An efficient metaheuristic method based on the BitTorrent communication protocol (EM-BT)

Abstract

In the last few years, considerable attention has been paid to metaheuristic techniques to solve hard optimization problems, as they can produce adequate solutions with less computational complexity. In this paper, an efficient metaheuristic technique based on the BitTorrent communication protocol (EM-BT) is proposed. Our EM-BT algorithm uses the same concept of the BitTorrent communication protocol to ameliorate the communication between the candidate solutions, and manage the way information is exchanged between them in order to provide sufficient exploration and exploitation of the search space. The presented algorithm is evaluated on a group of 23 classical benchmark functions defined in CEC 2005 and 10 functions defined in CEC-C06, 2019. The results show that our algorithm outperforms, in many functions, other metaheuristics such as particle swarm optimization (PSO), grey wolf optimizer (GWO), butterfly optimization algorithm (BOA), salp swarm algorithm (SSA), whale optimization algorithm (WOA), Jaya algorithm, differential evolution (DE) and genetic algorithm (GA). The difference between the solution obtained with our EM-BT method and the global optimum solution is very small, up to $1 \times 10^{-16}$, with a low standard deviation, which proves the accuracy, stability and superiority of the proposed method.

Appendices

Appendix A

Test functions	Dimension	Range	Maximum iteration
$F_{01}=\sum _{i=1}^{n}x_{i}^{2}$	30	$[-100,+100]$	500
$F_{02}=\sum _{i=1}^{n} |x_{i} |+ \prod _{i=1}^{n} |x_{i} |$	30	$[-10,+10]$	950
$F_{03}=\sum _{i=1}^{n}\left( \sum _{j=1}^{i}x_{j} \right) ^{2}$	30	$[-100,+100]$	500
$F_{04}= \max _{i}\{\vert x_{i} |, 1 \le i \le D \}$	30	$[-100,+100]$	1000
$F_{05}= \sum _{i=1}^{D-1} [100 \times (x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2} ]$	30	$[-30,+30]$	8000
$F_{06}= \sum _{i=1}^{D} ([x_{i}+0.5] )^{2}$	30	$[-100,+100]$	15
$F_{07}= \sum _{i=1}^{D} [i x_{i}^{4}+random [0,1)]$	30	$[-1.28,+1.28]$	1500
$F_{08}= \sum _{i=1}^{D} -x_{i} \sin (\sqrt{|x_{i}|})$	30	$[-500,+500]$	1500
$F_{09}= \sum _{i=1}^{D} [x_{i}^{2}-10 \cos (2\pi x_{i} )+10]$	30	$[-5.12,+5.12]$	40
$\begin{array}{ll} F_{10}= &{} -20 exp (-0.2 \sqrt{\frac{1}{D} \sum _{i=1}^{D} x_i^{2} } ) \\ &{} -exp\left( \frac{1}{D} \sum _{i=1}^{D} \cos (2\pi x_i)\right) +20+e \end{array}$	30	$[-32,+32]$	60
$F_{11}= \frac{1}{4000} \sum _{i=1}^{D} x_{i}^{2} -\prod _{i=1}^{D} \cos (\frac{x_{i}}{\sqrt{i} }) +1$	30	[-600,+600]	70
$\begin{array}{ll} F_{12}= &{} \frac{\pi }{D} \{ 10 \sin ^2 (\pi y_i) \\ &{} +\sum _{i=1}^{D-1} (y_i-1)^2 \, [1+10 \sin ^2 (\pi y_i+1)] \\ &{} +(yD-1)^2+\sum _{i=1} ^D u(x_i,10,100,4) \} \\ y_i= &{} 1+\frac{x_i+1}{4} \\ u(x_i,a,k,m) = &{} \{ \begin{array}{ll} k(x_i-a)^m &{} x_i > a \\ 0 &{} -a< x_i< a \\ k(-x_i-a)^m &{} x_i < -a \end{array} \end{array}$	30	$[-50,+50]$	2000
$\begin{array}{ll} F_{13}= &{} 0.1 \{ 10 \sin ^2 (\pi y_i) \\ &{} +\sum _{i=1}^{D-1} (y_i-1)^2 \, [1+10 \sin ^2 (\pi y_i+1)] \\ &{} +(yD-1)^2+\sum _{i=1} ^D u(x_i,10,100,4) \} \end{array}$	30	$[-50,+50]$	2000
$F_{14}= \left[ \frac{1}{500} + \sum _{j=1}^{25} \frac{1}{j+\sum _{i=1}^{2} (x_{i}-a_{ij} )^{6} }\right] ^{-1}$	2	[-65.53,+65.53]	150
$F_{15}= \sum _{i=1}^{11} \left[ a_i- \frac{x_1 (b_i^2+b_i x_i )}{b_i^2+b_1 x_3+x_4}\right] ^2$	4	[-5,+5]	400
$F _{16}= 4x_1^2-2.1x_i^4+ \frac{1}{3} x_1^6+x_1 x_2-4x_2^2+4x_2^4$	2	[-5,+5]	200
$F_{17}= \left( x_2-\frac{5.1}{4\pi ^2} x_1^2+\frac{5}{\pi } x_1-6\right) ^2+10\left( 1-\frac{1}{8\pi }\right) \cos (x_1)+10$	2	$[5,+10]\times [0,+15]$	180
$\begin{array}{ll} F_{18}= &{} [1+(x_1+x_2+1)^2 \\ &{} (19-14x_1+3x_1^2-14x_2+6x_1 x_2+3x_2^2 )] \\ &{} \times [30+( 2x_1-3x_2 )^2 \\ &{} (18-32x_1+12x_1^2+48x_2-36x_1 x_2+27x_2^2 ) ] \\ \end{array}$	2	$[-5,+5]$	200
$F_{19}=-\sum _{i=1}^4 c_i \, exp \, \left( -\sum _{j=1}^3 a_{ij} (x_{j}-p_{ij} )^2 \right)$	3	$[0,+1]$	100
$F_{20}=-\sum _{i=1}^4 c_i \, exp \, \left( -\sum _{j=1}^6 a_{ij} (x_{j}-p_{ij} )^2 \right)$	6	$[0,+1]$	250
$F_{21}=-\sum _{i=1}^5[(X-a_i ) (X-a_i )^T+c_i] ^{-1}$	4	$[0,+10]$	200
$F_{22}=-\sum _{i=1}^7[(X-a_i ) (X-a_i )^T+c_i] ^{-1}$	4	$[0,+10]$	200
$F_{23}=-\sum _{i=1}^{10}[(X-a_i ) (X-a_i )^T+c_i] ^{-1}$	4	$[0,+10]$	200

