A Free Search Krill Herd Algorithm for Functions Optimization

Table 1

Benchmark functions.

Category	Number	Name	Benchmark function	Scope	𝑓min
I	𝐹1	Sphere	𝑓(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖	[−5.12,5.12]	0
	𝐹2	Step	𝑓(𝑥)=∑^𝑛−1𝑖=1𝑥𝑖+0.5^2	[−100,100]	0
	𝐹3	Rosenbrock	𝑓(𝑥)=∑^𝑛−1𝑖=1𝑥𝑖−1^2+100𝑥^2𝑖−𝑥𝑖+1^2	[−2.048,2.048]	0
	𝐹4	Quartic	𝑓(𝑥)=∑^𝑛𝑖=1𝑥^4𝑖+random[0,1)	[−1.28,1.28]	0
	𝐹5	Rastrigin	𝑓(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	[−5.12,5.12]	0
	𝐹6	Ackley	𝑓(𝑥)=−20exp−0.21𝑛^𝑛𝑖∑=1𝑥^2𝑖1−exp𝑛^𝑛𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒	[−32.768,32.768]	0
	𝐹7	Quadric	𝑓(𝑥)=∑^𝑛∑𝑗=1^𝑗𝑖=1𝑥𝑖^2	[−100,100]	0
	𝐹8	Griewank	𝑓(𝑥)=14000^𝑛𝑖∑=1𝑥^2𝑖−∏^𝑛𝑖=1𝑥cos^𝑖√𝑖+1	[−600,600]	0
	𝐹9	Alpine	𝑓(𝑥)=∑^𝑛𝑖=1𝑥𝑖sin𝑥𝑖+0.1𝑥𝑖	[−10,10]	0
	𝐹10	Zakharov	𝑓(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖+∑^𝑛𝑖=10.5𝑖𝑥𝑖^2+∑^𝑛𝑖=10.5𝑖𝑥𝑖^4	[−5,10]	0
II	𝐹11	Schaffer’s F6	𝑓(𝑥)=sin^2(𝑥^21+𝑥^22)−0.51+0.001𝑥^21+𝑥^22^2−0.5	[−100,100]	−1
	𝐹12	Drop wave	𝑓(𝑥)=−1+cos12𝑥^21+𝑥^22(1/2)(𝑥^21+𝑥^22)+2	[−5.12,5.12]	−1
	𝐹13	Six-hump camel back	𝑓𝑥1,𝑥2=4−2.1𝑥^21+𝑥^413𝑥^21+𝑥1𝑥2+−4+4𝑥^22𝑥^22	[−3,3]	−1.0316
	𝐹14	Easom	𝑓(𝑥)=−(−1)∏^𝑛𝑛𝑖=1cos^2𝑥𝑖exp−∑^𝑛𝑖=1𝑥𝑖−𝜋^2	[−2𝜋,2𝜋]	−1