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Differential evolution algorithms with novel mutations, adaptive parameters, and Weibull flight operator

  • Optimization
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Abstract

Differential evolution (DE) is among the best evolutionary algorithms for global optimization. However, the basic DE has several shortcomings, like the slow convergence speed, and it is more likely to be stuck at local optima. Additionally, DE's performance is sensitive to its mutation strategies and control parameters for mutation and crossover. In this scope, we present in this paper three mechanisms to overcome DE limitations. First, two novel mutations called DE/mean-current/2 and DE/best-mean-current/2 are proposed and integrated in the DE algorithm, and they have both exploration ability and exploitation trend. On the other hand, to avoid being trapped in local minima of hard functions, a new exploration operator has been proposed called Weibull flight based on the Weibull distribution. Finally, new adapted control parameters based on the Weibull distribution are integrated. These parameters contribute to the optimization process by adjusting mutation scale and alleviating the parameter setting problem often encountered in various metaheuristics. The efficacy of the proposed algorithms called meanDE, MDEW, AMDE, and AMDEW is validated through intensive experimentations using classical tests, some challenging tests, the CEC2017, CEC2020, the most recent CEC2022, four constraint engineering problems, and the data clustering problem. Moreover, comparisons with several popular, recent, and high-performance optimization algorithms show a high effectiveness of the proposed algorithms in locating the optimal or near-optimal solutions with higher efficiency. The experiments clearly indicate the effectiveness of the new mutations compared to the standard DE mutations. Moreover, the proposed Weibull flight has a great capacity to deal with the hard composition functions of CEC benchmarks. Finally, the use of adapted control parameters for the mutation scale helps overcome the parameter setting problem commonly encountered in various metaheuristics.

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The code is available at: https://www.mathworks.com/matlabcentral/fileexchange/117725-differential-evolution-algorithms-with-novel-mutations?s_tid=srchtitle

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Authors and Affiliations

  1. LISIA Laboratory, Computer Science and Its Application Department, NTIC Faculty, University of Constantine 2, El Khroub, Algeria

    Abdesslem Layeb

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Appendices

Appendix

1.1 Pressure vessel design

figure a

Mathematical model

$${\text{Min}} \,\, f\left(\overrightarrow{x}\right)=0.6224{x}_{1}{x}_{3}{x}_{4}+1.7781{x}_{2}{x}_{3}^{2}+3.1661{x}_{1}^{2}{x}_{4}+19.84{x}_{1}^{2}{x}_{3}$$

subject to

$${g}_{1}\left(\overrightarrow{x}\right)={-x}_{1}+0.0193{x}_{3}\le 0,$$
$${g}_{2}\left(\overrightarrow{x}\right)=-{x}_{3}+0.00954{x}_{3}\le 0,$$
$${g}_{3}\left(\overrightarrow{x}\right)=-\pi {x}_{3}^{2}{x}_{4}-\frac{4}{3}\pi {x}_{3}^{3}+1296000\le 0,$$
$${g}_{4}\left(\overrightarrow{x}\right)={x}_{4}-240\le 0,$$

where

$$1*0.0625\le {x}_{i}\le 99*0.0625 i=\mathrm{1,2};$$
$$10\le {x}_{i}\le 200 i=\mathrm{3,4}.$$

1.2 Welded beam design

figure b

Mathematical model

$${\text{Min}} f(\overrightarrow{x})=1.10471{x}_{1}^{2}{x}_{2}+0.04811{x}_{3}{x}_{4}({x}_{2}+14)$$

subject to

$${g}_{1}\left(\overrightarrow{x}\right)=\tau \left(\overrightarrow{x}\right)-{\tau }_{{\text{max}}}\le 0,$$
$${g}_{2}\left(\overrightarrow{x}\right)=\sigma \left(\overrightarrow{x}\right)-{\sigma }_{{\text{max}}}\le 0,$$
$${g}_{3}\left(\overrightarrow{x}\right)={x}_{1}-{x}_{4}\le 0,$$
$${g}_{4}\left(\overrightarrow{x}\right)=1.10471{x}_{1}^{2}+0.04811{x}_{3}{x}_{4}\left(14.0+{x}_{2}\right)-5.0\le 0,$$
$${g}_{5}\left(\overrightarrow{x}\right)=0.125-{x}_{1}\le 0,$$
$${g}_{6}\left(\overrightarrow{x}\right)=\delta \left(\overrightarrow{x}\right)-{\delta }_{{\text{max}}}\le 0,$$
$${g}_{7}\left(\overrightarrow{x}\right)=P-{P}_{c}\left(\overrightarrow{x}\right)\le 0,$$

where \(0.1\le {x}_{1}\le \mathrm{2,0.1}\le {x}_{2}\le \mathrm{10,0.1}\le {x}_{3}\le \mathrm{10,0.1}\le {x}_{4}\le 2,\)

