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. 2021 May 10;27(6):3307–3326. doi: 10.1007/s00500-021-05839-6

Table 8.

Benchmark functions

Function Dim Range fmin
Unimodal functions
TF1(x)=i=1nxi+i=1nxi 30 [-10,10] 0
TF2(x)=i=1n-1[100(xi+1-xi2)2+(xi-1)2] 30 [-30,30] 0
Multimodal functions
TF3(x)=i=1n-xisin(xi) 30 [-500,500] -418.9829×Dim
TF4(x)=14000i=1nxi2-i=1ncosxii+1 30 [-600,600] 0
TF5(x)=0.1{sin2(3πx1)+i=1n(xi-1)2[1+sin2(3πxi+1)]+(xn-1)2[1+sin2(2πxn)]}+i=1nu(xi,5,100,4) 30 [− 50, 50] 0
Fixed-dimension multimodal functions
TF6(x)=1500+j=1251j+i=12(xi-aij)6-1 2 [− 65, 65] 1
TF7(x)=[1+(x1+x2+1)2(19-14x1+3x12-14x2+6x1x2+3x22)]×[30+(2x1-3x2)2×(18-32x1+12x12+48x2-36x1x2+27x22)] 2 [− 2, 2] 3
Composition function
TF8CF1f1,f2,f3,,f10:Sphere Functionσ1,σ2,σ3,,σ10=1,1,1,,1λ1,λ2,λ3,,λ10=5/100,5/100,5/100,,5/100 10 [− 5, 5] 0
TF9CF3f1,f2,f3,,f10:Griewank's Functionσ1,σ2,σ3,,σ10=1,1,1,,1λ1,λ2,λ3,,λ10=1,1,1,,1 10 [− 5, 5] 0