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The Role of grossone in Nonlinear Programming and Exact Penalty Methods

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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

Exact penalty methods form an important class of methods for solving constrained optimization problems. Using penalty functions, the original constrained optimization problem can be transformed in an “equivalent” unconstrained problem. In this chapter we show how grossone can be utilized in constructing exact differentiable penalty functions for the case of only equality constraints, the general case of equality and inequality constraints, and quadratic problems. These new penalty functions allow to recover the solution of the unconstrained problem from the finite term (in its grossone expansion) of the optimal solution of the unconstrained problem. Moreover, Lagrangian duals associated to the constraints are also automatically obtained from the infinitesimal terms. Finally a new algorithmic scheme is presented.

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Notes

  1. 1.

    We refer the reader to Chap. 1 for an in–depth description of this new applied approach to infinite and infinitesimal quantities and the arithmetics of infinity.

  2. 2.

    For further details we refer the reader to Chap. 2 and references therein.

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Authors and Affiliations

  1. School of Science and Technology, University of Camerino, Camerino, MC, Italy

    Renato De Leone

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  1. Renato De Leone
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Correspondence to Renato De Leone .

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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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De Leone, R. (2022). The Role of grossone in Nonlinear Programming and Exact Penalty Methods. 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_3

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