$$\tau (\overrightarrow{x})=\sqrt{{\left({\tau }^{\prime}\right)}^{2}+2{\tau }^{\prime}{\tau }^{{\prime}{\prime}}\frac{{x}_{2}}{2R}+{\left({\tau }^{{\prime}{\prime}}\right)}^{2}},$$
$${\tau }^{\prime}=\frac{P}{\sqrt{2}{x}_{1}{x}_{2}},{\tau }^{{\prime}{\prime}}=\frac{MR}{J}, M=P\left(L+\frac{{x}_{2}}{2}\right), R=\sqrt{\frac{{x}_{2}^{2}}{4}+{\left(\frac{{x}_{1}+{x}_{3}}{2}\right)}^{2}},$$
$$J=2\left\{\sqrt{2}{x}_{1}{x}_{2}\left[\frac{{x}_{2}^{2}}{4}+{\left(\frac{{x}_{1}+{x}_{3}}{2}\right)}^{2}\right]\right\}, \sigma \left(\overrightarrow{x}\right)=\frac{6PL}{{x}_{4}{x}_{3}^{2}}, \delta \left(\overrightarrow{x}\right)=\frac{6P{L}^{3}}{E{x}_{3}^{2}{x}_{4}},$$
$${P}_{{\text{c}}}(\overrightarrow{x})=\frac{4.013E\sqrt{\frac{{x}_{3}^{2}{x}_{4}^{6}}{36}}}{{L}^{2}}\left(1-\frac{{x}_{3}}{2L}\sqrt{\frac{E}{4G}} \right),$$
$$P=6000lb,L=14\text{ in}., E=30\times 1{0}^{6}\text{ psi}, G=12\times 1{0}^{6}{\text{psi}},$$
$${\tau }_{{\text{max}}}=13600\text{ psi},{ \sigma }_{{\text{max}}}=30000\text{ psi}, {\delta }_{{\text{max}}}=0.25\text{ in}.$$

Tension/compression spring design

figure c

Mathematical model

$${\text{Min}}\,\,\, f\left(\overrightarrow{x}\right)=({x}_{3}+2){x}_{2}{x}_{1}^{2}$$

Subject to

$${g}_{1}\left(\overrightarrow{x}\right)=1-\frac{{x}_{2}^{3}{x}_{3}}{71785{x}_{1}^{4}}\le 0,$$
$${g}_{2}\left(\overrightarrow{x}\right)=\frac{{4x}_{2}^{2}-{x}_{1}{x}_{2}}{12566({x}_{2}{x}_{1}^{3}-{x}_{1}^{4})}+\frac{1}{5108{x}_{1}^{2}}\le 0,$$
$${g}_{3}\left(\overrightarrow{x}\right)=1-\frac{140.41{x}_{1}}{{x}^{2}{x}_{3}}\le 0,$$
$${g}_{4}\left(\overrightarrow{x}\right)=\frac{{x}_{1}+{x}_{2}}{1.5}-1\le 0,$$

where \(0.05\le {x}_{1}\le 2.00, 0.25\le {x}_{2}\le 1.30\), \(2.0\le {x}_{3}\le 15.0.\)

Speed reducer design problem

figure d

Mathematical model

$${\text{Min}}\,\,\, f\left(\overrightarrow{x}\right)=0.7854{x}_{1}{x}_{2}^{2}\left(3.3333{x}_{3}^{2}+14.9334{x}_{3}-43.0934\right)-1.508{x}_{1}\left({x}_{6}^{2}+{x}_{7}^{2}\right)+7.4777\left({x}_{6}^{3}+{x}_{7}^{3}\right)+0.7854\left({x}_{4}{x}_{6}^{2}+{x}_{5}{x}_{7}^{2}\right)$$

subject to

$${g}_{1}\left(\overrightarrow{x}\right)=\frac{27}{{x}_{1}{x}_{2}^{2}{x}_{3}}-1\le 0$$
$${g}_{2}\left(\overrightarrow{x}\right)=\frac{397.5}{{x}_{1}{x}_{2}^{2}{x}_{3}^{2}}-1\le 0$$
$${g}_{3}\left(\overrightarrow{x}\right)=\frac{1.93{x}_{4}^{3}}{{x}_{2}{x}_{6}^{4}{x}_{3}}-1\le 0$$
$${g}_{4}\left(\overrightarrow{x}\right)=\frac{1.93{x}_{5}^{3}}{{x}_{2}{x}_{7}^{4}{x}_{3}}-1\le 0$$
$${g}_{5}\left(\overrightarrow{x}\right)=\frac{{\left[{\left(745\left({x}_{4}/{x}_{2}{x}_{3}\right)\right)}^{2}+16.9*{10}^{6}\right]}^{1/2}}{{110x}_{6}^{3}}-1\le 0$$
$${g}_{6}\left(\overrightarrow{x}\right)=\frac{{\left[{\left(745\left({x}_{5}/{x}_{2}{x}_{3}\right)\right)}^{2}+157.5*{10}^{6}\right]}^{1/2}}{{85x}_{7}^{3}}-1\le 0$$
$${g}_{7}\left(\overrightarrow{x}\right)=\frac{{x}_{2}{x}_{3}}{40}-1\le 0$$
$${g}_{8}\left(\overrightarrow{x}\right)=\frac{{5x}_{2}}{{x}_{1}}-1\le 0$$
$${g}_{9}\left(\overrightarrow{x}\right)=\frac{{x}_{1}}{12{x}_{2}}-1\le 0$$
$${g}_{10}\left(\overrightarrow{x}\right)=\frac{1.5{x}_{6}+1.9}{{x}_{4}}-1\le 0$$
$${g}_{11}\left(\overrightarrow{x}\right)=\frac{1.1{x}_{7}+1.9}{{x}_{5}}-1\le 0,$$

where \(2.6\le {x}_{1}\le 3.6\) \(0.7\le {x}_{2}\le 0.8\), \(17\le {x}_{3}\le 28\), \(7.3\le {x}_{4}\le 8.3\), \(7.3\le {x}_{5}\le 8.3\), \(2.9\le {x}_{6}\le 3.9\). \(5.0\le {x}_{7}\le 5.5\)

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Layeb, A. Differential evolution algorithms with novel mutations, adaptive parameters, and Weibull flight operator. Soft Comput 28, 7039–7091 (2024). https://doi.org/10.1007/s00500-023-09561-3

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