Modified Biogeography-Based Optimization with Local Search Mechanism

Table 8

Benchmark functions used in our experimental tests.

Functions	Mathematical formula	Name	Dimension	Search space	Optima
f1	𝑓1(𝑥)=∑^𝑛𝑖=1𝑥^2𝑖	Sphere	n	[−100, 100]	0
f2	𝑓2(𝑥)=∑^𝑛𝑖=1𝑥𝑖+∏^𝑛𝑖=1𝑥𝑖	Schwefel’s Problem 2.22	n	[10, 10]	0
f3	𝑓3(𝑥)=∑^𝑛𝑖=1(∑^𝑖𝑗=1𝑥𝑗)^2	Schwefel’s Problem 1.2	n	[−100, 100]	0
f4	𝑓4(𝑥)=max𝑖𝑥𝑖,1≤𝑖≤𝑛	Schwefel’s Problem 2.21	n	[−100, 100]	0
f5	𝑓5(𝑥)=∑^𝑛−1𝑖=1[100(𝑥𝑖+1−𝑥^2𝑖)^2+(𝑥𝑖−1)^2]	Rosenbrock function	n	[−30, 30]	0
f6	𝑓6(𝑥)=∑^𝑛𝑖=1(𝑥𝑖+0.5)^2	Step function	n	[−100, 100]	0
f7	𝑓7(𝑥)=∑^𝑛𝑖=1𝑖𝑥^4𝑖+random[0,1)	Quartic function	n	[−1.28, 1.28]	0
f8	𝑓8(𝑥)=−∑^𝑛𝑖=1𝑥𝑖sin(𝑥𝑖)	Schwefel function	n	[−512, 512]	−12569.5
f9	𝑓9(𝑥)=∑^𝑛𝑖=1[𝑥^2𝑖−10cos(2𝜋𝑥𝑖)+10]	Rastrigin function	n	[−5.12, 5.12]	0
f10	𝑓10(𝑥)=−20exp−0.21𝑛^𝑛𝑖∑=1𝑥^2𝑖1−exp𝑛^𝑛𝑖∑=1cos2𝜋𝑥𝑖+20+𝑒	Ackley function	n	[−32, 32]	0
f11	𝑓111(𝑥)=4000^𝑛𝑖∑=1𝑥^2𝑖−∏^𝑛𝑖=1𝑥cos^𝑖√𝑖+1	Griewank function	n	[−600, 600]	0
f12	𝑓12𝜋(𝑥)=𝑛{10sin^2(𝜋𝑦𝑖)+∑^𝑛−1𝑖=1(𝑦𝑖−1)^2[1+10sin^2(𝜋𝑦𝑖+1)]+(𝑦𝑛−1)^2}+∑^𝑛𝑖=1𝑢(𝑥𝑖𝑦,10,100,4),𝑖=1+(𝑥𝑖−1)4𝑢(𝑥𝑖,𝑎,𝑘,𝑚)=𝑘(𝑥𝑖−𝑎)^2,0,𝑘(−𝑥𝑖−𝑎)^2,𝑥𝑖>𝑎−𝑎≤𝑥𝑖𝑥≤𝑎𝑖<−𝑎	Penalized function 1	n	[−50, 50]	0
f13	𝑓13(𝑥)=0.1{sin^2(3𝜋𝑥1)+∑^𝑛−1𝑖=1(𝑥𝑖−1)^2[1+sin^2(3𝜋𝑥𝑖+1)]+(𝑥𝑛−1)^2[1+sin^2(2𝜋𝑥𝑛)]}+∑^𝑛𝑖=1𝑢(𝑥𝑖,5,100,4),	Penalized function 2	n	[−50, 50]	0
F1	𝐹1=∑^𝑛𝑖=1𝑧^2𝑖+𝑓_bias1,𝑍=𝑋−𝑂	Shifted sphere function	n	[−100, 100]	−450
F2	𝐹2=∑^𝑛𝑖=1(∑^𝑖𝑗=1𝑧𝑗)^2+𝑓_bias2,𝑍=𝑋−𝑂	Shifted Schwefel’s Problem 1.2	n	[−100, 100]	−450
F3	𝐹3=∑^𝑛𝑖=1(10^6)^(𝑖−1)/(𝐷−1)𝑧^2𝑖+𝑓_bias3,𝑍=𝑋−𝑂	Shifted rotated high conditioned elliptic function	n	[−100, 100]	−450
