An Algorithm for Global Optimization Inspired by Collective Animal Behavior

Table 14

Multimodal test functions with fixed dimensions.

Test function	S	𝑓opt
𝑓14(𝐱)=1/500+∑^25𝑗=11𝑗+∑^2𝑖=1(𝑥𝑖−𝑎𝑖𝑗)^6,^−1 𝑎𝑖𝑗=−32,−16,0,16,32,−32,…,0,16,32−32,−32,−32,−32,16,…,32,32,32	[−65.5,65.5]^2	1
𝑓15(𝐱)=∑^11𝑖=1𝑎𝑖−𝑥𝑖(𝑏^2𝑖+𝑏𝑖𝑥2)𝑏^2𝑖+𝑏𝑖𝑥3+𝑥4^2; 𝐚=[0.1957,0.1947,0.1735,0.1600,0.0844,0.0627,0.0456,0.0342,0.0342,0.0235,0.0246]; 𝐛=[0.25,0.5,1,2,4,6,8,10,12,14,16]	[−5,5]^4	0.00030
𝑓16(𝑥1,𝑥2)=4𝑥^21−2.1𝑥^41+(1/3)𝑥^61+𝑥1𝑥2−4𝑥^22+4𝑥^42	[−5,5]^2	−1.0316
𝑓17(𝑥1,𝑥2)=(𝑥2−(5.1/4𝜋^2))𝑥^21+(5/𝜋)𝑥1−6)^2+10(1−(1/8𝜋))cos𝑥1+10	𝑥1∈[−5,10]; 𝑥2∈[0,15]	0.398
𝑓18(𝑥1,𝑥2)=[1+(𝑥1+𝑥2+1)^2(19−14𝑥1+3𝑥^21−14𝑥2+6𝑥1𝑥2+3𝑥^22]×[30+(2𝑥1−3𝑥2)^2×(18−32𝑥1+12𝑥^21+48𝑥2−36𝑥1𝑥2+27𝑥^22]	[−5,5]^2	3
𝑓19(𝐱)=−∑^4𝑖=1𝑐𝑖exp(−∑^3𝑗=1𝑎𝑖𝑗(𝑥𝑗−𝑝𝑖𝑗)^2); 𝐚=310300.11035310300.11030, 𝐜=[1,1.2,3,3.2], 𝐏=0.36890.1170.26730.46990.43870.74700.10910.87320.55470.03810.57430.8828	[0,1]^3	−3.86
𝑓20(𝐱)=−∑^4𝑖=1𝑐𝑖exp(−∑^6𝑗=1𝑎𝑖𝑗(𝑥𝑗−𝑝𝑖𝑗)^2); 𝐚=103173.51.780.0510170.181433.517101781780.05100.114,𝐜=[1,1.2,3,3.2], 𝐏=0.1310.1690.5560.0120.8280.5880.2320.4130.8300.3730.1000.9990.2340.1410.3520.2880.3040.6650.4040.8820.8730.5740.1090.038	[0,1]^6	−3.32
𝑓21(𝐱)=−∑^5𝑖=1[(𝐱−𝑎𝑖)(𝐱−𝑎𝑖)^𝑇+𝑐𝑖]^−1, 𝐚=44441111888866663737292955338181626273.673.6,𝐜=[0.1,0.2,0.2,0.4,0.4,0.6,0.3,0.7,0.5,0.5]	[0,10]^4	−10.1532
𝑓22(𝐱)=−∑^7𝑖=1[(𝐱−𝑎𝑖)(𝐱−𝑎𝑖)^𝑇+𝑐𝑖]^−1, a and c, equal to 𝑓21	[0,10]^4	−10.4028
𝑓23(𝐱)=−∑^7𝑖=1[(𝐱−𝑎𝑖)(𝐱−𝑎𝑖)^𝑇+𝑐𝑖]^−1, a and c, equal to 𝑓21	[0,10]^4	−10.5363