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

Table 1

The test suite of multimodal functions for Experiment  4.2.

Function	Search space	Sketch
Deb’s function
5 optima
𝑓1=sin^6(5𝜋𝑥)	𝑥∈[0,1]
Deb’s decreasing function
5 optima
𝑓2(𝑥)=2^−2((𝑥−0.1)/0.9)2⋅sin(5𝜋𝑥)	𝑥∈[0,1]
Roots function
6 optima
𝑓31(𝑧)=1+𝑧^6+1	𝑧∈𝐶,𝑧=𝑥1+𝑖𝑥2𝑥1,𝑥2∈[−2,2]
Two dimensional multi-modal function
100 optima
𝑓4(𝑥1,𝑥2)=𝑥1sin(4𝜋𝑥1)−𝑥2sin(4𝜋𝑥2+𝜋)+1	𝑥1,𝑥2∈[−2,2]