Biogeography-Based Optimization with Orthogonal Crossover

Table 1

Benchmark functions used in our experimental tests.

Function	Name	Dim	Search space	Optima
𝑓1(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖	Sphere	𝑛	[−100,100]	0
𝑓2(𝑥)=∑^𝑛𝑖=1𝑥𝑖+∏^𝑛𝑖=1𝑥𝑖	Schwefel’s problem 2.22	𝑛	[10,10]	0
𝑓3(𝑥)=∑^𝑛𝑖=1∑^𝑖𝑗=1𝑥𝑗^2	Schwefel’s problem 1.2	𝑛	[−100,100]	0
𝑓4(𝑥)=max𝑖𝑥𝑖,1≤𝑖≤𝑛	Schwefel’s problem 2.21	𝑛	[−100,100]	0
𝑓5(𝑥)=∑^𝑛−1𝑖=1[100𝑥𝑖+1−𝑥^2𝑖^2+(𝑥𝑖−1)^2]	Rosenbrock function	𝑛	[−30,30]	0
𝑓6(𝑥)=∑^𝑛𝑖=1𝑥𝑖+0.5^2	Step function	𝑛	[−100,100]	0
𝑓7(𝑥)=∑^𝑛𝑖=1𝑖𝑥^4𝑖+random[0,1)	Quartic function	𝑛	[−1.28,1.28]	0
𝑓8(𝑥)=−∑^𝑛𝑖=1𝑥𝑖sin(𝑥𝑖)	Schwefel function	𝑛	[−512,512]	−12569.5
𝑓9(𝑥)=∑^𝑛𝑖=1[𝑥^2𝑖−10cos(2𝜋𝑥𝑖)+10]	Rastrigin function	𝑛	[−5.12,5.12]	0
𝑓10(𝑥)=−20exp−0.2(1/𝑛)∑^𝑛𝑖=1𝑥^2𝑖−exp(1/𝑛)∑^𝑛𝑖=1cos2𝜋𝑥𝑖+20+𝑒	Ackley function	𝑛	[−32,32]	0
𝑓11(𝑥)=(1/4000)∑^𝑛𝑖=1𝑥^2𝑖−∏^𝑛𝑖=1cos(𝑥𝑖/√𝑖)+1	Griewank function	𝑛	[−600,600]	0
𝑓12(𝑥)=(𝜋/𝑛)10sin^2𝜋𝑦𝑖+∑^𝑛−1𝑖=1𝑦𝑖−1^21+10sin^2𝜋𝑦𝑖+1+(𝑦𝑛−1)^2+∑^𝑛𝑖=1𝑢(𝑥𝑖,10,100,4),	Penalized function 1	𝑛	[−50,50]	0
𝑦𝑖=1+(𝑥𝑖−1)/4
𝑢𝑥𝑖,𝑎,𝑘,𝑚=𝑘(𝑥𝑖−𝑎)^2,𝑥𝑖>𝑎0,−𝑎≤𝑥𝑖≤𝑎𝑘(−𝑥𝑖−𝑎)^2,𝑥𝑖<−𝑎
𝑓13(𝑥)=0.1sin^2(3𝜋𝑥1)+∑^𝑛−1𝑖=1(𝑥𝑖−1)^2[1+sin^2(3𝜋𝑥𝑖+1)]+(𝑥𝑛−1)^2[1+sin^2(2𝜋𝑥𝑛)]+∑^𝑛𝑖=1𝑢(𝑥𝑖,5,100,4),	Penalized function 2	𝑛	[−50,50]	0
𝑓14(𝑥)=1/500+∑^25𝑗=11/𝑗+∑^2𝑖=1𝑥𝑖−𝑎𝑖𝑗^6−1	Shekel Foxhole	2	[−65.536,65.536]	1
𝑓15(𝑥)=∑^11𝑖=1𝑎𝑖−𝑥1(𝑏^2𝑖+𝑏𝑖𝑥2)/𝑏^2𝑖+𝑏𝑖𝑥3+𝑥4^2	Kowalik’s function	4	[−5,5]	0.003075
𝑓16(𝑥)=4𝑥^21−2.1𝑥^41+(1/3)𝑥^61+𝑥1𝑥2−4𝑥^22+4𝑥^42	Six-Hump camel-Back function	4	[−5,5]	−1.0316
𝑓17(𝑥)=(𝑥2−5.1/4𝜋^2𝑥^21+(5/𝜋)𝑥1−6)^2+10(1−1/8𝜋)cos𝑥1+10	Branin function	2	[−5,10]×[0,15]	0.398
𝑓18(𝑥)=[1+𝑥1+𝑥2+1^2(19−14𝑥1+3𝑥^21−14𝑥2+6𝑥1𝑥2+3𝑥^22)]×[30+(2𝑥1−3𝑥2)^2(18−32𝑥1+12𝑥^21+48𝑥2−36𝑥1𝑥2+27𝑥^22)]	Goldstein-Price function	2	[−2,2]	3
𝑓19(𝑥)=−∑^4𝑖=1𝑐𝑖exp−∑^4𝑗=1𝑎𝑖𝑗(𝑥𝑗−𝑝𝑖𝑗)^2	Hartman’s function 1	3	[0,1]	−3.86
𝑓20(𝑥)=−∑^4𝑖=1𝑐𝑖exp−∑^6𝑗=1𝑎𝑖𝑗(𝑥𝑗−𝑝𝑖𝑗)^2	Hartman’s function 2	6	[0,1]	−3.32
𝑓21(𝑥)=−∑^5𝑖=1[(𝑥−𝑎𝑖)(𝑥−𝑎𝑖)^𝑇+𝑐𝑖]^−1	Shekel’s Function 1	4	[0,10]	−10.1532
𝑓22(𝑥)=−∑^7𝑖=1[(𝑥−𝑎𝑖)(𝑥−𝑎𝑖)^𝑇+𝑐𝑖]^−1	Shekel’s Function 2	4	[0,10]	−10.4029
𝑓23(𝑥)=−∑^10𝑖=1[(𝑥−𝑎𝑖)(𝑥−𝑎𝑖)^𝑇+𝑐𝑖]^−1	Shekel’s Function 3	4	[0,10]	−10.5364

Note: Dim 𝑛 means these functions with changeable dimension.