An adaptive rejuvenation of bacterial foraging algorithm for global optimization

Abstract

Bacterial foraging algorithm (BFA) is a novel nature-inspired algorithm that mimics the social foraging behavior of E. coli. Bacteria. However, it gets stuck in the local optima trap and yields poor convergence in complex landscapes. To improve the exploration-exploitation balance and achieve the global optima quickly, this paper proposes a novel hybrid called the Bacterial foraging algorithm-firefly algorithm (BFA-FA). In this work, two strategies namely adaptive strategy and leadership strategy are applied on conventional BFA. The performance is examined on standard, non-linear and CEC_2017 benchmark functions over several evaluation parameters. The results on benchmark functions show that BFA-FA provides accurate solutions, avoids local optima, works well on multimodal and multidimensional landscapes, and converges faster. It also shows the statistically significant difference among other algorithms. The proposed algorithm is applied on two classical engineering problems to validate its robustness and applicability.

Appendix A

Standard benchmark functions used are as follows:

Type of Function	Function	Dim	Range	fmin value
Unimodal benchmark functions	${f}_1(x)=\sum \limits_{i=1}^s{x}_i^2$	30	[−100,100]	0
	${f}_2(x)=\sum \limits_{i=1}^s\left|{x}_i\right|+\prod \limits_{i=1}^s\left|{x}_i\right|$	30	[−10,10]	0
	${f}_3(x)=\sum \limits_{i=1}^s{\left(\sum \limits_{j=1}^i{x}_j\right)}^2$	30	[−100,100]	0
	f4(x) = maxi{|xi|, 1 ≤ i ≤ s}	30	[−100,100]	0
	${f}_5(x)=\sum \limits_{i=1}^s{\left(\left[{x}_i+0.5\right]\right)}^2$	30	[−100,100]	0
Multimodal benchmark functions	${f}_6(x)=\sum \limits_{i=1}^s-{x}_i\sin \left(\sqrt{\left|{x}_i\right|}\right)$	30	[−500,500]	-418.9829 X 5
	${f}_7(x)=\sum \limits_{i=1}^s\left[{x}_i^2-10\cos \left(2\pi {x}_i\right)+10\right]$	30	[−5.12,5.12]	0
	${f}_8(x)=-20\exp \left(-0.2\sqrt{\frac{1}{s}\left(\sum \limits_{i=1}^s{x}_i^2\right)}\right)-\exp \left(\frac{1}{s}\sum \limits_{i=1}^s\cos \left(2\pi {x}_i\right)\right)+20+e$	30	[−32,32]	0
	${f}_9(x)=\frac{\pi }{s}\left\{10\sin \left(\pi {y}_1\right)+\sum \limits_{i=1}^{s-1}{\left({y}_i-1\right)}^2\left[1+10{\mathit{\sin}}^2\left(\pi {y}_{i+1}\right)\right]+{\left({y}_s-1\right)}^2\right\}+\sum \limits_{i=1}^su\left({x}_i,\mathrm{10,100,4}\right)$ ${y}_i=1+\frac{x_i+1}{4}$ $u\left({x}_i,a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_i-a\right)}^m\kern0.5em {x}_i>a\\ {}\ 0\kern1.75em -a<{x}_i<a\\ {}k{\left(-{x}_i-a\right)}^m\kern0.5em {x}_i<-a\kern1.5em \end{array}\right.$	30	[−50,50]	0
	${f}_{10}(x)=0.1\left\{{\mathit{\sin}}^2\left(3\pi {x}_1\right)+\sum \limits_{i=1}^s{\left({x}_i-1\right)}^2\left[1+{\mathit{\sin}}^2\left(3\pi {x}_i+1\right)\right]+{\left({x}_s-1\right)}^2\left[1+{\mathit{\sin}}^2\left(2\pi {x}_s\right)\right]\right\}+\sum \limits_{i=1}^su\left({x}_i,\mathrm{5,100,4}\right)$	30	[−50,50]	0
	${f}_{11}(x)=\left[{e}^{-\sum \limits_{i=1}^s{\left(\raisebox{1ex}{${x}_i$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.\right)}^{2m}}-2{e}^{-\sum \limits_{i=1}^s{x}_i^2}\right].\prod \limits_{i=1}^s{\mathit{\cos}}^2\left({x}_i\right),m=5$	30	[−20,20]	−1
fixed dimension multimodal benchmark function	${f}_{12}(x)=\left[1+{\left({x}_1+{x}_2+1\right)}^2\left(19-14{x}_1+3{x}_1^2-14{x}_2+6{x}_1{x}_2+3{x}_2^2\right)\right]\times \left[30+{\left(2{x}_1-3{x}_2\right)}^2\times \left(18-32{x}_1+12{x}_1^2+48{x}_2-36{x}_1{x}_2+27{x}_2^2\right)\right]$	2	[−2,2]	3

