An Improved Harmony Search Based on Teaching-Learning Strategy for Unconstrained Optimization Problems

Table 1

Benchmark test functions.

Function name	Benchmark functions expression	Search range	Optimum value
𝐹1: Ackley function	𝐹1(𝑥)=−20𝑒^(1/5)(1/𝐷)∑𝐷𝑖=1𝑥2𝑖−𝑒^(1/𝐷)∑𝐷𝑖=1cos(2𝜋𝑥𝑖)+20−𝑒	[−15,30]^𝐷	𝑥^∗=(0,0,…,0), 𝐹1(𝑥^∗)=0
𝐹2: Dixon and Price function	𝐹2(𝑥)=(𝑥1+1)^2+∑^𝐷𝑖=2𝑖(2𝑥^2𝑖−𝑥𝑖−1)^2	[−10,10]^𝐷	𝐹2(𝑥^∗)=0
𝐹3: Griewank function	𝐹3=(1/4000)∑^𝐷𝑖=1𝑥^2𝑖−∏^𝐷𝑖=1(cos(𝑥𝑖/√𝑖))+1	[−600,600]^𝐷	𝑥^∗=(0,0,…,0), 𝐹3(𝑥^∗)=0
𝐹4: Levy function	𝐹4(𝑥)=sin^2𝜋𝑦1+∑^𝐷−1𝑖=1(𝑦𝑖−1)^21+10sin^2𝜋𝑦𝑖+1	[−10,10]^𝐷	𝑥^∗=(1,1,…,1), 𝐹4(𝑥^∗)=0
	+(𝑦𝐷−1)^21+10sin^22𝜋𝑦𝐷, where 𝑦𝑖=1+((𝑥𝑖−1)/4), 𝑖=1,2,…,𝐷
𝐹5: Michalewics function	𝐹5(𝑥)=−∑^𝐷𝑖=1sin𝑥𝑖sin𝑖𝑥^2𝑖/𝜋^2𝑚;𝑚=10	[−10,𝜋]^𝐷	𝑛=5;𝐹5(𝑥^∗)=−4.687658
𝐹6: Powell function	𝐹6(𝑥)=∑^𝐷/4𝑖=1𝑥4𝑖−3+10𝑥4𝑖−2^2+5𝑥4𝑖−1−𝑥4𝑖^2+𝑥4𝑖−2−𝑥4𝑖−1^2+10𝑥4𝑖−3−𝑥4𝑖^4	[−4,5]^𝐷	x* = (3, −1, 0, 1, …, 3, −1, 0, 1), 𝐹6(𝑥^∗)=0
𝐹7: Rastrigin function	𝐹7(𝑥)=10𝐷+∑^𝐷𝑖=1𝑥^2𝑖−10cos(2𝜋𝑥𝑖)	[−5.12,5.12]^𝐷	𝑥^∗=(0,0,…,0), 𝐹7(𝑥^∗)=0
𝐹8: Rosenbrock function	𝐹8(𝑥)=∑^𝐷−1𝑖=1100(𝑥^2𝑖−𝑥𝑖−1)^2+(𝑥𝑖−1)^2	[−10,10]^𝐷	𝑥^∗=(1,1,…,1), 𝐹8(𝑥^∗)=0
𝐹9: Schwefel2.26 function	𝐹9(𝑥)=418.9829𝐷+∑^𝐷𝑖=1𝑥^2𝑖sin(𝑥𝑖)	[−512,512]^𝐷	𝑥^∗=(420.9687,420.9687,…, 420.9687), 𝐹9(𝑥^∗)=0
𝐹10: Sphere function	𝐹10(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖	[−100,100]^𝐷	𝑥^∗=(0,0,…,0), 𝐹10(𝑥^∗)=0
𝐹11: Trid function	𝐹11(𝑥)=∑^𝐷𝑖=1(𝑥𝑖−1)^2−∑^𝐷𝑖=2𝑥𝑖𝑥𝑖−1	[−𝐷^2,𝐷^2]^𝐷	𝐷=5;𝐹11(𝑥^∗)=−30
𝐹12: Zakharov function	𝐹12(𝑥)=∑^𝐷𝑖=1𝑥^2𝑖+∑^𝐷𝑖=10.5𝑖𝑥𝑖^2+∑^𝐷𝑖=10.5𝑖𝑥𝑖^4	[−5,10]^𝐷	𝑥^∗=(0,0,…,0), 𝐹12(𝑥^∗)=0
𝐹13: Sphere shift function	𝐹13(𝑥)=∑^𝐷𝑖=1𝑧^2𝑖+𝑓_bias13, 𝑓_bias13=−450, 𝑧=𝑥−𝑜; 𝑜=[𝑜1,𝑜2,...,𝑜𝐷]	[−100,100]^𝐷	𝑥^∗=𝑜, 𝐹12(𝑥^∗)=−450
𝐹14: Schwefel shift function	𝐹14=max(|𝑧|)+𝑓_bias14, 𝑓_bias14=−450, 𝑧=𝑥−𝑜; 𝑜=[𝑜1,𝑜2,...,𝑜𝐷]	[−100,100]^𝐷	𝑥^∗=𝑜, 𝐹14(𝑥^∗)=−450
