An Improved Cockroach Swarm Optimization

Table 1

Benchmark test functions.

Number	Range	𝐷	C	Functions	Description
1	[−100, 100]	30	US	Step	𝑓(𝑥)=∑^𝑛𝑖=1𝑥𝑖+0.5^2
2	[−100, 100]	30	US	Sphere	𝑓(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖
3	[−10, 10]	30	US	Sumsquares	𝑓(𝑥)=∑^𝑛𝑖=1𝑖𝑥^2𝑖
4	[−100, 100]	2	MS	Bohachevsky1	𝑓(𝑥)=𝑥^21+2𝑥^22−0.3cos(3𝜋𝑥1)−0.4cos(4𝜋𝑥2)+0.7
5	[−100, 100]	2	MN	Bohachevsky2	𝑓(𝑥)=𝑥^21+2𝑥^22−0.3cos(3𝜋𝑥1)(4𝜋𝑥2)+0.3
6	[−100, 100]	2	MN	Bohachevsky3	𝑓(𝑥)=𝑥^21+2𝑥^22−0.3cos(3𝜋𝑥1+4𝜋𝑥2)+0.3
7	[0, 180]	20	UN	Sinusoidal20	𝑓(𝑥)=−𝐴∏^𝑛𝑖=1sin(𝑥𝑖−𝑧)+∏^𝑛𝑖=1sin𝐵𝑥𝑖−𝑧
					𝐴=2.5,𝐵=5,𝑧=30
8	[−100, 100]	30	UN	Quadric	𝑓(𝑥)=∑^𝑛∑𝑖=1^𝑛𝑖=1𝑥𝑗^2
9	[−100, 100]	2	UN	Easom	𝑓(𝑥)=−cos𝑥1cos𝑥2⋅exp−𝑥1−𝜋^2exp(−𝑥2−𝜋)^2
10	[−10, 10]	2	UN	Matyas	𝑓(𝑥)=0.26(𝑥1+𝑥2)−0.48𝑥1𝑥2
11	[−5, 10]	10	UN	Zakharov	𝑓(𝑥)=∑^𝑛𝑖=1𝑥𝑖^2+∑^𝑛𝑖=10.5𝑖𝑥𝑖^2+∑^𝑛𝑖=10.5𝑖𝑥𝑖^4
12	[−10, 10]	24	UN	Powell	𝑓(𝑥)=∑^𝑛/𝑘𝑖=1𝑥4𝑖−3+10𝑥4𝑖−2^2+5𝑥4𝑖−1−𝑥4𝑖^2+𝑥4𝑖−2−𝑥4𝑖−1^4+10𝑥4𝑖−3−𝑥4𝑖^4
13	[−10, 10]	30	UN	Schwefel2.22	𝑓(𝑥)=∑^𝑛𝑖=1𝑥𝑖+∏^𝑛𝑖=1𝑥𝑖
14	[−30, 30]	30	UN	Rosenbrock	𝑓(𝑥)=∑^𝑛−1𝑖=1100𝑥𝑖+1−𝑥^2𝑖^2+𝑥𝑖−1^2
15	[−5.12, 5.12]	30	MS	Rastrigin	𝑓(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖−10cos(2𝜋𝑥𝑖)+10
16	[−100, 100]	2	MN	Schaffer1	𝑓(𝑥)=0.5+sin^2𝑥^21+𝑥^22^2−0.51+0.001𝑥^21+𝑥^22^2
17	[−100, 100]	30	MN	Schaffer2	𝑓(𝑥)=𝑥^21+𝑥^22^0.25sin^250(𝑥^21+𝑥^22)^0.1+1
18	[−600, 600]	30	MN	Griewangk	1𝑓(𝑥)=4000^𝑛𝑖∑=1𝑥^2𝑖−∏^𝑛𝑖=1𝑥cos^𝑖√𝑖+1
19	[−32, 32]	30	MN	Ackley	𝑓(𝑥)=−20exp^(−0.2∑𝑛𝑖=1(𝑥2𝑖/𝑛))−exp^(∑𝑛𝑖=1cos(2𝜋𝑥𝑖/𝑛))+20+𝑒
20	[−5, 5]	2	MN	Three hump camel back	𝑓(𝑥)=2𝑥^21−1.05𝑥^41+16𝑥^61+𝑥1𝑥2+𝑥^22
21	[−5, 5]	2	MN	Six hump camel back	𝑓(𝑥)=4𝑥^21−2.1𝑥^41+13𝑥^61+𝑥1𝑥2−4𝑥^22+4𝑥^42
22	[−128, 128]^𝑛	9	UN	Storn's Tchebychev	𝑓(𝑥)=𝑝1+𝑝2+𝑝3,
23	[−32768, 32768]^𝑛	17		Storn's Tchebychev	𝑝where^1=(𝑢−𝑑)^2if𝑢<𝑑0if𝑢≥𝑑𝑢=^𝑛𝑖=1(1.2)^𝑛−𝑖𝑥𝑖𝑝2=(𝑣−𝑑)^2if𝑣<𝑑0if𝑣≥𝑑𝑣=^𝑛𝑖=1(−1.2)^𝑛−𝑖𝑥𝑖𝑝3=∑^𝑚𝑗=0(𝑤𝑗−1)^2if𝑤𝑗>1(𝑤𝑗+1)^2if𝑤𝑗<−10if−1≤𝑤𝑗𝑤≤1^𝑗=∑^𝑛𝑖=12𝑗𝑚−1^𝑛−𝑖𝑥𝑖,; for 𝑛=9: 𝑑=72.661, and 𝑚=60; for 𝑛=17: 𝑑=10558.145, and 𝑚=100.

𝐷∶ dimension; C: characteristic; U: unimodal; S: seperable; N: non-separable.