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MrDIRECT: a multilevel robust DIRECT algorithm for global optimization problems

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Abstract

Although DIRECT global optimization algorithm quickly gets close to the basin of the optimum, it often takes much longer to refine the solution to a high degree of accuracy. This behavior of DIRECT is similar to the “smooth mode phenomenon” encountered when solving linear systems discretized from partial differential equation (PDE). In the case of PDE, this smooth mode phenomenon can be eliminated efficiently by the multigrid algorithm in which the PDE solver is applied at different levels of discretization. In this paper we adapt the multigrid approach to a robust version of DIRECT algorithm, obtaining a “multilevel” robust DIRECT (MrDIRECT) algorithm. Although additional parameters are needed, our numerical results show that MrDIRECT is insensitive to the parameters, and the parameters setting proposed in this paper performs very well on the tested sets of benchmark problems, in terms of the speed with which the global optimum is found to a high degree of accuracy.

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Acknowledgments

We would like to thank an anonymous reviewer for the helpful comments and suggestions, which improve this paper greatly. We would like to thank Doctor Finkel D.E. and Professor Kelley C.T. for their DIRECT codes. We would also like to thank Professor Sergeyev Ya. D. for the GKLS codes.

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Authors and Affiliations

  1. School of Computing, Dongguan University of Technology, Dongguan, 523808, China

    Qunfeng Liu, Jinping Zeng & Gang Yang

Authors
  1. Qunfeng Liu
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  2. Jinping Zeng
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  3. Gang Yang
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Corresponding author

Correspondence to Qunfeng Liu.

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 11271069) and MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Project No. 13YJC630095).

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Liu, Q., Zeng, J. & Yang, G. MrDIRECT: a multilevel robust DIRECT algorithm for global optimization problems. J Glob Optim 62, 205–227 (2015). https://doi.org/10.1007/s10898-014-0241-8

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