Improved Particle Swarm Optimization with a Collective Local Unimodal Search for Continuous Optimization Problems

Table 3

Benchmark problems.

Number	Problem	Formulation	Feature	Search range
1	Ackley	𝑓(𝑥)=−20exp−0.21𝑛^𝑑𝑖∑=1𝑥^2𝑖1−exp𝑛^𝑑𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒	MN	±32
2	Booth	𝑓(𝑥)=𝑥1+2𝑥2−7^2+2𝑥1+𝑥2−5^2	MN	±10
3	Easom	𝑓(𝑥)=−cos𝑥1cos𝑥2exp−(𝑥1−𝜋)^2−(𝑥2−𝜋)^2	UN	±100
4	Griewank	𝑓(𝑥)=14000^𝑑𝑖∑=1𝑥^2𝑖−∏^𝑑𝑖=1𝑥cos^𝑖√𝑖+1	MN	±600
5	Dixon-Price	𝑓(𝑥)=𝑥1−1^2+∑^𝑑𝑖=2𝑖2𝑥^2𝑖−𝑥𝑖−1^2	UN	±10
6	Levy	𝑓(𝑥)=sin^2𝜋𝑦1+∑^𝑑−1𝑖=1(𝑦𝑖−1)^21+10sin^2𝜋𝑦𝑖+1+𝑦𝑑−1^21+sin^22𝜋𝑥𝑑,; where 𝑦𝑖𝑥=1+^𝑖−14,and𝑖=1,2,…,𝑑	MN	±10
7	Michalewicz	𝑓(𝑥)=−∑^𝑑𝑖=1sin𝑥𝑖sin𝑖𝑥^2𝑖𝜋^2𝑚,where𝑚=10	MS	[0, 𝜋]
8	Noisy Quartic	𝑓(𝑥)=∑^𝑑𝑖=1𝑖𝑥^4𝑖+random(0,1)	US	±1.28
9	Noncontinous Rastrigin	𝑓(𝑥)=∑^𝑑𝑖=1𝑦^2𝑖−10cos2𝜋𝑦𝑖+10; 𝑦𝑖=𝑥𝑖if𝑥𝑖<0.5round(2𝑥𝑖)2if𝑥𝑖≥0.5	MS	±5.12
10	Rastrigin	𝑓(𝑥)=∑^𝑑𝑖=1𝑥^2𝑖−10cos2𝜋𝑥𝑖+10	MS	±5.12
11	Rosenbrock	𝑓(𝑥)=∑^𝑑−1𝑖=1100𝑥𝑖+1−𝑥^2𝑖^2+𝑥𝑖−1^2	UN	±30
12	Rotated Ellipsoid	𝑓(𝑥)=∑^𝑑∑𝑖=1^𝑖𝑗=1𝑥𝑗^2	UN	±100
13	Salomon	𝑓(𝑥)=−cos2𝜋∑^𝑑𝑖=1𝑥^2𝑖+0.1∑^𝑑𝑖=1𝑥^2𝑖+1	MN	±100
14	Schaffer's f6	𝑓(𝑥)=∑^𝑑−1𝑖=10.5+sin^2𝑥^2𝑖+1+𝑥^2𝑖−0.50.001𝑥^2𝑖+1+𝑥^2𝑖+1^2	MN	±100
15	Schwefel	𝑓(𝑥)=∑^𝑛𝑖=1−𝑥𝑖sin𝑥𝑖	MS	±500
16	Schwefel P2.22	𝑓(𝑥)=∑^𝑑𝑖=1𝑥𝑖+∏^𝑑𝑖=1𝑥𝑖	UN	±10
17	Shubert	𝑓(𝑥)=∏^𝑑∑𝑖=1^5𝑗=1𝑗cos𝑗+1𝑠𝑥𝑖+𝑗	MN	±10
18	Sphere	𝑓(𝑥)=∑^𝑑𝑖=1𝑥^2𝑖	US	±100
19	Step	𝑓(𝑥)=∑^𝑑𝑖=1𝑥𝑖+0.5^2	US	±10
20	SumSquares		US	±10
21	Trid	𝑓(𝑥)=∑^𝑑𝑖=1𝑥𝑖−1^2−∑^𝑑𝑖=2𝑥𝑖𝑥𝑖−1	UN	±d ^2