F4	𝐹4=∑^𝑛∑𝑖=1^𝑖𝑗=1𝑧𝑗^2∗(1+0.4|𝑁(0,1)|)+𝑓_bias4,𝑍=𝑋−𝑂	Shifted Schwefel’s Problem 1.2 with noise in fitness	n	[−100, 100]	−450
F5	𝐹5=max𝐴𝑖𝑥−𝐵𝑖+𝑓_bias5	Schwefel’s Problem 2.6 with global optimum on bounds	n	[−30, 30]	−310
F6	𝐹6=∑^𝑛−1𝑖=1[100(𝑧𝑖+1−𝑧^2𝑖)^2+(𝑧𝑖−1)^2]+𝑓_bias6,𝑍=𝑋−𝑂+1	Shifted rosenbrock’s function	n	[−100, 100]	390
F7	𝐹7=14000^𝑛𝑖∑=1𝑧^2𝑖−∏^𝑛𝑖=1𝑧cos^𝑖√𝑖+1+𝑓_bias7,𝑍=(𝑋−𝑂)∗𝑀	Shifted rotated Griewank’s function without bounds	n	Outside of initialization range [0, 600]	−180
F8	𝐹8=−20exp−0.21𝑛^𝑛𝑖∑=1𝑧^2𝑖1−exp𝑛^𝑛𝑖∑=1cos2𝜋𝑧𝑖+20+𝑒+𝑓_bias8,𝑍=(𝑋−𝑂)∗𝑀	Shifted rotated Ackley’s function with global optimum on bounds	n	[−32, 32]	−140
F9	𝐹9=∑^𝑛𝑖=1[𝑧^2𝑖−10cos(2𝜋𝑧𝑖)+10]+𝑓_bias8,𝑍=𝑋−𝑂	Shifted Rastrigin’s function	n	[−5.12, 5.12]	−330
F10	𝐹9=∑^𝑛𝑖=1[𝑧^2𝑖−10cos(2𝜋𝑧𝑖)+10]+𝑓_bias8,𝑍=(𝑋−𝑂)∗𝑀	Shifted rotated Rastrigin’s function	n	[−5.12, 5.12]	−330
F11	𝐹11=∑^𝑛∑𝑖=1^𝑘max𝑘=0𝑎^𝑘cos(2𝜋𝑏^𝑘(𝑧𝑖+0.5))−𝑛∑^𝑘max𝑘=0𝑎^𝑘cos(2𝜋𝑏^𝑘⋅0.5)+𝑓_bias11𝑎=0.5,𝑏=3,𝑘max=20,𝑍=(𝑋−𝑂)∗𝑀	Shifted rotated weierstrass Function	n	[−600, 600]	90
F12	𝐹12=∑^𝑛𝑖=1(𝐴𝑖−𝐵𝑖(𝑥))^2+𝑓_bias12𝐴𝑖=∑^𝑛𝑖=1𝑎𝑖𝑗sin𝛼𝑗+𝑏𝑖𝑗cos𝛼𝑗,𝐵𝑖=∑^𝑛𝑖=1𝑎𝑖𝑗sin𝑥𝑗+𝑏𝑖𝑗cos𝑥𝑗	Schwefel’s Problem 2.13	n	[−100, 100]	−460
F13	𝐹13=𝑓11(𝑓5(𝑧1,𝑧2))+𝑓11(𝑓5(𝑧2,𝑧3))+⋯+𝑓11(𝑓5(𝑧𝑛−1,𝑧𝑛))+𝑓11(𝑓5(𝑧𝑛,𝑧1))+𝑓_bias13𝑍=𝑋−𝑂+1	Expanded extended Griewank’s plus Rosenbrock’s function (F8F2)	n	[−50, 50]	−130
F14	𝐹(𝑥,𝑦)=0.5+sin^2𝑥^2+𝑦^2−0.51+0.001𝑥^2+𝑦^22𝐹14=𝐹(𝑧1,𝑧2)+𝐹(𝑧2,𝑧3)+⋯+𝐹(𝑧𝑛−1,𝑧𝑛)+𝐹(𝑧𝑛,𝑧1)+𝑓_bias14,𝑍=(𝑋−𝑂)∗𝑀	Shifted rotated expanded Scaffer’s F6	n	[−100, 100]	−300