Hybrid particle swarm-differential evolution algorithm and its engineering applications

Abstract

Differential evolution (DE) has been applied to solve various optimization problems due to its simplicity and high search efficiency. However, researchers have confirmed that it still has some shortcomings such as premature convergence and slow convergence, especially when dealing with complex optimization problems. To address these concerning issues, this paper proposes a hybrid particle swarm-differential evolution algorithm (HPSDE). Firstly, to enhance the optimization performance, a modified updating scheme named particle-swarm mutation strategy is designed and an improved control parameters adaption is developed. Then, DE/rand-to-rand/1 mutation strategy is adopted to increase the population diversity and enhance the ability of particles escaping away from local optima. To achieve an improved DE variant with rapid convergence and fine stability, a random mutation framework is designed to combine the two mutation strategies mentioned above. To evaluate the efficiency of HPSDE algorithm, four different experiments have been taken on twenty-nine benchmark functions. The numerical results validate that HPSDE has better overall performance than the other competitors. Additionally, HPSDE is successfully applied to solve five typical engineering optimization problems.

Appendix

1.1 A.1 Unimodal benchmark functions

Function	Dim	Search range	Global optimum
${f}_{1}\left(x\right)={\sum }_{i=1}^{D}{x}_{i}^{2}$	30	[− 100,100]	0
${f}_{2}\left(x\right)={\sum }_{i=1}^{D}\left|{x}_{i}\right|+{\prod }_{i=1}^{D}\left|{x}_{i}\right|$	30	[− 10,10]	0
${f}_{3}\left(x\right)={\sum }_{i=1}^{D}{({\sum }_{j=1}^{i}{x}_{j})}^{2}$	30	[− 100,100]	0
${f}_{4}\left(x\right)={max}_{i}\{\left|{x}_{i}\right|,1\le i\le D\}$	30	[− 100,100]	0
${f}_{5}\left(x\right)={\sum }_{i=1}^{D-1}[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{({x}_{i}-1)}^{2}]$	30	[− 30,30]	0
${f}_{6}\left(x\right)={\sum }_{i=1}^{D}{([{x}_{i}+0.5])}^{2}$	30	[− 100,100]	0
${f}_{7}\left(x\right)={\sum }_{i=1}^{D}i{x}^{4}+random[\mathrm{0,1})$	30	[− 1.28,1.28]	0
${f}_{8}\left(x\right)={x}_{1}^{2}+{10}^{6}\sum_{i=2}^{D}{x}_{i}^{2}$	30	[− 100,100]	0
${f}_{9}\left(x\right)={10}^{6}*{x}_{1}^{2}+\sum_{i=2}^{D}{x}_{i}^{2}$	30	[− 100,100]	0
${f}_{10}\left(x\right)=\sum_{i=1}^{D}{\left({10}^{6}\right)}^{\frac{i-1}{D-1}}*{x}_{i}^{2}$	30	[− 100,100]	0

