Large-Scale Nonlinear Optimization, Springer, Boston (2006), 35–59.zbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.DELTA SPE LLCKyivUkraine
  2. 2.Institute of MathematicsNational Academy of Sciences of UkraineKyivUkraine

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