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Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization

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Abstract

A separable cubic model, for smooth unconstrained minimization, is proposed and evaluated. The cubic model uses some novel secant-type choices for the parameters in the cubic terms. A suitable hard-case-free trust-region strategy that takes advantage of the separable cubic modeling is also presented. For the convergence analysis of our specialized trust region strategy we present as a general framework a model \(q\)-order trust region algorithm with variable metric and we prove its convergence to \(q\)-stationary points. Some preliminary numerical examples are also presented to illustrate the tendency of the specialized trust region algorithm, when combined with our cubic modeling, to escape from local minimizers.

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Authors and Affiliations

  1. Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, Rua Sérgio Buarque de Holanda, 651 Cidade Universitária “Zeferino Vaz”, Distrito Barão Geraldo, 13083-859, Campinas, SP, Brazil

    J. M. Martínez

  2. Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Ap. 89000, Caracas, 1080, Venezuela

    M. Raydan

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  1. J. M. Martínez
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  2. M. Raydan
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Correspondence to M. Raydan.

Additional information

This work was supported by PRONEX-CNPq/FAPERJ (E-26/111.449/2010-APQ1), CEPID–Industrial Mathematics/FAPESP (Grant 2011/51305-02), FAPESP (Projects 2013/05475-7 and 2013/07375-0), and CNPq (Project 400926/2013-0).

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Martínez, J.M., Raydan, M. Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization. J Glob Optim 63, 319–342 (2015). https://doi.org/10.1007/s10898-015-0278-3

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