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Simplicial Lipschitz optimization without the Lipschitz constant

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  • Remigijus Paulavičius
  • Julius Žilinskas

Abstract

In this paper we propose a new simplicial partition-based deterministic algorithm for global optimization of Lipschitz-continuous functions without requiring any knowledge of the Lipschitz constant. Our algorithm is motivated by the well-known Direct algorithm which evaluates the objective function on a set of points that tries to cover the most promising subregions of the feasible region. Almost all previous modifications of Direct algorithm use hyper-rectangular partitions. However, other types of partitions may be more suitable for some optimization problems. Simplicial partitions may be preferable when the initial feasible region is either already a simplex or may be covered by one or a manageable number of simplices. Therefore in this paper we propose and investigate simplicial versions of the partition-based algorithm. In the case of simplicial partitions, definition of potentially optimal subregion cannot be the same as in the rectangular version. In this paper we propose and investigate two definitions of potentially optimal simplices: one involves function values at the vertices of the simplex and another uses function value at the centroid of the simplex. We use experimental investigation to compare performance of the algorithms with different definitions of potentially optimal partitions. The experimental investigation shows, that proposed simplicial algorithm gives very competitive results to Direct algorithm using standard test problems and performs particularly well when the search space and the numbers of local and global optimizers may be reduced by taking into account symmetries of the objective function. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Remigijus Paulavičius & Julius Žilinskas, 2014. "Simplicial Lipschitz optimization without the Lipschitz constant," Journal of Global Optimization, Springer, vol. 59(1), pages 23-40, May.
  • Handle: RePEc:spr:jglopt:v:59:y:2014:i:1:p:23-40
    DOI: 10.1007/s10898-013-0089-3
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    References listed on IDEAS

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    1. Giampaolo Liuzzi & Stefano Lucidi & Veronica Piccialli, 2010. "A partition-based global optimization algorithm," Journal of Global Optimization, Springer, vol. 48(1), pages 113-128, September.
    2. R. Horst, 2010. "Bisecton by Global Optimization Revisited," Journal of Optimization Theory and Applications, Springer, vol. 144(3), pages 501-510, March.
    3. D. Serafino & G. Liuzzi & V. Piccialli & F. Riccio & G. Toraldo, 2011. "A Modified DIviding RECTangles Algorithm for a Problem in Astrophysics," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 175-190, October.
    4. Anatoly Zhigljavsky & Antanas Žilinskas, 2008. "Stochastic Global Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-74740-8, June.
    5. Krivy, Ivan & Tvrdik, Josef & Krpec, Radek, 2000. "Stochastic algorithms in nonlinear regression," Computational Statistics & Data Analysis, Elsevier, vol. 33(3), pages 277-290, May.
    6. Antanas Žilinskas & Julius Žilinskas, 2013. "A hybrid global optimization algorithm for non-linear least squares regression," Journal of Global Optimization, Springer, vol. 56(2), pages 265-277, June.
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    Citations

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    Cited by:

    1. Nazih-Eddine Belkacem & Lakhdar Chiter & Mohammed Louaked, 2024. "A Novel Approach to Enhance DIRECT -Type Algorithms for Hyper-Rectangle Identification," Mathematics, MDPI, vol. 12(2), pages 1-24, January.
    2. Albertas Gimbutas & Antanas Žilinskas, 2018. "An algorithm of simplicial Lipschitz optimization with the bi-criteria selection of simplices for the bi-section," Journal of Global Optimization, Springer, vol. 71(1), pages 115-127, May.
    3. G. Liuzzi & S. Lucidi & V. Piccialli, 2016. "Exploiting derivative-free local searches in DIRECT-type algorithms for global optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 449-475, November.
    4. Christopher M. Cotnoir & Balša Terzić, 2017. "Decoupling linear and nonlinear regimes: an evaluation of efficiency for nonlinear multidimensional optimization," Journal of Global Optimization, Springer, vol. 68(3), pages 663-675, July.
    5. Donald R. Jones & Joaquim R. R. A. Martins, 2021. "The DIRECT algorithm: 25 years Later," Journal of Global Optimization, Springer, vol. 79(3), pages 521-566, March.
    6. Linas Stripinis & Remigijus Paulavičius, 2023. "Novel Algorithm for Linearly Constrained Derivative Free Global Optimization of Lipschitz Functions," Mathematics, MDPI, vol. 11(13), pages 1-19, June.
    7. Jonas Mockus & Remigijus Paulavičius & Dainius Rusakevičius & Dmitrij Šešok & Julius Žilinskas, 2017. "Application of Reduced-set Pareto-Lipschitzian Optimization to truss optimization," Journal of Global Optimization, Springer, vol. 67(1), pages 425-450, January.
    8. Stefan C. Endres & Carl Sandrock & Walter W. Focke, 2018. "A simplicial homology algorithm for Lipschitz optimisation," Journal of Global Optimization, Springer, vol. 72(2), pages 181-217, October.
    9. Remigijus Paulavičius & Yaroslav Sergeyev & Dmitri Kvasov & Julius Žilinskas, 2014. "Globally-biased Disimpl algorithm for expensive global optimization," Journal of Global Optimization, Springer, vol. 59(2), pages 545-567, July.

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