Lévy-Flight Krill Herd Algorithm

Table 1

Benchmark functions.

Number	Name	Definition
F01	Ackley	𝑓(^→𝑥)=20+𝑒−20⋅𝑒^−0.2⋅(1/𝑛)∑𝑛𝑖=1𝑥2𝑖−𝑒^(1/𝑛)∑𝑛𝑖=1cos(2𝜋𝑥𝑖)
F02	Fletcher-Powell	𝑓(^→𝑥)=∑^𝑛𝑖=1(𝐴𝑖−𝐵𝑖)^2,𝐴𝑖=∑^𝑛𝑗=1𝑎𝑖𝑗sin𝛼𝑗+𝑏𝑖𝑗cos𝛼𝑗𝐵𝑖=∑^𝑛𝑗=1𝑎𝑖𝑗sin𝑥𝑗+𝑏𝑖𝑗cos𝑥𝑗
F03	Griewank	𝑓(^→𝑥)=∑^𝑛𝑖=1𝑥^2𝑖−4000^𝑛∏𝑖=1𝑥cos^𝑖√𝑖+1
F04	Penalty #1	𝑓^→𝑥=𝜋3010sin^2𝜋𝑦1+∑^𝑛−1𝑖=1𝑦𝑖−1^2⋅1+10sin^2𝜋𝑦𝑖+1+𝑦𝑛−1^2+∑^𝑛𝑖=1𝑢𝑥𝑖,10,100,4,𝑦𝑖=1+0.25𝑥𝑖+1
F05	Penalty #2	𝑓(^→𝑥)=0.1sin^23𝜋𝑥1+∑^𝑛−1𝑖=1𝑥𝑖−1^2⋅1+sin^23𝜋𝑥𝑖+1+𝑥𝑛−1^21+sin^22𝜋𝑥𝑛+∑^𝑛𝑖=1𝑢𝑥𝑖,5,100,4
F06	Quartic; with noise	𝑓(^→𝑥)=∑^𝑛𝑖=1(𝑖⋅𝑥^4𝑖+𝑈(0,1))
F07	Rastrigin	𝑓(^→𝑥)=10⋅𝑛+∑^𝑛𝑖=1(𝑥^2𝑖−10⋅cos(2𝜋𝑥𝑖))
F08	Rosenbrock	𝑓(^→𝑥)=∑^𝑛−1𝑖=1100𝑥𝑖+1−𝑥^2𝑖^2+𝑥𝑖−1^2
F09	Schwefel 2.26	𝑓(^→𝑥)=418.9829×𝐷−∑^𝐷𝑖=1𝑥𝑖sin𝑥𝑖^1/2
F10	Schwefel 1.2	𝑓(^→𝑥)=∑^𝑛∑𝑖=1^𝑖𝑗=1𝑥𝑗^2
F11	Schwefel 2.22	𝑓(^→𝑥)=∑^𝑛𝑖=1|𝑥𝑖|+∏^𝑛𝑖=1|𝑥𝑖|
F12	Schwefel 2.21	𝑓(^→𝑥)=max𝑖𝑥𝑖,1≤𝑖≤𝑛
F13	Sphere	𝑓(^→𝑥)=∑^𝑛𝑖=1𝑥^2𝑖
F14	Step	𝑓(^→𝑥)=6⋅𝑛+∑^𝑛𝑖=1𝑥𝑖

^∗In benchmark function F02, the matrix elements 𝐚_𝑛×𝑛,𝐛_𝑛×𝑛∈(−100,100),and𝜶_𝑛×1∈(−𝜋,𝜋) are draw from uniform distribution.
*In benchmark functions F04 and F05, the definition of the function 𝑢(𝑥_𝑖,𝑎,𝑘,𝑚) is as follows: 𝑢(𝑥_𝑖,𝑎,𝑘,𝑚)={𝑘(𝑥_𝑖−𝑎)^𝑚,𝑥_𝑖>𝑎;0,−𝑎≤𝑥_𝑖≤𝑎;𝑘(−𝑥_𝑖−𝑎)^𝑚,𝑥_𝑖<−𝑎}.