Exponential moving average weights N=20 (α≈0.095)
TEST FUNCTIONS
: Shifted Sphere Function : Shifted Schwefel’s Problem 1.2
: Shifted Rotated High Conditioned Elliptic Function
: Shifted Schwefel’s Problem 1.2 with Noise in Fitness : Schwefel’s Problem 2.6 with Global Optimum on Bounds : Shifted Rosenbrock’s Function
: Shifted Rotated Griewank’s Function without Bounds
: Shifted Rotated Ackley’s Function with Global Optimum on Bounds : Shifted Rastrigin’s Function
: Shifted Rotated Rastrigin’s Function
: Shifted Rotated Weierstrass Function
: Schwefel’s Problem 2.13
: Expanded Extended Griewank’s plus Rosenbrock’s Function (F8F2)
: Shifted Rotated Expanded Scaffer’s F6
: Hybrid Composition Function
: Rotated Hybrid Composition Function
: Rotated Hybrid Composition Function with Noise in Fitness
: Rotated Hybrid Composition Function
: Rotated Hybrid Composition Function with a Narrow Basin for the Global Optimum
: Rotated Hybrid Composition Function with the Global Optimum on the Bounds
: Rotated Hybrid Composition Function
: Rotated Hybrid Composition Function with High Condition Number Matrix
: Non-Continuous Rotated Hybrid Composition Function
: Rotated Hybrid Composition Function
: Rotated Hybrid Composition Function without Bounds
5. RESULTS
All results for the two algorithms (DE and VDE-algorithms) are summarized in the following chapters. In addition to the benchmark checkpoints, the results also include extended performance for 500000 (5.0e5) function evaluations. Tables 9-14 summarize the function error values for 10-dimensional functions, and Tables 18-23 summarize the function error values for 30-dimensional functions. Tables 15-16 summarize successful runs which reach the fixed accuracy level and success performance for 10-dimensional functions, and Tables 24-25 summarize successful runs which reach the fixed accuracy level and success performance for 30-dimensional functions. Best values are marked in bold. If all algorithms have reached the global optimum or have achieved the same value at given point, the values are not bolded.
Results for 10-dimensional functions 5.1.
Table 9. Error values achieved for - (10D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 1.57E+021.96E+021.41E+021.55E+021.40E+037.99E+027.59E+024.55E+021.02E+075.74E+066.92E+063.79E+061.84E+031.91E+032.13E+038.84E+027th3.27E+023.32E+023.37E+023.36E+022.62E+031.78E+032.89E+032.31E+032.40E+072.25E+072.37E+072.56E+073.57E+033.23E+033.82E+031.99E+0313th4.59E+024.16E+024.38E+024.37E+023.28E+032.31E+033.73E+032.80E+033.05E+073.01E+073.69E+073.67E+075.70E+034.30E+034.68E+033.14E+0319th5.87E+025.99E+025.60E+025.29E+024.59E+033.27E+035.21E+034.14E+034.31E+074.38E+075.64E+074.53E+077.63E+036.09E+035.99E+034.44E+0325th7.90E+021.04E+031.41E+037.71E+026.78E+037.24E+038.68E+036.36E+035.96E+077.19E+071.38E+089.09E+071.09E+048.33E+038.93E+037.00E+03 