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Enhanced Lichtenberg algorithm: a discussion on improving meta-heuristics

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Abstract

Meta-heuristics have been successfully applied to many complex optimization problems. One of the main reasons for its success is its ability to handle non-convex, nonlinear, multimodal, multi-variable, and multi-objective problems with easy implementation. However, the quality of the response of these algorithms to an optimization problem is highly susceptible to the control parameters, and few works aim to tune them or find tools that can improve the algorithms. The literature is rich in proposals for new algorithms, but not for improving existing ones. This paper presents different strategies for tuning and accelerating meta-heuristics using the first hybrid algorithm in the literature. The Lichtenberg algorithm is inspired by lightning and Lichtenberg figures and has been increasingly successfully applied to various optimization problems. However, a study of its best parameters has never been presented until now. After a discussion of the best tuning tools, its tuning parameters are performed using response surface methodology. Then, 14 versions are studied through 10 test functions using chaos theory and Lévy flights scenarios. After 13,500 simulations, the chaotic Lichtenberg algorithm equipped with the piecewise function and tuned parameters proved the best version with only 16% similarity to the original algorithm. Then, it was compared to the genetic algorithm, particle swarm optimization, gray wolf optimizer, salp swarm optimization, whale optimization algorithm, and dragonfly algorithm. The proposed algorithm had both the best average accuracy, lower computational cost, and the smallest standard deviation.

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Abbreviations

CCD:

Central composite design

CLA1:

Chaotic Lichtenberg algorithm with Chebyshev series

CLA2:

Chaotic Lichtenberg algorithm with circle series

CLA3:

Chaotic Lichtenberg algorithm with Gauss series

CLA4:

Chaotic Lichtenberg algorithm with iterative series

CLA5:

Chaotic Lichtenberg algorithm with logistic series

CLA6:

Chaotic Lichtenberg algorithm with piecewise series

CLA7:

Chaotic Lichtenberg algorithm with sine series

CLA8:

Chaotic Lichtenberg algorithm with singer series

CLA9:

Chaotic Lichtenberg algorithm with sinusoidal series

CLA10:

Chaotic Lichtenberg algorithm with tent series

DA:

Dragonfly algorithm

DLA:

Diffusion limited aggregation

DoE:

Design of experiments

F opt :

Theoretical optimum

GA:

Genetic algorithm

GWO:

Grey wolf optimizer

LA:

Lichtenberg algorithm

LF:

Lévy flights

LFG:

Lichtenberg figure

LLA1:

Chaotic Lichtenberg algorithm with Lévy flights series (β = ½)

LLA2:

Chaotic Lichtenberg algorithm with Lévy flights series (β = 1)

M :

Figure switching parameter in LA

n :

Dimension or number of design variables (of problems)

N p :

Number of particles (DLA and LA)

OLA:

Original Lichtenberg algorithm

Pop:

Number of population used in LA

PSO:

Particle swarm optimization

R c :

Creation radius (DLA and LA)

Ref:

LA refinement

rand:

Random number between 0 and 1

S :

Stickiness coefficient (DLA and LA)

SSA:

Salp swarm algorithm

TLA:

Tuned Lichtenberg algorithm

WOA:

Whale optimization algorithm

μ :

Mean

σ :

Standard deviation

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Acknowledgements

The authors would like to acknowledge the financial support from the Brazilian agencies FAPESP (São Paulo Research Foundation, Grant 2022/10683-7), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico—Process Number 150117/2021-3), and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais—APQ-00385-18).

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Authors and Affiliations

  1. Computer Science Division, Aeronautics Institute of Technology (ITA), São José dos Campos, Brazil

    João Luiz Junho Pereira

  2. Mechanical Engineering Institute, Federal University of Itajubá (UNIFEI), Itajubá, Brazil

    Matheus Brendon Francisco, Sebastião Simões Cunha Jr & Guilherme Ferreira Gomes

  3. Institute of Electrical Systems and Energy, Federal University of Itajubá (UNIFEI), Itajubá, Brazil

    Fabricio Alves de Almeida

  4. Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong (HKU), Hong Kong, PR China

    Benedict Jun Ma

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  1. João Luiz Junho Pereira
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Appendices

Appendix A. Simulation results for the tuning of the Lichtenberg algorithm parameters

Tables 14 and 15.

Table 14 Means results for the 10 objective functions using the matrix of experiments
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Table 15 Standard deviations results for the 10 objective functions using the matrix of experiments
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Appendix B. Lichtenberg algorithm optimization results for each modified version

See Tables 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and 29.

Table 16 Performance of the original Lichtenberg algorithm (LA)
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Table 17 Performance of the tuned Lichtenberg algorithm (TLA)
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Table 18 Performance of the chaotic Lichtenberg algorithm with Chebyshev map (CLA1)
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Table 19 Performance of the chaotic Lichtenberg algorithm with circle map (CLA2)
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Table 20 Performance of the Chaotic Lichtenberg algorithm with Gauss map (CLA3)
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Table 21 Performance of the Chaotic Lichtenberg algorithm with iterative map (CLA4)
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Table 22 Performance of the chaotic Lichtenberg algorithm with logistic map (CLA5)
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Table 23 Performance of the chaotic Lichtenberg algorithm with piecewise map (CLA6)
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Table 24 Performance of the Chaotic Lichtenberg algorithm with Sine map (CLA7)
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Table 25 Performance of the chaotic Lichtenberg algorithm with singer map (CLA8)
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Table 26 Performance of the Chaotic Lichtenberg algorithm with Sinusoidal map (CLA9)
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Table 27 Performance of the chaotic Lichtenberg algorithm with tent map (CLA10)
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Table 28 Performance of the Lévy Lichtenberg algorithm with β = 3/2 (LLA1)
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Table 29 Performance of the Lévy Lichtenberg algorithm with β = 1 (LLA2)
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Appendix C. Results of other algorithms for test functions

See Tables 30, 31, 32, 33, 34 and 35.

Table 30 Genetic algorithm (GA)
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Table 31 Particle swarm optimization (PSO)
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Table 32 Grey wolf optimizer (GWO)
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Table 33 Salp swarm algorithm (SSA)
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Table 34 Whale optimization algorithm (WOA)
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Table 35 Dragonfly algorithm (DA)
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Pereira, J.L.J., Francisco, M.B., de Almeida, F.A. et al. Enhanced Lichtenberg algorithm: a discussion on improving meta-heuristics. Soft Comput 27, 15619–15647 (2023). https://doi.org/10.1007/s00500-023-08782-w

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