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Exact Numerical Differentiation on the Infinity Computer and Applications in Global Optimization

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Numerical Infinities and Infinitesimals in Optimization

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 43))

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Abstract

There exist many applications where it is necessary to approximate numerically derivatives of a function f(x) which is given by a computer procedure. A novel way to efficiently compute exact derivatives (the word “exact” means here with respect to the accuracy of the implementation of f(x)) is presented in this Chapter. It uses a new kind of a supercomputer—the Infinity Computer—able to work numerically with different finite, infinite, and infinitesimal numbers. Numerical examples illustrating these concepts and numerical tools are given. In particular, the field of Lipschitz global optimization having a special interest in exact numerical differentiation is considered in cases where there exists a code for computing f(x) but a code for its derivative \(f'(x)\) is not available. In addition, it is supposed that the first derivative \(f'(x)\) satisfies the Lipschitz condition. Algorithms using smooth piece-wise quadratic support functions and their convergence conditions are discussed. All the methods are implemented both in the traditional floating-point arithmetic and in the Infinity Computing framework.

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Authors and Affiliations

  1. University of Calabria, Rende, CS, Italy

    Maria Chiara Nasso & Yaroslav D. Sergeyev

  2. Lobachevsky State University, Nizhny Novgorod, Russia

    Yaroslav D. Sergeyev

Authors
  1. Maria Chiara Nasso
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  2. Yaroslav D. Sergeyev
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Correspondence to Maria Chiara Nasso .

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Editors and Affiliations

  1. Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. Scuola di Scienze e Tecnologie, Università degli Studi di Camerino, Camerino, Italy

    Renato De Leone

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Nasso, M.C., Sergeyev, Y.D. (2022). Exact Numerical Differentiation on the Infinity Computer and Applications in Global Optimization. In: Sergeyev, Y.D., De Leone, R. (eds) Numerical Infinities and Infinitesimals in Optimization. Emergence, Complexity and Computation, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-030-93642-6_9

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