Mean4.54E+024.71E+024.64E+024.19E+023.71E+032.94E+033.99E+033.25E+033.39E+073.39E+074.20E+074.10E+075.78E+034.57E+035.09E+033.41E+03Std1.74E+022.05E+022.38E+021.58E+021.50E+031.67E+031.87E+031.57E+031.34E+071.71E+072.78E+072.28E+072.49E+031.71E+031.97E+031.65E+031st 2.68E-110.00E+003.30E-110.00E+004.92E-021.41E-061.42E-021.32E-059.75E+053.56E+041.00E+061.16E+031.76E-018.21E-059.84E-035.57E-057th1.08E-100.00E+001.54E-100.00E+001.09E-019.38E-058.95E-024.24E-051.57E+061.21E+051.53E+067.12E+036.27E-016.57E-041.76E-013.42E-0313th1.50E-101.48E-122.99E-101.59E-122.81E-013.17E-043.55E-014.94E-042.05E+061.94E+051.91E+062.14E+041.58E+002.27E-034.60E-011.00E-0219th3.20E-102.24E-107.06E-103.30E-116.46E-018.44E-041.31E+005.97E-032.45E+065.34E+052.35E+061.36E+052.19E+006.72E-031.61E+005.98E-0225th7.99E-107.31E-083.62E-093.61E-091.25E+002.21E-028.82E+001.10E-013.77E+061.56E+063.92E+062.21E+069.32E+002.77E-022.30E+012.60E+00
Mean2.32E-103.23E-096.49E-101.89E-104.19E-012.80E-031.29E+001.42E-022.11E+063.80E+052.05E+062.16E+052.08E+005.22E-032.98E+002.04E-01Std1.98E-101.43E-088.76E-107.05E-103.54E-015.71E-032.11E+003.06E-027.11E+054.17E+057.82E+054.64E+052.32E+007.36E-036.13E+005.63E-011st 0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+001.28E+000.00E+001.22E-020.00E+000.00E+000.00E+000.00E+000.00E+007th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+003.32E+000.00E+004.03E-020.00E+000.00E+000.00E+000.00E+000.00E+0013th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+005.01E+000.00E+006.55E-020.00E+000.00E+000.00E+000.00E+000.00E+0019th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+001.01E+013.41E-121.58E-010.00E+000.00E+000.00E+000.00E+000.00E+0025th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+003.11E+018.69E-115.23E-010.00E+000.00E+000.00E+000.00E+000.00E+00 Mean0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+007.35E+005.98E-121.21E-010.00E+000.00E+000.00E+000.00E+000.00E+00Std0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+006.55E+001.70E-111.24E-010.00E+000.00E+000.00E+000.00E+000.00E+001st 0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+007th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+0013th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+0019th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+0025th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
Mean0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00Std0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
1E+0 5 5E+0
5 10D 1243ffff
1E+0 3 1E+0
4
