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On the convergence rate issues of general Markov search for global minimum

Author

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  • Dawid Tarłowski

    (Jagiellonian University)

Abstract

This paper focuses on the convergence rate problem of general Markov search for global minimum. Many of existing methods are designed for overcoming a very hard problem which is how to efficiently localize and approximate the global minimum of the multimodal function f while all information which can be used are the f-values evaluated for generated points. Because such methods use poor information on f, the following problem may occur: the closer to the optimum, the harder to generate a “better” (in sense of the cost function) state. This paper explores this issue on theoretical basis. To do so the concept of lazy convergence for a globally convergent method is introduced: a globally convergent method is called lazy if the probability of generating a better state from one step to another goes to zero with time. Such issue is the cause of very undesired convergence properties. This paper shows when an optimization method has to be lazy and the presented general results cover, in particular, the class of simulated annealing algorithms and monotone random search. Furthermore, some attention is put on accelerated random search and evolution strategies.

Suggested Citation

  • Dawid Tarłowski, 2017. "On the convergence rate issues of general Markov search for global minimum," Journal of Global Optimization, Springer, vol. 69(4), pages 869-888, December.
    • Handle: RePEc:spr:jglopt:v:69:y:2017:i:4:d:10.1007_s10898-017-0544-7
    DOI: 10.1007/s10898-017-0544-7
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    References listed on IDEAS

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    1. M. Locatelli, 2001. "Convergence and first hitting time of simulated annealing algorithms for continuous global optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(2), pages 171-199, December.
    2. Mathieu Gerber & Luke Bornn, 2017. "Improving simulated annealing through derandomization," Journal of Global Optimization, Springer, vol. 68(1), pages 189-217, May.
    3. Luis Rios & Nikolaos Sahinidis, 2013. "Derivative-free optimization: a review of algorithms and comparison of software implementations," Journal of Global Optimization, Springer, vol. 56(3), pages 1247-1293, July.
    4. R. L. Yang, 2000. "Convergence of the Simulated Annealing Algorithm for Continuous Global Optimization," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 691-716, March.
    5. Anatoly Zhigljavsky & Antanas Žilinskas, 2008. "Stochastic Global Optimization," Springer Optimization and Its Applications, Springer, number 978-0-387-74740-8, July.
    6. Zheng Peng & Donghua Wu & Wenxing Zhu, 2016. "The robust constant and its applications in random global search for unconstrained global optimization," Journal of Global Optimization, Springer, vol. 64(3), pages 469-482, March.
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