1.2 A. 2 Multimodal benchmark functions

Function	Dim	Search range	Global optimum
${f}_{11}\left(x\right)=\sum_{i=1}^{D}\left(\sum_{k=0}^{kmax}\left[{a}^{k}\mathrm{cos}\left(2\pi {b}^{k}\left({x}_{i}+0.5\right)\right)\right]\right)-D\sum_{k=0}^{kmax}\left[{a}^{k}\mathrm{cos}\left(\pi {b}^{k}\right)\right]$ $a=0.5,b=3,kmax=20$	30	[− 0.5,0.5]	0
${f}_{12}\left(x\right)=1-\mathrm{cos}\left(2\pi \sqrt{\sum_{i=1}^{D}{x}_{i}^{2}}\right)+0.1*\left(2\pi \sqrt{\sum_{i=1}^{D}{x}_{i}^{2}}\right)$	30	[− 100,100]	0
${f}_{13}\left(x\right)=10*D+{\sum }_{i=1}^{D}[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)]$	30	[− 0.5,0.5]	0
${f}_{14}\left(x\right)={\sum }_{i=1}^{D}-{x}_{i}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})$	30	[− 500,500]	− 418.9829*D
${f}_{15}\left(x\right)={\sum }_{i=1}^{D}[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10]$	30	[− 5.12,5.12]	0
${f}_{16}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{D}{\sum }_{i=1}^{D}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{D}{\sum }_{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$	30	[− 32,32]	0
${f}_{17}\left(x\right)=\frac{1}{4000}{\sum }_{i=1}^{D}{x}_{i}^{2}-{\prod }_{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$	30	[− 600,600]	0
${f}_{18}\left(x\right)=\frac{\pi }{D}\left\{10\mathrm{sin}\left(\pi {y}_{i}\right)+{\sum }_{i=1}^{D-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{sin}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{D}-1\right)}^{2}\right\} +{\sum }_{i=1}^{D}u\left({x}_{i},\mathrm{10,100,4}\right)$ ${y}_{i}=1+\frac{{x}_{i}+1}{4}$, $\mathrm{u}\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{({x}_{i}-a)}^{m}{x}_{i}>a\\ -a<{x}_{i}<a\\ k{(-{x}_{i}-a)}^{m}{x}_{i}<-a\end{array}\right.$	30	[− 50,50]	0
$\begin{array}{c}{f}_{19}\left(x\right)=0.1\{{sin}^{2}\left(3\pi {x}_{1}\right)+{\sum }_{i=1}^{D}{\left({x}_{i}-1\right)}^{2}\left[1+{sin}^{2}\left(3\pi {x}_{1}+1\right)\right]\\ +{({x}_{D}-1)}^{2}[1+{sin}^{2}(2\pi {x}_{D})]\}+{\sum }_{i=1}^{D}u({x}_{i},\mathrm{5,100,4})\end{array}$	30	[− 50,50]	0

1.3 A.3 Fixed-dimension multimodal benchmark functions

Function	Dim	Search range	Global optimum
${f}_{20}\left(x\right)={(\frac{1}{500}+{\sum }_{j=1}^{25}{(j+{\sum }_{i=1}^{2}{({x}_{i}-{a}_{ij})}^{6})}^{-1})}^{-1}$	2	[− 65,65]	0.9980
${f}_{21}\left(x\right)={\sum }_{i=1}^{11}{[{a}_{i}-\frac{{x}_{1}({b}_{i}^{2}+{b}_{i}{x}_{2})}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}]}^{2}$	4	[− 5,5]	0.0003
${f}_{22}\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$	2	[− 5,5]	− 1.0316
${f}_{23}\left(x\right)={({x}_{2}-\frac{5.1}{4{\pi }^{2}}{x}_{1}^{2}+\frac{5}{\pi }{x}_{1}-6)}^{2}+10\left(1-\frac{1}{8\pi }\right)cos{x}_{1}+10$	2	[− 5,5]	0.3980
$\begin{array}{l}{f}_{24}\left(x\right)=\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)\right]\\ \times [30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\times (18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2})]\end{array}$	2	[− 2,2]	3.0000
${f}_{25}\left(x\right)=-{\sum }_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\sum }_{j=1}^{3}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2})$	3	[1,3]	− 3.8600
${f}_{26}\left(x\right)=-{\sum }_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\sum }_{j=1}^{6}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2})$	6	[0,1]	− 3.3200
${f}_{27}\left(x\right)=-{\sum }_{i=1}^{5}{[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}]}^{-1}$	4	[0,10]	− 10.1532
${f}_{28}\left(x\right)=-{\sum }_{i=1}^{7}{[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}]}^{-1}$	4	[0,10]	− 10.4028
${f}_{29}\left(x\right)=-{\sum }_{i=1}^{10}{[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+{c}_{i}]}^{-1}$	4	[0,10]	− 10.5363