Table 10. Error values achieved for - (10D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 3.00E+032.97E+033.49E+031.97E+031.21E+068.50E+053.30E+065.47E+051.19E+023.53E+021.28E+022.55E+022.04E+012.05E+012.03E+012.02E+017th4.51E+034.08E+034.31E+033.80E+039.77E+063.29E+068.60E+063.15E+062.60E+025.49E+022.86E+025.29E+022.06E+012.07E+012.07E+012.06E+0113th5.12E+034.61E+035.09E+034.50E+031.58E+071.18E+072.25E+075.48E+063.57E+026.80E+024.09E+026.41E+022.08E+012.07E+012.07E+012.08E+0119th6.03E+035.52E+035.84E+035.49E+032.50E+072.01E+074.01E+071.34E+074.96E+028.02E+025.17E+027.23E+022.08E+012.08E+012.08E+012.08E+0125th9.88E+037.01E+037.81E+037.47E+039.03E+071.16E+081.27E+085.88E+078.51E+021.30E+037.72E+021.08E+032.09E+012.09E+012.09E+012.09E+01 Mean5.32E+034.85E+035.15E+034.47E+032.27E+071.89E+073.12E+079.91E+064.17E+027.00E+024.14E+026.28E+022.07E+012.07E+012.07E+012.07E+01Std1.43E+031.11E+031.07E+031.33E+032.07E+072.59E+073.26E+071.17E+071.93E+022.39E+021.66E+022.06E+021.37E-011.17E-011.34E-011.72E-011st 2.51E-011.77E-032.90E-013.56E-046.55E+002.91E-014.85E+002.74E-025.56E-019.85E-019.65E-025.78E-012.04E+012.03E+012.03E+012.02E+017th1.47E+004.67E-036.87E-011.23E-031.10E+013.85E+001.60E+014.78E+007.14E-011.20E+007.72E-011.12E+002.05E+012.05E+012.05E+012.05E+0113th2.18E+001.14E-021.11E+006.40E-031.74E+015.02E+002.35E+015.37E+008.01E-011.55E+008.36E-011.25E+002.06E+012.05E+012.06E+012.06E+0119th3.75E+003.09E-022.49E+001.57E-025.36E+016.37E+009.13E+016.56E+008.60E-011.95E+008.95E-012.28E+002.06E+012.06E+012.06E+012.06E+0125th1.28E+011.76E-011.37E+012.21E+002.11E+022.29E+012.34E+022.84E+029.32E-015.87E+001.11E+006.32E+002.07E+012.06E+012.08E+012.07E+01
Mean3.00E+003.23E-022.23E+001.05E-014.34E+016.27E+005.91E+012.97E+017.87E-012.10E+008.22E-011.73E+002.05E+012.05E+012.06E+012.05E+01Std2.61E+004.52E-022.84E+004.31E-015.30E+014.91E+006.47E+017.33E+019.57E-021.43E+001.81E-011.20E+008.79E-028.20E-021.03E-011.07E-011st 0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+007.40E-039.86E-039.86E-031.23E-022.03E+012.02E+012.03E+012.02E+017th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+005.16E-025.17E-026.40E-027.39E-022.04E+012.03E+012.04E+012.03E+0113th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+009.10E-026.64E-029.61E-021.40E-012.05E+012.03E+012.05E+012.04E+0119th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+001.35E-013.84E-011.30E-015.03E-012.05E+012.04E+012.06E+012.04E+0125th3.64E-120.00E+000.00E+000.00E+000.00E+003.99E+003.99E+003.99E+004.61E-015.92E-015.09E-016.56E-012.06E+012.05E+012.07E+012.05E+01 Mean2.18E-130.00E+000.00E+000.00E+000.00E+007.97E-017.97E-014.79E-011.10E-011.88E-011.27E-012.65E-012.05E+012.03E+012.05E+012.04E+01Std7.84E-130.00E+000.00E+000.00E+000.00E+001.59E+001.59E+001.30E+009.41E-022.07E-011.06E-012.18E-018.38E-027.19E-021.20E-017.66E-021st 