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The DIRECT algorithm: 25 years Later

Author

Listed:
  • Donald R. Jones

    (University of Michigan)

  • Joaquim R. R. A. Martins

    (University of Michigan)

Abstract

Introduced in 1993, the DIRECT global optimization algorithm provided a fresh approach to minimizing a black-box function subject to lower and upper bounds on the variables. In contrast to the plethora of nature-inspired heuristics, DIRECT was deterministic and had only one hyperparameter (the desired accuracy). Moreover, the algorithm was simple, easy to implement, and usually performed well on low-dimensional problems (up to six variables). Most importantly, DIRECT balanced local and global search (exploitation vs. exploration) in a unique way: in each iteration, several points were sampled, some for global and some for local search. This approach eliminated the need for “tuning parameters” that set the balance between local and global search. However, the very same features that made DIRECT simple and conceptually attractive also created weaknesses. For example, it was commonly observed that, while DIRECT is often fast to find the basin of the global optimum, it can be slow to fine-tune the solution to high accuracy. In this paper, we identify several such weaknesses and survey the work of various researchers to extend DIRECT so that it performs better. All of the extensions show substantial improvement over DIRECT on various test functions. An outstanding challenge is to improve performance robustly across problems of different degrees of difficulty, ranging from simple (unimodal, few variables) to very hard (multimodal, sharply peaked, many variables). Opportunities for further improvement may lie in combining the best features of the different extensions.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jglopt:v:79:y:2021:i:3:d:10.1007_s10898-020-00952-6
    DOI: 10.1007/s10898-020-00952-6
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    References listed on IDEAS

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    1. Qunfeng Liu & Guang Yang & Zhongzhi Zhang & Jinping Zeng, 2017. "Improving the convergence rate of the DIRECT global optimization algorithm," Journal of Global Optimization, Springer, vol. 67(4), pages 851-872, April.
    2. 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.
    3. Remigijus Paulavičius & Julius Žilinskas, 2014. "Simplicial Lipschitz optimization without the Lipschitz constant," Journal of Global Optimization, Springer, vol. 59(1), pages 23-40, May.
    4. G. Di Pillo & G. Liuzzi & S. Lucidi & V. Piccialli & F. Rinaldi, 2016. "A DIRECT-type approach for derivative-free constrained global optimization," Computational Optimization and Applications, Springer, vol. 65(2), pages 361-397, November.
    5. Jingxian Hao & Zhuoping Yu & Zhiguo Zhao & Peihong Shen & Xiaowen Zhan, 2016. "Optimization of Key Parameters of Energy Management Strategy for Hybrid Electric Vehicle Using DIRECT Algorithm," Energies, MDPI, vol. 9(12), pages 1-24, November.
    6. 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.
    7. Na, Jonggeol & Lim, Youngsub & Han, Chonghun, 2017. "A modified DIRECT algorithm for hidden constraints in an LNG process optimization," Energy, Elsevier, vol. 126(C), pages 488-500.
    8. YA. D. Sergeyev, 2000. "Efficient Strategy for Adaptive Partition of N-Dimensional Intervals in the Framework of Diagonal Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 107(1), pages 145-168, October.
    9. Qunfeng Liu & Jinping Zeng & Gang Yang, 2015. "MrDIRECT: a multilevel robust DIRECT algorithm for global optimization problems," Journal of Global Optimization, Springer, vol. 62(2), pages 205-227, June.
    10. Qinghua Tao & Xiaolin Huang & Shuning Wang & Li Li, 2017. "Adaptive block coordinate DIRECT algorithm," Journal of Global Optimization, Springer, vol. 69(4), pages 797-822, December.
    11. Ratko Grbić & Emmanuel Nyarko & Rudolf Scitovski, 2013. "A modification of the DIRECT method for Lipschitz global optimization for a symmetric function," Journal of Global Optimization, Springer, vol. 57(4), pages 1193-1212, December.
    12. 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.
    13. C. Kokail & C. Maier & R. van Bijnen & T. Brydges & M. K. Joshi & P. Jurcevic & C. A. Muschik & P. Silvi & R. Blatt & C. F. Roos & P. Zoller, 2019. "Self-verifying variational quantum simulation of lattice models," Nature, Nature, vol. 569(7756), pages 355-360, May.
    14. Remigijus Paulavičius & Lakhdar Chiter & Julius Žilinskas, 2018. "Global optimization based on bisection of rectangles, function values at diagonals, and a set of Lipschitz constants," Journal of Global Optimization, Springer, vol. 71(1), pages 5-20, May.
    15. Gianni Pillo & Stefano Lucidi & Francesco Rinaldi, 2015. "A Derivative-Free Algorithm for Constrained Global Optimization Based on Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 862-882, March.
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    4. Riccardo Pellegrini & Andrea Serani & Giampaolo Liuzzi & Francesco Rinaldi & Stefano Lucidi & Matteo Diez, 2022. "A Derivative-Free Line-Search Algorithm for Simulation-Driven Design Optimization Using Multi-Fidelity Computations," Mathematics, MDPI, vol. 10(3), pages 1-13, February.
    5. 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.
    6. Kaiwen Ma & Luis Miguel Rios & Atharv Bhosekar & Nikolaos V. Sahinidis & Sreekanth Rajagopalan, 2023. "Branch-and-Model: a derivative-free global optimization algorithm," Computational Optimization and Applications, Springer, vol. 85(2), pages 337-367, June.
    7. Slicker, Gerilyn & Hustedt, Jason T., 2022. "Predicting participation in the child care subsidy system from provider features, community characteristics, and use of funding streams," Children and Youth Services Review, Elsevier, vol. 136(C).
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