Appendix B

Test functions	Dimension	Range	Optimum fitness
$CEC_{1}$: Storn’s Chebyshev polynomial fitting problem	9	$[-8192, 8192]$	1
$CEC_{2}$: Inverse Hilbert matrix problem	16	$[-16384, 16384]$	1
$CEC_{3}$: Lennard–Jones minimum energy cluster	18	$[-4,4]$	1
$CEC_{4}$: Rastrigin’s function	10	$[-100, 100]$	1
$CEC_{5}$: Griewangk’s function	10	$[-100, 100]$	1
$CEC_{6}$: Weierstrass function	10	$[-100,+100]$	1
$CEC_{7}$: Modified Schwefel’s function	10	$[-100, 100]$	1
$CEC_{8}$: Expanded Schaffer’s $F_{06}$ Function	10	$[-100, 100]$	1
$CEC_{9}$: Happy cat function	10	$[-100, 100]$	1
$CEC_{10}$: Ackley function	10	$[-100, 100]$	1

Appendix C

Some of the source codes used in this paper were downloaded from these addresses:

GWO: https://www.mathworks.com/matlabcentral/fileexchange/47258-grey-wolf-optimizer-toolbox.

SSA: https://www.mathworks.com/matlabcentral/fileexchange/63745-ssa-salp-swarm-algorithm.

WOA: https://www.mathworks.com/matlabcentral/fileexchange/55667-the-whale-optimization-algorithm.