0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+007.40E-039.86E-039.86E-031.23E-022.03E+012.02E+012.03E+012.02E+017th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+005.16E-024.19E-026.40E-025.66E-022.04E+012.02E+012.04E+012.02E+0113th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+009.10E-025.41E-029.61E-021.03E-012.05E+012.03E+012.05E+012.03E+0119th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+001.35E-017.38E-021.30E-011.50E-012.05E+012.03E+012.06E+012.03E+0125th0.00E+000.00E+000.00E+000.00E+000.00E+003.99E+003.99E+003.99E+004.61E-011.45E-015.09E-015.26E-012.06E+012.04E+012.07E+012.04E+01
Mean0.00E+000.00E+000.00E+000.00E+000.00E+007.97E-017.97E-014.79E-011.10E-016.04E-021.27E-011.47E-012.05E+012.03E+012.05E+012.03E+01Std0.00E+000.00E+000.00E+000.00E+000.00E+001.59E+001.59E+001.30E+009.41E-023.43E-021.06E-011.34E-018.59E-025.49E-021.20E-015.23E-02
1E+0 5 5E+0
5 10D 5687ffff
1E+0 3 1E+0
4
Table 11. Error values achieved for - (10D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 1.90E+011.80E+011.79E+012.13E+017.13E+018.52E+017.75E+017.92E+019.47E+009.70E+008.82E+009.49E+001.82E+041.70E+042.85E+043.17E+047th2.95E+012.61E+012.60E+012.52E+019.55E+019.37E+019.19E+011.01E+021.16E+011.14E+011.10E+011.10E+016.53E+045.08E+045.92E+044.94E+0413th3.13E+012.98E+013.12E+013.03E+011.04E+021.05E+021.05E+021.07E+021.21E+011.18E+011.20E+011.20E+017.39E+046.76E+046.49E+046.62E+0419th3.38E+013.36E+013.40E+013.26E+011.13E+021.13E+021.19E+021.17E+021.24E+011.23E+011.23E+011.23E+019.14E+048.27E+048.46E+047.35E+0425th3.97E+013.91E+014.39E+014.20E+011.42E+021.30E+021.34E+021.28E+021.33E+011.38E+011.29E+011.30E+011.17E+051.07E+051.16E+058.76E+04 Mean3.15E+012.97E+013.06E+013.04E+011.05E+021.05E+021.05E+021.08E+021.19E+011.18E+011.16E+011.17E+017.20E+046.57E+046.91E+046.28E+04Std4.01E+005.49E+006.50E+005.33E+001.43E+011.26E+011.52E+011.27E+018.89E-018.20E-011.04E+009.42E-012.63E+042.30E+041.90E+041.53E+041st 3.47E-100.00E+006.67E-110.00E+004.36E+014.04E+014.44E+012.89E+018.88E+008.87E+008.82E+008.60E+007.92E+034.68E+038.01E+031.14E+037th2.26E-093.33E-116.44E-102.10E-105.54E+014.87E+015.29E+014.43E+019.74E+009.53E+009.98E+009.49E+001.30E+047.64E+031.11E+044.63E+0313th4.18E-091.62E-091.77E-094.30E-105.99E+015.50E+015.60E+014.91E+011.00E+011.02E+011.03E+019.99E+001.59E+049.12E+031.50E+048.32E+0319th1.21E-083.58E-083.18E-091.65E-096.30E+015.71E+016.06E+015.42E+011.05E+011.07E+011.07E+011.08E+011.83E+041.31E+041.76E+041.02E+0425th9.95E-012.33E-077.78E-089.95E-016.75E+016.32E+017.01E+016.71E+011.14E+011.15E+011.11E+011.14E+012.09E+042.33E+042.06E+041.85E+04
