Theoretical and Empirical Analyses of an Improved Harmony Search Algorithm Based on Differential Mutation Operator

Table 1

Benchmark test functions.

Function name	Benchmark functions expression	Search range	Optimum value
𝐹1 Ackley function	𝐹1(𝑥)=−20𝑒^(1/5)(1/𝑛)∑𝑛𝑖=1𝑥2𝑖−𝑒^(1/𝑛)∑𝑛𝑖=1cos(2𝜋𝑥𝑖)+20−𝑒	[−15,30]^𝑛	𝑥^∗=(0,0,…,0), 𝐹1(𝑥^∗)=0
𝐹2 Dixon and Price function	𝐹2(𝑥)=𝑥1+1^2+∑^𝑛𝑖=2𝑖2𝑥^2𝑖−𝑥𝑖−1^2	[−10,10]^𝑛	𝐹2(𝑥^∗)=0
𝐹3 Levy function	𝐹3(𝑥)=sin^2𝜋𝑦1+∑^𝑛−1𝑖=1(𝑦𝑖−1)^21+10sin^2𝜋𝑦𝑖+1+(𝑦𝑛−1)^21+10sin^22𝜋𝑦𝑛; 𝑦𝑖𝑥=1+^𝑖−14,𝑖=1,2,…,𝑛	[−10,10]^𝑛	𝑥^∗=(1,1,…,1), 𝐹3(𝑥^∗)=0
𝐹4 Michalewics function	𝐹4𝑥1,𝑥2=−∑^2𝑗=1sin(𝑥𝑗)sin(𝑗𝑥^2𝑗/𝜋)^2𝑚;𝑚=10	[0,𝜋]^𝑛	𝑛=5;𝐹4(𝑥^∗)=−4.687658
𝐹5 Perm function	𝐹5(𝑥)=∑^𝑛𝑘=1∑^𝑛𝑖=1𝑖^𝑘+𝛽(𝑥𝑖/𝑘)^𝑘−1^2	[−𝑛,𝑛]	𝑥^∗=(1,2,…,𝑛), 𝐹5(𝑥^∗)=0
𝐹6 Powell function	𝐹6(𝑥)=∑^𝑛/4𝑖=1[𝑥4𝑖−3+10𝑥4𝑖−2^2+5𝑥4𝑖−1−𝑥4𝑖^2+𝑥4𝑖−2−𝑥4𝑖−1^2; +10𝑥4𝑖−3−𝑥4𝑖^4]	[−4,5]^𝑛	𝑥^∗=(3,−1,0,1,…,3,−1,0,1), 𝐹6(𝑥^∗)=0
𝐹7 Rastrigin function	𝐹7(𝑥)=10𝑛+∑^𝑛𝑖=1𝑥^2𝑖−10cos(2𝜋𝑥𝑖)	[−5.12,5.12]^𝑛	𝑥^∗=(0,0,…,0), 𝐹7(𝑥^∗)=0
𝐹8 Rosenbrock function	𝐹8(𝑥)=∑^𝑛−1𝑖=1100(𝑥^2𝑖−𝑥𝑖−1)^2+(𝑥𝑖−1)^2	[−5,10]^𝑛	𝑥^∗=(1,1,…,1), 𝐹8(𝑥^∗)=0
𝐹9 Sphere function	𝐹9(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖	[−5.12,5.12]^𝑛	𝑥^∗=(0,0,…,0), 𝐹9(𝑥^∗)=0
𝐹10; Trid function	𝐹10(𝑥)=∑^𝑛𝑖=1(𝑥𝑖−1)^2−∑^𝑛𝑖=2𝑥𝑖𝑥𝑖−1	[−𝑛^2,𝑛^2]^𝑛	𝑛=5;𝐹10(𝑥^∗)=−30
𝐹11 Zakharov function	𝐹11(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖+∑^𝑛𝑖=10.5𝑖𝑥𝑖^2+∑^𝑛𝑖=10.5𝑖𝑥𝑖^4	[−5,10]^𝑛	𝑥^∗=(0,0,…,0), 𝐹11𝑥^∗=0
𝐹12 Beale function	𝐹12(𝑥)=(1.5−𝑥1+𝑥1𝑥2)^2+(2.25−𝑥1+𝑥1𝑥^22)^2+(2.625−𝑥1+𝑥1𝑥^32)^2	[−4.5,4.5]^2	𝑥^∗=(3,0.5), 𝐹12(𝑥^∗)=0
𝐹13 Easom function	𝐹13(𝑥)=−cos(𝑥1)cos(𝑥2)exp−(𝑥1−𝜋)^2−(𝑥2−𝜋)^2	[−100,100]^2	𝑥^∗=(𝜋,𝜋), 𝐹13(𝑥^∗)=−1
𝐹14 Goldstein and Price function	𝐹14(𝑥)=1+𝑥1+𝑥2+1^219−14𝑥1+3𝑥^21−14𝑥2+6𝑥1𝑥2+3𝑥^22; ×30+2𝑥1−3𝑥2^218−32𝑥1+12𝑥^21+48𝑥2−36𝑥1𝑥2+27𝑥^22	[−2,2]^2	𝑥^∗=(0,−1), 𝐹14(𝑥^∗)=3
𝐹15 Hump function	𝐹15(𝑥)=4𝑥^21−2.1𝑥^41+(1/3)𝑥^61+𝑥1𝑥2−4𝑥^22+4𝑥^42	[−5,5]^2	𝑥^∗=(0.0898,−0.7126) or; 𝑥^∗=(−0.0898,0.7126),𝐹15(𝑥^∗)=0
𝐹16 My function	𝐹16(𝑥)=(𝑥^21+𝑥2−11)^2+(𝑥1+𝑥^22−7)^2+0.1(𝑥1−3)^2+(𝑥2−2)^2	[−6,6]^2	𝑥^∗=(3,2),; 𝐹16𝑥^∗=0