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A single-solution–compact hybrid algorithm for continuous optimization

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Abstract

This research paper proposes a memetic algorithm based on a hybridization of two metaheuristic approaches, a single-solution method and a compact optimization algorithm. The hybrid algorithm is thus a bi-module framework, where each module encapsulates a different search logic. Both modules use the Non-Uniform Mutation, although with different flavors: the first one acting on a single variable at a time, the second one acting on multiple variables. Hence, the algorithm is dubbed “compact Single/Multi Non-Uniform Mutation” (in short, cSM). It is designed for being suitable for tackling optimization problems on memory-constrained devices, i.e., devices for which the available memory may be not enough to run population-based metaheuristics. The performance of cSM is evaluated by an extensive comparative analysis including 12 state-of-the-art memory-saving (also called “lightweight”) algorithms on three well-known testbeds, namely the BBOB, the CEC-2014, and CEC-2017 benchmarks, as well as seven real-world optimization problems included in the CEC-2011 benchmark. In the case of the CEC benchmarks, our method is also compared against the top (population-based) algorithms that participated in respective competitions. The numerical results indicate that, compared to all the other lightweight algorithms under study, the proposed algorithm is better at handling most functions at different dimensionalities, especially in the case of non-separable problems.

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Data Availability

The datasets generated and/or analyzed in the current study are available on reasonable request.

Notes

  1. In preliminary experiments on the BBOB benchmark, not reported here for brevity, three cSM variants have been compared: one, where Module-2 was initialized with \(-\varvec{elite}\); one, where Module-2 was initialized with \(\varvec{elite}\); one, where the initialization in line 22 of Algorithm 1) was not present, thus \(\varvec{\mu }\) was kept at its initial values (zero). When looking at all the BBOB functions in 20 and 40 dimensions together, the numerical results obtained with the initialization at \(-\varvec{elite}\) were statistically better than those obtained with the other two variants. Hence, this variant is presented in the rest of this paper. Our intuition is that the initialization from an opposite \(\varvec{elite}\) allows Module-2 to explore a different part of the search space, thus counterbalancing the exploitation achieved in Module-1.

  2. https://coco.gforge.inria.fr.

  3. In particular, concerning L-SHADE (please see the original paper available at http://metahack.org/CEC2014-Tanabe-Fukunaga.pdf, Sec. III.A and Table 2), it results that the size (in terms of number of D-dimensional vectors) of the population and archive explicitly depends on D. Likewise, for JSO both the original paper and its publicly available source code (please see https://github.com/justinjk007/JSO/blob/master/include/JSO.hpp, line 32) indicate that the population size explicitly depends on D.

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Author information

Authors and Affiliations

  1. Department of Fundamental Informatics and Its Applications, Constantine 2 University, Constantine, Algeria

    Souheila Khalfi & Amer Draa

  2. Department of Mathematics and Computer Science, Mila University Center, Mila, Algeria

    Souheila Khalfi

  3. Department of Information Engineering and Computer Science, University of Trento, Trento, Italy

    Giovanni Iacca

Authors
  1. Souheila Khalfi
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  2. Giovanni Iacca
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  3. Amer Draa
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Contributions

The authors’ contribution in the different parts of the paper is as follows: SK: Conceptualization; Methodology; Software; Validation; Formal analysis; Investigation; Visualization; Resources; Data Curation; Writing—Original Draft; Writing—Review & Editing. GI: Conceptualization; Methodology; Writing—Review & Editing. AD: Conceptualization; Methodology; Writing—Review & Editing.

Corresponding author

Correspondence to Giovanni Iacca.

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The authors did not receive support from any organization for this work. The authors declare that they have no conflict of interest.

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A Detailed results on the CEC-2014 and CEC-2017 benchmarks

A Detailed results on the CEC-2014 and CEC-2017 benchmarks

See Tables 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and 29.

Table 11 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2014 in 10 dimensions
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Table 12 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2014 in 10 dimensions
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Table 13 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2014 in 30 dimensions
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Table 14 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2014 in 30 dimensions
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Table 15 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2014 in 50 dimensions
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Table 16 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2014 in 50 dimensions
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Table 17 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2014 in 100 dimensions
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Table 18 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2014 in 100 dimensions
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Table 19 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2017 in 10 dimensions
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Table 20 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2017 in 10 dimensions
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Table 21 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2017 in 30 dimensions
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Table 22 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2017 in 30 dimensions
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Table 23 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2017 in 50 dimensions
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Table 24 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2017 in 50 dimensions
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Table 25 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against compact algorithms on CEC-2017 in 100 dimensions
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Table 26 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against single-solution algorithms on CEC-2017 in 100 dimensions
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Table 27 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against L-SHADE on CEC-2014 in 10, 30, 50 and 100 dimensions
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Table 28 Average error ± standard deviation and Wilcoxon rank-sum test (reference: cSM) for cSM against JSO on CEC-2017 in 10, 30, 50 and 100 dimensions
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Table 29 List of symbols used in the paper
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Khalfi, S., Iacca, G. & Draa, A. A single-solution–compact hybrid algorithm for continuous optimization. Memetic Comp. 15, 155–204 (2023). https://doi.org/10.1007/s12293-022-00386-5

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