Mean1.19E-012.90E-088.78E-093.98E-025.87E+015.29E+015.62E+014.97E+011.01E+011.01E+011.02E+011.00E+011.51E+041.08E+041.43E+048.23E+03Std3.23E-015.27E-081.96E-081.95E-016.21E+006.15E+006.56E+007.63E+005.86E-017.80E-016.25E-017.56E-013.90E+034.30E+033.71E+034.58E+031st 0.00E+000.00E+000.00E+000.00E+002.36E+012.00E+012.92E+012.26E+013.00E-016.02E-042.54E-011.07E-031.00E+028.61E-061.88E+029.15E-127th0.00E+000.00E+000.00E+000.00E+003.47E+012.73E+013.53E+012.91E+013.37E+009.93E-026.37E-011.50E+003.88E+026.43E-057.08E+023.22E-1013th0.00E+000.00E+000.00E+000.00E+003.86E+013.21E+013.78E+013.13E+016.42E+003.95E+003.23E+007.98E+007.34E+022.41E-041.38E+031.90E-0819th0.00E+000.00E+000.00E+000.00E+004.05E+013.54E+014.12E+013.30E+018.57E+008.73E+008.50E+008.85E+009.60E+024.15E-032.03E+033.27E-0725th9.95E-010.00E+000.00E+009.95E-014.38E+013.91E+014.64E+013.79E+019.26E+009.77E+009.47E+009.53E+002.37E+031.06E+013.97E+031.02E+01 Mean1.19E-010.00E+000.00E+003.98E-023.74E+013.13E+013.80E+013.06E+015.93E+004.00E+004.20E+005.32E+007.73E+028.30E-011.61E+038.15E-01Std3.23E-010.00E+000.00E+001.95E-014.78E+004.83E+004.33E+003.59E+002.72E+003.79E+003.69E+003.97E+005.25E+022.79E+001.12E+032.74E+001st 0.00E+000.00E+000.00E+000.00E+001.99E+002.98E+001.99E+003.98E+000.00E+000.00E+000.00E+000.00E+001.48E-110.00E+000.00E+000.00E+007th0.00E+000.00E+000.00E+000.00E+004.97E+004.97E+005.07E+008.95E+000.00E+000.00E+000.00E+000.00E+003.27E-100.00E+006.67E-110.00E+0013th0.00E+000.00E+000.00E+000.00E+006.96E+009.95E+008.95E+001.99E+010.00E+000.00E+001.74E-109.84E-013.11E-090.00E+001.53E-100.00E+0019th0.00E+000.00E+000.00E+000.00E+008.95E+002.26E+012.35E+012.38E+011.00E+000.00E+009.78E-011.58E+004.03E-080.00E+009.06E-100.00E+0025th9.95E-010.00E+000.00E+009.95E-012.68E+012.59E+013.19E+012.84E+016.42E+004.06E+004.78E+003.52E+005.96E-071.00E+019.46E-011.00E+01
Mean1.19E-010.00E+000.00E+003.98E-028.75E+001.36E+011.35E+011.66E+011.01E+004.06E-017.03E-019.51E-016.17E-088.00E-013.78E-028.00E-01Std3.23E-010.00E+000.00E+001.95E-016.02E+008.74E+001.01E+018.07E+001.87E+009.97E-011.16E+001.00E+001.37E-072.71E+001.85E-012.71E+00
1E+0 5 5E+0
5 10D 9101211ffff
1E+0 3 1E+0
4
Table 12. Error values achieved for - (10D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 8.07E+008.44E+008.52E+007.06E+004.12E+003.87E+004.07E+004.09E+005.79E+026.30E+026.00E+026.74E+023.17E+023.13E+023.19E+022.95E+027th1.02E+019.73E+001.03E+011.02E+014.27E+004.19E+004.30E+004.28E+007.20E+026.99E+026.85E+027.56E+023.71E+023.72E+023.61E+023.94E+0213th1.15E+011.19E+011.32E+011.14E+014.31E+004.29E+004.35E+004.34E+007.42E+027.41E+027.31E+027.90E+023.93E+023.87E+023.87E+024.21E+0219th1.32E+011.32E+011.49E+011.22E+014.37E+004.38E+004.43E+004.42E+007.84E+027.63E+027.61E+028.14E+024.13E+024.06E+024.22E+024.49E+0225th1.45E+011.86E+011.83E+011.78E+014.50E+004.51E+004.54E+004.52E+008.41E+028.16E+028.96E+028.87E+024.80E+024.69E+025.06E+025.02E+02 Mean1.16E+011.18E+011.31E+011.12E+014.32E+004.29E+004.34E+004.34E+007.38E+027.32E+027.23E+027.82E+023.95E+023.87E+023.95E+024.20E+02Std1.68E+002.63E+002.79E+002.24E+008.92E-021.53E-011.12E-011.10E-016.30E+014.80E+016.27E+015.14E+013.87E+013.78E+014.99E+014.05E+011st 