A Method for Global Minimization of Functions Using the Krawczyk Operator
Article
First Online:
- 2 Downloads
Abstract
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.
Keywords
global minimization Krawczyk operator rootfinding HessianPreview
Unable to display preview. Download preview PDF.
References
- 1.J. Dennis and R. Schnabel, Numerical Methods for Unconditional Optimization and Nonlinear Equations [Russian translation], Mir, Moscow (1988).Google Scholar
- 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.A. Neumaier, “Complete search in continuous global optimization and constraint satisfaction,” Acta Numerica, Vol. 13, 271–369 (2004).MathSciNetCrossRefGoogle Scholar
- 4.R. E. Moore, Interval Arithmetic and Automatic Error Analysis in Digital Computing, Ph.D. Thesis, Stanford University (1962).Google Scholar
- 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.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.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.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.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.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.J. Makhoul, S. Roucos, and H Gish, “Vector quantization in speech coding,” Proc. IEEE, Vol. 73, No. 11, 19–61 (1985).CrossRefGoogle Scholar
- 12.A. Neculai, “An unconstrained optimization test functions collection,” Advanced Modeling and Optimization, Vol. 10, No. 1, 147–161 (2008).Google Scholar
- 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.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
Copyright information
© Springer Science+Business Media, LLC, part of Springer Nature 2019