Journal of Global Optimization

, Volume 67, Issue 4, pp 851–872 | Cite as

Improving the convergence rate of the DIRECT global optimization algorithm

  • Qunfeng Liu
  • Guang Yang
  • Zhongzhi Zhang
  • Jinping Zeng


DIRECT is derivative-free global-search algorithm has been found to perform robustly across a wide variety of low-dimensional test problems. The reason DIRECT’s robustness is its lack of algorithmic parameters that need be “tuned” to make the algorithm perform well. In particular, there is no parameter that determines the relative emphasis on global versus local search. Unfortunately, the same algorithmic features that enable DIRECT to perform so robustly have a negative side effect. In particular, DIRECT is usually quick to get close to the global minimum, but very slow to refine the solution to high accuracy. This is what we call DIRECT’s “eventually inefficient behavior.” In this paper, we outline two root causes for this undesirable behavior and propose modifications to eliminate it. The paper builds upon our previously published “MrDIRECT” algorithm, which we can now show only addressed the first root cause of the “eventually inefficient behavior.” The key contribution of the current paper is a further enhancement that allows MrDIRECT to address the second root cause as well. To demonstrate the effectiveness of the enhanced MrDIRECT, we have identified a set of test functions that highlight different situations in which DIRECT has convergence issues. Extensive numerical work with this test suite demonstrates that the enhanced version of MrDIRECT does indeed improve the convergence rate of DIRECT.


Global optimization Multilevel algorithm DIRECT algorithm MrDIRECT algorithm Convergence rate 



We would like to thank two anonymous reviewers for their very helpful suggestions, which improve this paper greatly. We would like to thank Doctor Finkel D.E., Professor Kelley C.T. and Professor Sergeyev Ya. D. for their DIRECT code and the GKLS codes, respectively.

Supplementary material

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Qunfeng Liu
    • 1
  • Guang Yang
    • 1
  • Zhongzhi Zhang
    • 1
  • Jinping Zeng
    • 1
  1. 1.College of Computer Science and TechnologyDongguan University of TechnologyDongguanChina

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