2.43E+002.74E+003.87E+002.65E+003.71E+003.47E+003.29E+003.42E+003.99E+024.22E+023.85E+024.50E+022.08E+021.63E+021.73E+022.01E+027th4.12E+003.68E+004.46E+003.37E+003.91E+003.77E+003.89E+003.96E+004.63E+025.39E+024.80E+025.06E+022.27E+021.93E+022.14E+022.21E+0213th4.94E+003.96E+005.04E+003.72E+004.01E+004.05E+004.01E+004.01E+005.45E+025.51E+025.50E+025.29E+022.36E+022.05E+022.31E+022.32E+0219th5.51E+004.75E+005.53E+004.12E+004.15E+004.09E+004.11E+004.12E+005.79E+025.71E+025.81E+025.66E+022.49E+022.18E+022.41E+022.45E+0225th6.06E+005.47E+006.20E+005.53E+004.22E+004.23E+004.18E+004.21E+006.23E+025.87E+026.10E+026.22E+022.67E+022.34E+022.58E+022.69E+02
Mean4.73E+004.11E+005.01E+003.79E+004.01E+003.94E+003.97E+003.99E+005.23E+025.39E+025.29E+025.33E+022.37E+022.03E+022.27E+022.33E+02Std9.70E-016.89E-016.40E-016.61E-011.40E-012.04E-011.88E-011.65E-016.59E+014.57E+015.89E+014.57E+011.52E+011.89E+012.15E+011.59E+011st 3.22E-014.75E-014.29E-015.55E-012.15E+002.32E+001.55E+002.66E+001.57E+021.11E+021.05E+021.36E+021.66E+021.49E+021.37E+021.23E+027th7.41E-011.22E+001.34E+001.68E+003.30E+003.46E+003.25E+003.27E+001.96E+022.07E+021.53E+021.99E+021.66E+021.59E+021.69E+021.60E+0213th1.62E+002.28E+002.26E+002.09E+003.59E+003.62E+003.63E+003.44E+002.24E+024.36E+021.93E+022.31E+021.73E+021.65E+021.76E+021.69E+0219th2.70E+002.61E+002.87E+002.34E+003.78E+003.70E+003.70E+003.55E+004.61E+024.69E+024.57E+022.56E+021.84E+021.72E+021.83E+021.72E+0225th3.63E+003.12E+003.33E+002.73E+003.93E+003.83E+003.92E+003.69E+005.26E+024.97E+025.15E+024.63E+021.95E+021.87E+021.94E+021.78E+02 Mean1.71E+001.98E+002.09E+002.00E+003.47E+003.52E+003.42E+003.35E+002.90E+023.39E+022.59E+022.50E+021.74E+021.65E+021.74E+021.65E+02Std1.02E+008.34E-019.03E-014.63E-014.23E-013.01E-015.02E-012.89E-011.33E+021.42E+021.50E+028.56E+011.34E+011.00E+011.35E+011.16E+011st 3.13E-019.87E-034.10E-012.76E-016.69E-011.52E-012.07E-011.19E+000.00E+000.00E+000.00E+000.00E+009.16E+015.09E+019.09E+011.21E+027th5.36E-014.11E-015.51E-015.90E-012.70E+001.22E+003.18E+001.75E+004.59E-054.04E+010.00E+002.33E-099.69E+019.80E+019.70E+011.36E+0213th7.41E-014.90E-018.67E-011.31E+003.17E+002.12E+003.27E+002.17E+003.00E-034.00E+020.00E+002.85E-031.02E+021.05E+021.02E+021.46E+0219th9.55E-016.45E-011.60E+001.47E+003.52E+002.69E+003.42E+002.62E+004.04E+024.00E+024.00E+028.05E+011.05E+021.10E+021.35E+021.48E+0225th1.62E+002.13E+002.41E+002.09E+003.93E+003.19E+003.92E+003.29E+004.71E+024.07E+024.01E+024.14E+021.51E+021.46E+021.63E+021.64E+02
Mean7.51E-016.05E-011.13E+001.11E+002.86E+002.01E+003.08E+002.21E+001.25E+022.44E+021.12E+026.34E+011.06E+021.09E+021.15E+021.43E+02Std2.86E-014.26E-016.36E-015.04E-019.41E-018.71E-018.48E-015.92E-011.96E+021.82E+021.80E+021.16E+021.49E+012.21E+012.37E+011.06E+01
