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Evaluating Lipschitz Constants for Functions Given by Algorithms

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Abstract

This paper describes an efficient method (O(n)) to evaluate the Lipschitz constant for functions described in some algorithmic language. Considering arithmetical operations as the basis of the algorithmic language and supported by control structures, the rules to evaluate such Lipschitz constants are presented and their correctness is proved. An extension of the method to evaluate Lipschitz constants over interval domains is also presented. Examples are presented, but the effectiveness of the method is doubtful when compared to other approaches, and effective enhancements based on slope evaluations are also explored.

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References

  1. G. Alefeld and J. Herzberger, Introduction to Interval Computations, Academic Press: New York, 1983.

    Google Scholar 

  2. H. Bauch, K.-U. Jahn, D. Oelschlaägel, H. Süsse, and V.Wiebigke, “Intervallmathematik,” in Mathematisch-Naturwissentschaftliche Bibliothek, Band 72, B.G. Teubner (Ed.), 1987.

  3. T. Beck and H. Fischer, “The if-problem in automatic differentiation,” Journal of Computational and Applied Mathematics, vol. 50, pp. 119–131, 1994.

    Google Scholar 

  4. A. Griewank and G. Corliss (Eds.), “Automatic differentiation of algorithms: Theory, implementation and application,” in SIAM Proceeding Series, Philadelphia, 1991.

  5. G. Kedem, “Automatic differentiation of computer programs,” ACMTrans. on Mathematical Software, vol. 6, no. 2, pp. 150–165, 1980.

    Google Scholar 

  6. R. Krawczyk and A. Neumaier, “Interval slopes for rational functions and associated centered forms,” SIAM Journal of Numerical Analysis, vol. 22, no. 3, pp. 604–616, 1985.

    Google Scholar 

  7. R.E. Moore, Interval Analysis, Prentice-Hall: Englewood Cliffs, 1966.

    Google Scholar 

  8. R.E. Moore, “Methods and applications of interval analysis,” in SIAM Studies in Applied Mathematics, Philadelphia, 1979.

  9. J.B. Oliveira, “Slope methods for sharper interval inclusions and their application to global optimization,” Ph.D. Thesis, Informatik III, Technische Universität Hamburg-Harburg, Hamburg, Germany, 1996.

    Google Scholar 

  10. J.B. Oliveira, “New slope methods for sharper interval functions and a note on Fischer's acceleration method,” Reliable Computing, vol. 2, no. 3, pp. 299–320, 1996.

    Google Scholar 

  11. L.B. Rall, Automatic Differentiation: Techniques and Applications, Springer, 1981 Lecture, Notes in Computer Science, vol. 120.

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Authors and Affiliations

  1. Faculdade de Informática, Pontifícia Universidade Católica do Rio Grande do Sul, 90619-900, Porto Alegre, Brazil

    João Batista Oliveira

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  1. João Batista Oliveira
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Oliveira, J.B. Evaluating Lipschitz Constants for Functions Given by Algorithms. Computational Optimization and Applications 16, 215–229 (2000). https://doi.org/10.1023/A:1008791528195

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  • DOI: https://doi.org/10.1023/A:1008791528195

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