A Dynamic Multistage Hybrid Swarm Intelligence Optimization Algorithm for Function Optimization

Table 1

Properties of the test function.

Function name	Expression	Search range	Optimization value
Sphere function	𝑓1=∑^𝐷𝑖=1𝑥^2𝑖	[−100,100]	Min(𝑓1)=𝑓1(0,0,…,0)=0
Rosenbrock function	𝑓2=∑^𝐷𝑖=1100𝑥𝑖+1−𝑥^2𝑖+𝑥𝑖−1^2	[−2.048,2.048]	Min(𝑓2)=𝑓2(1,1,…,1)=0
Ackley function	𝑓3(𝑥)=−20exp−0.21𝐷^𝐷𝑖∑=1𝑥^2𝑖	[−32.768,32.768]	Min(𝑓3)=𝑓3(0,0,…,0)=0
	1−𝑝𝐷^𝐷𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒
Griewank function	𝑓41(𝑥)=400^𝑛𝑖∑=1𝑥𝑖−100^2	[−600,600]	Min(𝑓4)=𝑓4(0,0,…,0)=0
	−∏^𝑛𝑖=1cos𝑥𝑖−100√𝑖+1
Rastrigin function	𝑓5=∑^𝐷𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	[−5.12,5.12]	Min(𝑓5)=𝑓5(0,0,…,0)=0
Noncontinuous Rastrigin function	𝑓6(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−10cos2𝜋𝑦𝑖+10	[−5.12,5.12]	Min(𝑓6)=𝑓6(0,0,…,0)=0
	𝑦𝑖=𝑥𝑖𝑥𝑖<12round2𝑥𝑖2𝑥𝑖≥12for𝑖=1,2,…,𝐷.
Schwefel function	𝑓7(𝑥)=418.9829∗𝐷−∑^𝐷𝑖=1𝑥𝑖∗sin𝑥𝑖^1/2	[−500,500]	Min(𝑓7)=𝑓7(420.96,420.96,…,420.96)=0
Weierstrass function	𝑓8(𝑥)=∑^𝐷∑𝑖=1^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑥𝑖+0.5	[−0.5,0.5]	Min(𝑓8)=𝑓8(0,0,…,0)=0
	−𝐷∑^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘∗0.5,
	𝑎=0.5,𝑏=3,𝑘max=20
Rotated Ackley function	𝑓9(𝑥)=−20exp−0.21𝐷^𝐷𝑖∑=1𝑥^2𝑖	[ −32.768, 32.768]	Min(𝑓9)=𝑓9(0,0,…,0)=0
	1−𝑝𝐷^𝐷𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒,𝑦=𝑀∗𝑥
Rotated Griewank function	𝑓101(𝑥)=400^𝑛𝑖∑=1𝑥𝑖−100^2	[−600,600]	Min(𝑓10)=𝑓10(0,0,…,0)=0
	−∏^𝑛𝑖=1cos𝑥𝑖−100√𝑖+1,𝑦=𝑀∗𝑥
Rotated Rastrigin function	𝑓11(𝑥)=∑^𝐷𝑖=1(𝑥^2𝑖−10cos(2𝜋𝑥𝑖)+10),𝑦=𝑀∗𝑥	[−5.12,5.12]	Min(𝑓11)=𝑓11(0,0,…,0)=0
Rotated noncontinuous Rastrigin function	𝑓12(𝑥)=∑^𝐷𝑖=1𝑦^2𝑖−10cos(2𝜋𝑦𝑖)+10	[−5.12,5.12]	Min(𝑓12)=𝑓12(0,0,…,0)=0
	𝑦𝑖=𝑥𝑖|𝑥𝑖1|<2round(2𝑥𝑖)2|𝑥𝑖1|≥2
	for𝑖=1,2,…,𝐷,𝑦=𝑀∗𝑥
	𝑓13(𝑥)=418.9829∗𝐷−∑^𝐷𝑖=1𝑦𝑖
Rotated Schwefel function	𝑦𝑖=𝑥𝑖sin𝑥𝑖^1/20.001𝑥𝑖−500^2if𝑥𝑖𝑥≤500if𝑖>500	[−500,500]	Min(𝑓13)=𝑓13(420.96,420.96,…,420.96)=0
	for𝑖=1,2,…,𝐷𝑥=𝑥^′+420.96,
	𝑥^′=𝑀∗(𝑧−420.96)
Rotated Weierstrass function	𝑓14(𝑥)=∑^𝐷∑𝑖=1^𝑘max𝑘=0𝑎^𝑘cos2𝜋𝑏^𝑘𝑥𝑖+0.5	[−0.5,0.5]	Min(𝑓14)=𝑓14(0,0,…,0)=0
	−𝐷∑^𝑘max𝑘=0𝑎^𝑘cos(2𝜋𝑏^𝑘∗0.5)
	𝑎=0.5,𝑏=3,𝑘max=20,𝑦=𝑀∗𝑥
Hybrid composition function 1 (CF 1)	𝑓15 marked as 𝑓16(𝑥) in the CEC2005 benchmark problem set [24] and is composed of ten sphere functions	[−500,500]	Min(CF1) = 0
Hybrid composition function 2 (CF2)	𝑓16 marked as 𝑓16(𝑥) in the CEC2005 benchmark problem set [24] and is composed of two rotated Rastrigin functions, two rotated Weierstrass functions, two rotated Griewank functions, two rotated Ackley functions, and two sphere functions.	[−500,500]	Min(CF2) = 0