A Comparison of Selected Modifications of the Particle Swarm Optimization Algorithm

Table 2

Summary of benchmark functions.

Function		Range	𝑓(𝐗^𝐨𝐩𝐭)
Sphere	𝑓1(𝑥)=∑^𝑑𝑖=1𝑧^2𝑖	[−100, 100]^𝑑	−450
Schwefel 1.2	𝑓2(𝑥)=∑^𝑑∑𝑖=1^𝑖𝑗=1𝑧𝑗^2	[−100, 100]^𝑑	−450
Elliptic rotated	𝑓3(𝑥)=∑^𝑑𝑖=110^6(𝑖−1)/(𝑑−1)𝑧^2𝑖	[−100, 100]^𝑑	−450
Schwefel 1.2 noise	𝑓4(𝑥)=∑^𝑑∑𝑖=1^𝑖𝑗=1𝑧𝑗^2⋅(1+0.4|𝑁(0,1)|)	[−100, 100]^𝑑	−450
Schwefel 2.6	𝑓5(𝑥)=max𝐴𝑖𝑥−𝐵𝑖	[−100, 100]^𝑑	−310
Rosenbrock	𝑓6(𝑥)=∑^𝑑−1𝑖=1100𝑧^2𝑖−𝑧𝑖+1^2+𝑧𝑖−1^2	[−100, 100]^𝑑	390
Griewank rotated	𝑓7(𝑥)=∑^𝑑𝑖=1𝑧^2𝑖−4000^𝑑∏𝑖=1𝑧cos^𝑖√𝑖+1	[0, 600]^𝑑	−180
Ackley rotated	𝑓8(𝑥)=−20exp−0.21𝑑^𝑑𝑖∑=1𝑧^2𝑖1−exp𝑑^𝑑𝑖∑=1cos2𝜋𝑧𝑖+20+𝑒	[−32, 32]^𝑑	−140
Rastrigin	𝑓9(𝑥)=∑^𝑑𝑖=1𝑧^2𝑖−10cos2𝜋𝑧𝑖+10	[−5, 5]^𝑑	−330
Rastrigin rotated	𝑓10(𝑥)=∑^𝑑𝑖=1𝑧^2𝑖−10cos2𝜋𝑧𝑖+10	[−5, 5]^𝑑	−330
Weierstrass rotated	𝑓11(𝑥)=∑^𝑑∑𝑖=1^20𝑘=00.5^𝑘cos2𝜋3^𝑘𝑧𝑖+0.5−𝑑∑^20𝑘=00.5^𝑘cos2𝜋3^𝑘⋅0.5	[−0.5, 0.5]^𝑑	90