A Swarm Optimization Algorithm for Multimodal Functions and Its Application in Multicircle Detection

Table 2

The test suite of multimodal functions used in the Experiment  4.3.

Function	Search space	Sketch
Rastringin’s function
100 optima
𝑓5(𝑥1,𝑥2)=−(20+𝑥^21+𝑥^22−10(cos(2𝜋𝑥1)+cos(2𝜋𝑥2)))	𝑥1,𝑥2∈[−10,10]
Shubert function
18 optima
𝑓6(𝑥1,𝑥2)=−∏^25𝑖=1∑𝑗=1cos((𝑗+1)𝑥𝑖+𝑗)	𝑥1,𝑥2∈[−10,10]
Griewank function
100 optima
𝑓7(𝑥1,𝑥21)=4000^2𝑖∑=1𝑥^2𝑖−∏^2𝑖=1𝑥cos^𝑖√2+1	𝑥1,𝑥2∈[−100,100]
Modified Griewank function
100 optima
𝑓8(𝑥1,𝑥2)=cos(0.5𝑥1)+cos(0.5𝑥2)4000+cos(10𝑥1)cos(10𝑥2)	𝑥1,𝑥2∈[0,120]