1.4 B.1 Tension spring design problem (TSD)

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

Subject to:

$${g}_{1}\left(x\right)=1-\frac{{x}_{2}^{3}{x}_{3}}{71785{x}_{1}^{4}}\le 0$$

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

$${g}_{3}\left(x\right)=1-\frac{140.45{x}_{1}}{{x}_{2}^{2}{x}_{3}}\le 0$$

$${g}_{4}\left(x\right)=\frac{{x}_{1}+{x}_{2}}{1.5}-1\le 0$$

$0.25\le {x}_{1}\le 1.3$, $0.05\le {x}_{2}\le 2.0$, $2\le {x}_{3}\le 15$

1.5 B.2 Pressure vessel design problem (PVD)

Min $f\left(x\right)=0.6224{x}_{1}{x}_{3}{x}_{4}+1.7881{x}_{2}{x}_{3}^{2}+3.1661{x}_{1}^{2}{x}_{4}+19.84{x}_{1}^{2}{x}_{3}$

Subject to:

$${g}_{1}\left(x\right)=-{x}_{1}+0.0193{x}_{3}\le 0$$

$${g}_{2}\left(x\right)=-{x}_{2}+0.00954{x}_{3}\le 0$$

$${g}_{3}\left(x\right)=-\pi {x}_{3}^{2}{x}_{4}-\frac{3}{4}\pi {x}_{3}^{3}+129600\le 0$$

$${g}_{4}\left(x\right)={x}_{4}-240\le 0$$

${x}_{1}\in \left\{\mathrm{1,2},3,\dots ,99\right\}\times 0.0625$, ${x}_{2}\in \left\{\mathrm{1,2},3,\dots ,99\right\}\times 0.0625$, ${x}_{3}\in \left[\mathrm{10,200}\right]$,${x}_{4}\in [\mathrm{10,200}]$

1.6 B.3 Welded beam design problem (WBD)

Min $f\left(x\right)=1.10471{x}_{1}^{2}{x}_{2}+0.04811{x}_{3}{x}_{4}(14.0+{x}_{2})$

Subject to:

$${g}_{1}\left(x\right)=\tau \left(x\right)-{\tau }_{max}\le 0$$

$${g}_{2}\left(x\right)=\sigma \left(x\right)-{\sigma }_{max}\le 0$$

$${g}_{3}\left(x\right)=\delta \left(x\right)-{\delta }_{max}\le 0$$

$${g}_{4}\left(x\right)={x}_{1}-{x}_{4}\le 0$$

$${g}_{5}\left(x\right)=P-{P}_{c}(x)\le 0$$

$${g}_{6}\left(x\right)=0.125-{x}_{1}\le 0$$

$${g}_{7}\left(x\right)=1.10471{x}_{1}^{2}+0.04811{x}_{3}{x}_{4}\left(14.0+{x}_{2}\right)-5.0\le 0$$

$0.1\le {x}_{1}\le 2$; $0.1\le {x}_{2}\le 10$; $0.1\le {x}_{3}\le 10$; $0.1\le {x}_{4}\le 2$

$\uptau \left(x\right)=\sqrt{{({\uptau }^{\mathrm{^{\prime}}})}^{2}+2{\uptau }^{\mathrm{^{\prime}}}{\uptau }^{\mathrm{^{\prime}}\mathrm{^{\prime}}}\frac{{x}_{2}}{2R}+{({\uptau }^{\mathrm{^{\prime}}\mathrm{^{\prime}}})}^{2}}$, ${\uptau }^{\mathrm{^{\prime}}}=\frac{P}{\sqrt{2}{x}_{1}{x}_{2}}$, ${\uptau }^{\mathrm{^{\prime}}\mathrm{^{\prime}}}=\frac{MR}{J}$, $M=P(L+\frac{{x}_{2}}{2})$, $R=\sqrt{\frac{{x}_{2}^{2}}{4}+{(\frac{{x}_{1}+{x}_{3}}{2})}^{2}}$, $J=2\{\sqrt{2}{x}_{1}{x}_{2}[\frac{{x}_{2}^{2}}{4}+{(\frac{{\mathrm{x}}_{1}+{\mathrm{x}}_{3}}{2})}^{2}]\}$, $\upsigma \left(x\right)=\frac{6PL}{{x}_{4}{x}_{3}^{2}}$, $\updelta \left(x\right)=\frac{6P{L}^{3}}{E{x}_{3}^{2}{x}_{4}}$, ${P}_{c}\left(x\right)=\frac{4.013\mathrm{E}\sqrt{\frac{{x}_{3}^{2}{x}_{4}^{6}}{36}}}{{\mathrm{L}}^{2}}(1-\frac{{x}_{3}}{2L}\sqrt{\frac{E}{4G}} )$, $P=6000lb$, $L=14in$, ${\delta }_{max}=0.25in$, $E=30\times {10}^{6}psi$, $G=12\times {10}^{6}psi$, ${\uptau }_{\mathrm{max}}=13600psi$, ${\sigma }_{max}=30000psi$