CEC_2017 optimization functions are as follows:

Function Type	Function Number	Function Name
Unimodal function	C01	Shifted and Rotated Bent Cigar
	C02	Shifted and Rotated Sum of Different Power
	C03	Shifted and Rotated Zakharov
Simple Multimodal function	C04	Shifted and Rotated Rosenbrock
	C05	Shifted and Rotated Rastrigin
	C06	Shifted and Rotated Expanded Schaffer F6
	C07	Shifted and Rotated Lunacek Bi-Rastrigin
	C08	Shifted and Rotated Non-Continuous Rastrigin
	C09	Shifted and Rotated Levy
	C10	Shifted and Rotated Schwefel
Hybrid function	C11	Zakharov; Rosenbrock; Rastrigin
	C12	High-conditioned Elliptic; Modified Schwefel; Bent Cigar
	C13	Bent Cigar; Rosenbrock; Lunacek bi-Rastrigin
	C14	High-conditioned Elliptic; Ackley; Schaffer F7; Rastrigin
	C15	Bent Cigar; HGBat; Rastrigin; Rosenbrock
	C16	Expanded Schaffer F6; HGBat; Rosenbrock; Modified Schwefel
	C17	Katsuura; Ackley; Expanded Griewank plus Rosenbrock; Schwefel; Rastrigin
	C18	High-conditioned Elliptic; Ackley; Rastrigin; HGBat; Discus
	C19	Bent Cigar; Rastrigin; Griewank plus Rosenbrock; Weierstrass; Expanded Schaffer F6
	C20	HappyCat; Katsuura; Ackley; Rastrigin; Modified Schwefel; Schaffer F7
Composite functions	C21	Rosenbrock; High-conditioned Elliptic; Rastrigin
	C22	Rastrigin; Griewank; Modified Schwefel
	C23	Rosenbrock; Ackley; Modified Schwefel; Rastrigin
	C24	Ackley; High-conditioned Elliptic; Griewank; Rastrigin
	C25	Rastrigin; HappyCat; Ackley; Discus; Rosenbrock
	C26	Expanded Schaffer F6; Modified Schwefel; Griewank; Rosenbrock; Rastrigin
	C27	HGBat; Rastrigin; Modified Schwefel; Bent Cigar; High-conditioned Elliptic; Expanded Schaffer F6
	C28	8 Ackley; Griewank; Discus; Rosenbrock; HappyCat; Expanded Schaffer F6

Noisy non-linear functions used are as follows:

