Abstract
This paper addresses the issue of estimating the computational complexity of optimizing real-coded multimodal functions where the aim is to find all global optima. The proposed complexity method provides a partial answer to this question in the form of the estimated sample size needed to sample all basins of attraction of all global optima at least once. The rationale behind the approach is that, in optimization, in order to locate all possible optima of a multimodal function, we should first locate all its basins of attraction and then exploit them using, e.g., gradient information. Therefore, estimating the cost of locating all basins of attraction provides a lower bound on the computational budget necessary to optimize a multimodal function. This lower bound can serve as a measure of the computational complexity of the problem. From the conducted experimentation, we show that the proposed model can be very useful in determining the computational complexity of specialized benchmarks and can also be used as a heuristic in case of having some partial knowledge of the features of the targeted function.
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- 1.
HillVallEA is the winner of GECCO’19 Competition on Niching Methods for Multimodal Optimization https://cs.adelaide.edu.au/~markus/temp/gecco2019_certificates2019-ALL.pdf and NMMSO winner of the competition at IEEE CEC 2015 https://titan.csit.rmit.edu.au/~e46507/cec15-niching/competition/NichingCEC2015.pdf. Accessed on November 2019.
- 2.
Note here that for other number of dimensions we can equivalently talk about length (1 dimension), volume (3 dimensions) or hypervolume (\(>3\) dimensions).
- 3.
Code for the experiments is available at https://www-apps.univ-lehavre.fr/forge/jimenezj/multistartnomad published under GNU Public License v3.0.
- 4.
Results for HillVallEA, which is available from https://github.com/scmaree/HillVallEA, were obtained by making 50 independent runs of the algorithm.
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Acknowledgments
This paper has been supported in part by projects DeepBio (TIN2017-85727-C4-2-P) funded by the Spanish Ministry of Economy, Industry and Competitiveness and FCT project (UID/EEA/50009/2013). We would also like to thank Petr Pošík for his valuable comments on this paper.
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Jiménez Laredo, J.L., Merelo Guervós, J.J., Fernandes, C.M., Sanlaville, E. (2020). A Method for Estimating the Computational Complexity of Multimodal Functions. In: Castillo, P.A., Jiménez Laredo, J.L., Fernández de Vega, F. (eds) Applications of Evolutionary Computation. EvoApplications 2020. Lecture Notes in Computer Science(), vol 12104. Springer, Cham. https://doi.org/10.1007/978-3-030-43722-0_13
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