𝐹15: Rosenbrock shift function	𝐹15(𝑥)=∑^𝐷−1𝑖=1100(𝑧^2𝑖−𝑧𝑖−1)^2+(𝑧𝑖−1)^2+𝑓_bias15,; 𝑓_bias15=390, 𝑧=𝑥−𝑜; 𝑜=[𝑜1,𝑜2,...,𝑜𝐷]	[−100,100]^𝐷	𝑥^∗=𝑜, 𝐹15(𝑥^∗)=390
𝐹16: Griewank shift function	𝐹16=(1/4000)∑^𝐷𝑖=1𝑧^2𝑖−∏^𝐷𝑖=1(cos(𝑧𝑖/√𝑖))+1+𝑓_bias16,; 𝑓_bias16=−180, 𝑧=(𝑥−𝑜)×𝑀, 𝑜=[𝑜1,𝑜2,...,𝑜𝐷]	[−600,600]^𝐷	𝑥^∗=𝑜, 𝐹16(𝑥^∗)=−180
𝐹17: Rastrigin shift function	𝐹17(𝑥)=10𝐷+∑^𝐷𝑖=1𝑧^2𝑖−10cos(2𝜋𝑧𝑖)+𝑓_bias17,; 𝑓_bias17=−330, 𝑧=𝑥−𝑜; 𝑜=[𝑜1,𝑜2,...,𝑜𝐷]	[−5.12,5.12]^𝐷	𝑥^∗=𝑜, 𝐹17(𝑥^∗)=−330
𝐹18: Ackley shift function	𝐹18(𝑥)=−20𝑒^(1/5)(1/𝐷)∑𝐷𝑖=1𝑧2𝑖−𝑒^(1/𝐷)∑𝐷𝑖=1cos(2𝜋𝑧𝑖)+20−𝑒,; 𝑓_bias18=−140, 𝑧=𝑥−𝑜; 𝑜=[𝑜1,𝑜2,...,𝑜𝐷]	[−32,32]^𝐷	𝑥^∗=𝑜, 𝐹18(𝑥^∗)=−140
𝐹19: Fast Fractal “Double Dip” function	𝐹19(𝑥)=∑^𝐷𝑖=1fractal1𝐷(𝑥𝑖+twist(𝑥(𝑖mod𝐷)+1))	Unknown	[−1,1]^𝐷
	twist(𝑦)=4(𝑦^4−2𝑦^3+𝑦^2)
	fractal1𝐷(𝑥)≈∑^3𝑘=1∑^2𝑘−11∑^ran21^(𝑜)1doubledip(𝑥,ran1(𝑜),2^𝑘−1((𝑜)2−ran1))
	doubledip(𝑥,𝑐,𝑠)=(−6144(𝑥−𝑐)^6+3088(𝑥−𝑐)^4−392(𝑥−𝑐)^2+1)𝑠0,otherwise
	ran1(𝑜): double, pseudo randomly chosen, with seed 𝑜, with equal probability from the interval [0,1]; ran2(𝑜): integer, pseudo randomly chosen, with seed 𝑜, with equal probability from the set {0,1,2}; fractal1𝐷(𝑥) is an approximation to a recursive algorithm, it does not take account of wrapping at the boundaries, or local reseeding of the random generators.
𝐹20: Schwefel2.22 function	𝐹20(𝑥)=∑^𝐷𝑖=1𝑥𝑖+∏^𝐷𝑖=1𝑥𝑖	[−10,10]^𝐷	𝑥^∗=(0,0,...,0), 𝐹20(𝑥^∗)=0
𝐹21: Extended_f10 shift function	𝐹21(𝑥)=∑^𝐷−1𝑖=1𝑓10(𝑧𝑖,𝑧𝑖+1)+𝑓10(𝑧𝐷,𝑧1𝑓),𝑧=𝑥−𝑜10=(𝑥^2+𝑦^2)^0.25⋅(sin^2(50⋅(𝑥^2+𝑦^2)^0.1)+1)	[−100,100]^𝐷	𝑥^∗=𝑜, 𝐹21(𝑥^∗)=0
𝐹22: Bohachevsky function	𝐹22(𝑥)=∑^𝐷𝑖=1(𝑧^2𝑖+2𝑧^2𝑖+1−0.3cos(3𝜋𝑧𝑖)−0.4cos(4𝜋𝑧𝑖+1)+0.7),𝑧=𝑥−𝑜	[−15,15]^𝐷	𝑥^∗=𝑜, 𝐹22(𝑥^∗)=0
𝐹23: Schaffer shift function	𝐹23(𝑥)=∑^𝐷−1𝑖=1(𝑧^2𝑖+𝑧^2𝑖+1)^0.25(sin^2(50⋅(𝑧^2𝑖+𝑧^2𝑖+1)^0.1)+1),𝑧=𝑥−𝑜	[−100,100]^𝐷	𝑥^∗=𝑜, 𝐹23(𝑥^∗)=0