Type of Function	Function
Four Peak Function	$NL1\left(x,y\right)={e}^{-{\left(x-4\right)}^2-{\left(y-4\right)}^2}+{e}^{-{\left(x+4\right)}^2-{\left(y-4\right)}^2}+2\left[{e}^{-{x}^2-{y}^2}+{e}^{-{x}^2-{\left(y+4\right)}^2}\right]$
Parabolic Function	$NL2\left(x,y\right)=12-\raisebox{1ex}{$\left({x}^2+{y}^2\right)$}\!\left/ \!\raisebox{-1ex}{$100$}\right.$
Camelback Function	$NL3\left(x,y\right)=10-\log \left({x}^2-\left(4-2.1{x}^2+\left(\frac{1}{3}\right){x}^4\right)\right)+ xy+4{y}^2\left({y}^2-1\right)$
Styblinski Function	$NL4\left(x,y\right)=275-\left[\left(\frac{x^4-16{x}^2+5x}{2}\right)+\left(\frac{y^4-16{y}^2+5y}{2}\right)+3\right]$
Goldstein-Price Function	$NL5\left(x,y\right)=10+\log \left[\frac{1}{\left\{\left(1+{\left(1+x+y\right)}^2\left(19-14x+3{x}^2-14y+6 xy+3{y}^2\right)\right)\right\}}\ast \frac{1}{\left(30+{\left(2x-3y\right)}^2\left(18-32x+12{x}^2+48y-36 xy+27{y}^2\right)\right)}\right]$
Rosenbrock Function	$NL6\left(x,y\right)=70\left[\frac{\left[\left[20-\left\{{\left(1-\frac{x}{-7}\right)}^2+{\left(\left(\frac{y}{6}\right)+{\left(\frac{x}{-7}\right)}^2\right)}^2\right\}\right]+150\right]}{170}\right]+10$
Rastrigin Function	NL7(x, y) = 80 − [20 + x2 + y2 − 10(cos(2πx) + cos(2πy))]

Single peak and multi peak functions used are as follows:

Type of Function	Function name	Function	Range	Fitness Value
Single Peak	Needle-in-haystack	$SP1\left(x,y\right)={\left(\frac{3}{0.05+{x}^2+{y}^2}\right)}^2+{\left({x}^2+{y}^2\right)}^2$	x, yϵ(5, −5)	3600
Single Peak	Branin	$SP2\left(x,y\right)=a{\left({x}_2-b{x}_1^2+c{x}_1-r\right)}^2+s\left(1-t\right)\cos \left({x}_1\right)+s$	${x}_1\epsilon \left[-5,10\right],{x}_2\epsilon \left[0,15\right],a=1,b=\frac{5.1}{\left(4{\pi}^2\right)},c=\frac{5}{\pi },r=6,s=10,t=\frac{1}{8\pi }$	0.397887
Multi peak	Extended Beale	$MP3\left(x,y\right)=\sum \limits_{i=1}^{\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left(1.5-{x}_{2i}-\left(1-{x}_{2i}\right)\right)}^2+{\left(2.25-{x}_{2i-1}\left(1-{x}_{2i}^2\right)\right)}^2$	xϵ[−4.5,4.5]	f(x∗) = 0, x ∗  = (3,0.5, …, 3,0.5)
Multi peak	Griewank	$MP4\left(x,y\right)=\frac{1}{4000}\sum \limits_{i=1}^n{x}_i^2-\prod \limits_{i=1}^n\mathit{\cos}\left|\frac{x_i}{\sqrt{i}}\right|+1$	xϵ(300, −300)	f(x∗) = 0, x ∗  = (0, …, 0)
Multi peak	MCCORMICK	$MP5\left(x,y\right)=\sum \limits_{i=1}^{n-1}\left(\left(-1.5{x}_i+2.5{x}_{i+1}+1+{\left({x}_{i+1}+{x}_i\right)}^2\right)+\sin \left({x}_{i+1}+{x}_i\right)\right)$	x0 = [1, 1, …, 1]	f(x∗) =  − 1.9133, x ∗  = (−0.54719, −1.54719, …, −0.54719, −1.54719)
Multi peak	Drop-Wave	$MP6\left(x,y\right)=\frac{1+\cos \left(12\sqrt{x_1^2+{x}_2^2}\right)}{0.5\left({x}_1^2+{x}_2^2\right)+2}$	xϵ[−5.12,5.12]	f(x∗) =  − 1, x ∗  = (0, …, 0)