1.7 B.4 Three-bar truss design problem (TTD)

Min $f\left(x\right)=(2\sqrt{2}{x}_{1}+{x}_{2})\times l$

Subject to:

$${g}_{1}\left(x\right)=\frac{\sqrt{2}{x}_{1}+{x}_{2}}{\sqrt{2}{x}_{1}^{2}+2{x}_{1}{x}_{2}}P-\sigma \le 0$$

$${g}_{2}\left(x\right)=\frac{{x}_{2}}{\sqrt{2}{x}_{1}^{2}+2{x}_{1}{x}_{2}}P-\sigma \le 0$$

$${g}_{3}\left(x\right)=\frac{1}{{x}_{1}+\sqrt{2}{x}_{2}}P-\sigma \le 0$$

$l$=100 cm, $P=2KN/{cm}^{2}$, $\upsigma =2KN/{cm}^{2}$, $0\le {x}_{1}\le 1$, $0\le {x}_{2}\le 1$

1.8 B.5 Cantilever beam design problem (CBD)

Min $f\left(x\right)=0.0624*({x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5})$

Subject to:

$$0.01\le x=[{x}_{1} {x}_{2} {x}_{3} {x}_{4 }{x}_{5}]\le 100$$

1.9 C.1 Full names and abbreviations

Full name	Abbreviation	Full name	Abbreviation
Hybrid particle swarm-differential evolution	HPSDE	Differential evolution	DE
DE variant with a self-adaptive strategy	SLADE	Particle swarm optimization	PSO
Symmetric Latin hypercube design	SLHD	adaptive DE	aDE
Parameters with adaptive learning mechanism	PLAM	Self-adaptive DE variant	SaDE
Differential evolution algorithm with ensemble of parameters	EPSDE	Improved DE variant	IDE
Estimation of distribution algorithm	EDA	Multiple sub-populations DE	MPADE
Time-varying acceleration coefficients	TVAC	Self-adaptive mutation DE	DEPSO
Sine cosine acceleration coefficients	SCAC	Hybrid DE and PSO	DEMPSO
Nonlinear dynamic acceleration coefficients	NDAC	Sine cosine algorithm	SCA
Sigmoid-based acceleration coefficients	SBAC	Opposition-based SCA	OBSCA
Cosine-based acceleration coefficients	CBAC	Grey wolf optimizer	GWO
DE with dual mutation strategies collaboration	DMCDE	Slap swarm algorithm	SSA
DE with elite archive and mutation strategies collaboration	EASCDE	Whale optimization algorithm	WOA
Self-adaptive DE with improved mutation strategy	IMSaDE	Multi-verse optimizer	MVO
DE with ensemble of parameters and mutation strategies	EPSDE	Tension spring design	TSD
DE with composite trial vector generation strategies and control parameters	CoDE	Pressure vessel design	PVD
DE with self-adapting control parameters	jDE	Welded beam design	WBD
Riesz fractional derivative Elite-guided SCA	RFSCA	Three-bar truss design	TTD
Opposition-based PSO with Cauchy mutation	OPSO	Cantilever beam design	CBD
Moth-flame optimization algorithm	MFO	Equilibrium optimizer	EO
Co-evolutionary differential evolution	CEDE	Harmony search algorithm	HS
Social mimic optimization algorithm	SMO	Improved sine cosine algorithm	ISCA
Improved accelerated PSO algorithm	IAPSO	Mine blast algorithm	MBA
Ludo game-based swarm intelligence algorithm	LGSI	Ant lion optimizer	ALO
Grasshopper optimization algorithm	GOA	League championship algorithm	LCA