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Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization

Computational Optimization and Applications volume 65, pages 449–475 (2016)Cite this article

Abstract

In this paper we consider bound constrained global optimization problems where first-order derivatives of the objective function can be neither computed nor approximated explicitly. For the solution of such problems the DIRECT algorithm has been proposed which has a good ability to locate promising regions of the feasible domain and convergence properties based on the generation of a dense set of points over the feasible domain. However, the efficiency of DIRECT deteriorates as the dimension and the ill-conditioning of the objective function increase. To overcome these limits, we propose DIRECT-type algorithms enriched by the efficient use of derivative-free local searches combined with nonlinear transformations of the feasible domain and, possibly, of the objective function. We report extensive numerical results both on test problems from the literature and on an application in structural proteomics.

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Acknowledgments

We thank two anonymous Reviewers whose helpful comments and suggestions helped up to improve the paper.

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Authors and Affiliations

  1. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, CNR, via dei Taurini 19, 00185, Rome, Italy

    G. Liuzzi

  2. Dipartimento di Ingegneria Informatica Automatica e Gestionale “A. Ruberti”, “Sapienza” Università di Roma, via Ariosto 25, 00185, Rome, Italy

    S. Lucidi

  3. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università degli Studi di Roma “Tor Vergata”, viale del Politecnico 1, 00133, Rome, Italy

    V. Piccialli

Authors
  1. G. Liuzzi
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  2. S. Lucidi
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  3. V. Piccialli
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Corresponding author

Correspondence to G. Liuzzi.

Appendix

Appendix

1.1 The derivative-free local algorithm

In this section we report the sketch of a derivative-free procedure for unconstrained local minimization [23].

figure f
Table 5 Test problems
Full size table

In particular, the actual implementation of Algorithm DF that we use is based on the one proposed in [23] where \(d^i = e^i\), \(i=1,\dots ,n\), where \(e^i\) denotes the \(i\)-th coordinate direction in \(\mathfrak {R}^n\).

1.2 Test set description

In the following table, for each problem of our test set, we report its name, the adopted number of variables and the value of the known global minimum point.

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Liuzzi, G., Lucidi, S. & Piccialli, V. Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization. Comput Optim Appl 65, 449–475 (2016). https://doi.org/10.1007/s10589-015-9741-9

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  • DOI: https://doi.org/10.1007/s10589-015-9741-9

Keywords

  • Global optimization
  • DIRECT-type algorithms
  • Local minimizations

Mathematics Subject Classification

  • 65K05
  • 90C26