An Adaptive Cauchy Differential Evolution Algorithm for Global Numerical Optimization

Table 1

Benchmark functions used in the performance evaluation.

Benchmark function	Dim	Search space	Global min.
𝐹1(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖	30	[−100,100]^𝐷	0
𝐹2(𝑥)=∑^𝐷𝑖=1|𝑥𝑖|+∏^𝐷𝑖=1|𝑥𝑖|	30	[−10,10]^𝐷	0
𝐹3(𝑥)=max𝑖(|𝑥𝑖|,1≤𝑖≤𝐷)	30	[−100,100]^𝐷	0
𝐹4(𝑥)=∑^𝐷𝑖=1𝑥𝑖+0.5^2	30	[−100,100]^𝐷	0
𝐹5(𝑥)=∑^𝐷𝑖=1𝑖𝑥^4𝑖+random[0,1)	30	[−1.28,1.28]^𝐷	0
𝐹6(𝑥)=∑^𝐷𝑖=1−𝑥𝑖sin𝑥𝑖	30	[−500,500]^𝐷	−12569.5
𝐹7(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	30	[−5.12,5.12]^𝐷	0
𝐹8(𝑥)=−20exp−0.21𝐷^𝐷𝑖∑=1𝑥^2𝑖1−exp𝐷^𝐷𝑖∑=1cos2𝜋𝑥𝑖+20+exp(1)	30	[−32,32]^𝐷	0
𝐹91(𝑥)=4000^𝐷𝑖∑=1𝑥^2𝑖−∏^𝐷𝑖=1𝑥cos^𝑖√𝑖+1	30	[−600,600]^𝐷	0
𝐹10(𝑥)=𝜋𝐷10sin^2𝜋𝑦1+∑^𝐷−1𝑖=1𝑦𝑖−1^21+10sin^2𝜋𝑦𝑖+1+𝑦𝐷−1^2; +∑^𝐷𝑖=1𝑢𝑥𝑖,10,100,4	30	[−50,50]^𝐷	0
𝐹11(𝑥)=0.1sin^23𝜋𝑥1+∑^𝐷−1𝑖=1𝑥𝑖−1^21+sin^23𝜋𝑥𝑖+1+𝑥𝐷−1^21+sin^22𝜋𝑥𝐷; +∑^𝐷𝑖=1𝑢𝑥𝑖,5,100,4	30	[−50,50]^𝐷	0
𝐹12(𝑥)=𝑓12(𝑥𝑛,𝑥1)+∑^𝐷−1𝑖=1𝑓12(𝑥𝑖,𝑥𝑖+1)	30	[−100,100]^𝐷	0
𝑓12(𝑥,𝑦)=𝑥^2+𝑦^20.25sin^250𝑥^2+𝑦^20.1+1
𝐹13(𝑥)=∑^𝐷−1𝑖=1𝑥^2𝑖+2𝑥^2𝑖+1−0.3cos3𝜋𝑥𝑖−0.4cos4𝜋𝑥𝑖+1+0.7	30	[−15,15]^𝐷	0
𝐹14(𝑥)=∑^𝐷−1𝑖=1𝑥^2𝑖+𝑥^2𝑖+1^0.25sin^250𝑥^2𝑖+𝑥^2𝑖+1^0.1+1	30	[−100,100]^𝐷	0