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Table 1:
The comparison between IPOP-CMA-ES-ncb and IPOP-CMA-ES-acb over 25 independent runs for each of the CEC’05 functions except f7 and f25 (these two functions are excluded as for these only an initialization range is specified). denotes that all 25 final solutions are outside the bounds. denotes that some but not all of the 25 final solutions are outside the bounds. Symbols <, ≈, and > denote whether the performance of IPOP-CMA-ES-ncb is statistically better, indifferent, or worse than that of IPOP-CMA-ES-acb according to a two-sided Wilcoxon matched-pairs signed-rank test at the .05 -level. The average errors that correspond to a statistically better result are highlighted. The numbers in parentheses at the bottom of the table represent the frequency of <, ≈, and >, respectively.
10 dimensions30 dimensions50 dimensions
fcecncbacbncbacbncbacb
f1 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f2 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f3 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f4 1.00E−08    ≈ 1.00E−08 2.44E+03 ≈ 6.58E+02 1.32E+05 1.43E+04 
f5 1.00E−08 ≈ 1.00E−08 2.30E+01 1.00E−08 7.91E+02 7.41E−02 
f6 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f8 2.01E+01 ≈ 2.00E+01 2.07E+01 2.04E+01 2.11E+01 2.09E+01 
f9 1.59E−01    ≈ 1.59E−01 1.01E+00    < 1.87E+00 1.12E+00 4.36E+00 
f10 1.19E−01    ≈ 3.18E−01 1.37E+00    ≈ 1.44E+00 2.36E+00    ≈ 2.89E+00 
f11 6.44E−01 1.00E−08 6.36E+00 7.17E−02 1.49E+01 9.94E−02 
f12 6.77E+01 4.07E+03 1.38E+03 1.19E+04 7.38E+03 4.25E+04 
f13 6.78E−01    ≈ 6.49E−01 2.47E+00    ≈ 2.63E+00 4.31E+00    ≈ 4.44E+00 
f14 2.61E+00 1.96E+00 1.28E+01 ≈ 1.26E+01 2.34E+01 2.28E+01 
f15 2.00E+02 ≈ 2.15E+02 2.01E+02 2.00E+02 2.01E+02 2.00E+02 
f16 9.02E+01    ≈ 9.04E+01 7.95E+01 1.48E+01 1.36E+02 1.10E+01 
f17 1.33E+02 ≈ 1.17E+02 4.31E+02 2.52E+02 7.69E+02 1.91E+02 
f18 7.48E+02 3.16E+02 8.16E+02 9.04E+02 8.36E+02 9.13E+02 
f19 7.75E+02 3.20E+02 8.16E+02 9.04E+02 8.36E+02 9.13E+02 
f20 7.62E+02 3.20E+02 8.16E+02 9.04E+02 8.36E+02 9.15E+02 
f21 1.06E+03 5.00E+02 8.57E+02 5.00E+02 7.15E+02 ≈ 6.64E+02 
f22 6.38E+02 7.28E+02 5.98E+02 8.10E+02 5.00E+02 8.19E+02 
f23 1.09E+03 5.86E+02 8.69E+02 5.34E+02 7.27E+02 ≈ 6.97E+02 
f24 4.05E+02 2.33E+02 2.10E+02 2.00E+02 2.14E+02 2.00E+02 
 (<, ≈, >): (2, 13, 8) (<, ≈, >): (6, 8, 9) (<, ≈, >): (6, 8, 9) 
 <or> : 10/23 (43%) <or> : 15/23 (65%) <or> : 15/23 (65%) 
 functions or : 14/23 (61%) functions or : 16/23 (70%) functions or : 17/23 (74%) 
10 dimensions30 dimensions50 dimensions
fcecncbacbncbacbncbacb
f1 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f2 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f3 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f4 1.00E−08    ≈ 1.00E−08 2.44E+03 ≈ 6.58E+02 1.32E+05 1.43E+04 
f5 1.00E−08 ≈ 1.00E−08 2.30E+01 1.00E−08 7.91E+02 7.41E−02 
f6 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 1.00E−08    ≈ 1.00E−08 
f8 2.01E+01 ≈ 2.00E+01 2.07E+01 2.04E+01 2.11E+01 2.09E+01 
f9 1.59E−01    ≈ 1.59E−01 1.01E+00    < 1.87E+00 1.12E+00 4.36E+00 
f10 1.19E−01    ≈ 3.18E−01 1.37E+00    ≈ 1.44E+00 2.36E+00    ≈ 2.89E+00 
f11 6.44E−01 1.00E−08 6.36E+00 7.17E−02 1.49E+01 9.94E−02 
f12 6.77E+01 4.07E+03 1.38E+03 1.19E+04 7.38E+03 4.25E+04 
f13 6.78E−01    ≈ 6.49E−01 2.47E+00    ≈ 2.63E+00 4.31E+00    ≈ 4.44E+00 
f14 2.61E+00 1.96E+00 1.28E+01 ≈ 1.26E+01 2.34E+01 2.28E+01 
f15 2.00E+02 ≈ 2.15E+02 2.01E+02 2.00E+02 2.01E+02 2.00E+02 
f16 9.02E+01    ≈ 9.04E+01 7.95E+01 1.48E+01 1.36E+02 1.10E+01 
f17 1.33E+02 ≈ 1.17E+02 4.31E+02 2.52E+02 7.69E+02 1.91E+02 
f18 7.48E+02 3.16E+02 8.16E+02 9.04E+02 8.36E+02 9.13E+02 
f19 7.75E+02 3.20E+02 8.16E+02 9.04E+02 8.36E+02 9.13E+02 
f20 7.62E+02 3.20E+02 8.16E+02 9.04E+02 8.36E+02 9.15E+02 
f21 1.06E+03 5.00E+02 8.57E+02 5.00E+02 7.15E+02 ≈ 6.64E+02 
f22 6.38E+02 7.28E+02 5.98E+02 8.10E+02 5.00E+02 8.19E+02 
f23 1.09E+03 5.86E+02 8.69E+02 5.34E+02 7.27E+02 ≈ 6.97E+02 
f24 4.05E+02 2.33E+02 2.10E+02 2.00E+02 2.14E+02 2.00E+02 
 (<, ≈, >): (2, 13, 8) (<, ≈, >): (6, 8, 9) (<, ≈, >): (6, 8, 9) 
 <or> : 10/23 (43%) <or> : 15/23 (65%) <or> : 15/23 (65%) 
 functions or : 14/23 (61%) functions or : 16/23 (70%) functions or : 17/23 (74%) 
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