1E+0 5 5E+0
5 10D 13141615ffff
1E+0 3 1E+0
4
Table 13. Error values achieved for - (10D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 2.95E+023.33E+023.22E+023.56E+021.06E+031.06E+031.06E+031.06E+031.09E+031.07E+031.11E+031.10E+031.11E+031.09E+031.09E+031.07E+037th3.62E+024.05E+023.63E+024.13E+021.16E+031.14E+031.15E+031.16E+031.13E+031.13E+031.15E+031.16E+031.13E+031.14E+031.13E+031.17E+0313th3.79E+024.21E+023.87E+024.41E+021.17E+031.16E+031.17E+031.19E+031.15E+031.15E+031.16E+031.19E+031.16E+031.17E+031.16E+031.18E+0319th4.17E+024.46E+024.05E+024.83E+021.19E+031.18E+031.20E+031.22E+031.19E+031.16E+031.18E+031.20E+031.20E+031.19E+031.18E+031.20E+0325th4.95E+025.37E+024.87E+025.27E+021.22E+031.24E+031.28E+031.27E+031.22E+031.19E+031.21E+031.30E+031.26E+031.21E+031.22E+031.27E+03 Mean3.88E+024.25E+023.89E+024.45E+021.17E+031.16E+031.17E+031.19E+031.16E+031.14E+031.17E+031.18E+031.17E+031.16E+031.16E+031.18E+03Std4.64E+013.94E+013.78E+014.32E+013.30E+013.73E+014.81E+014.90E+013.60E+013.05E+012.52E+014.32E+014.12E+013.32E+013.00E+013.98E+011st 2.01E+022.14E+021.97E+022.03E+026.27E+024.70E+026.72E+026.50E+026.38E+025.01E+026.40E+026.43E+026.46E+024.70E+026.31E+026.49E+027th2.29E+022.39E+022.47E+022.32E+026.97E+025.85E+027.40E+028.31E+027.25E+025.93E+027.35E+028.42E+027.30E+025.35E+027.79E+029.15E+0213th2.52E+022.48E+022.61E+022.51E+027.67E+026.43E+028.75E+029.18E+028.22E+026.69E+027.95E+028.94E+027.77E+026.27E+028.02E+029.71E+0219th2.58E+022.62E+022.66E+022.62E+028.29E+028.36E+029.36E+029.64E+028.93E+028.51E+029.12E+029.41E+028.43E+027.15E+029.06E+029.94E+0225th2.76E+022.85E+022.82E+022.83E+029.90E+021.00E+031.01E+031.01E+039.97E+029.82E+029.98E+021.01E+031.01E+038.97E+021.01E+031.05E+03
Mean2.45E+022.50E+022.56E+022.47E+027.74E+026.91E+028.44E+028.89E+028.20E+027.00E+028.10E+028.82E+027.95E+026.36E+028.37E+029.43E+02Std2.02E+011.86E+011.93E+012.29E+019.93E+011.48E+021.07E+029.57E+011.09E+021.52E+021.07E+029.19E+019.53E+011.12E+021.01E+028.64E+011st 1.29E+021.74E+021.24E+021.66E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+027th1.64E+021.84E+021.80E+021.77E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+0213th1.83E+021.92E+021.89E+021.86E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+0219th2.01E+021.97E+021.97E+021.94E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+0225th2.18E+022.12E+022.07E+022.07E+023.00E+028.00E+028.00E+028.00E+028.00E+028.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+02 Mean1.80E+021.92E+021.83E+021.86E+023.00E+023.40E+023.20E+023.20E+023.20E+023.40E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+02Std2.43E+011.07E+012.16E+011.25E+015.09E-021.36E+029.80E+019.80E+019.80E+011.36E+024.14E-026.37E-036.27E-027.34E-092.59E-026.33E-031st 9.11E+011.03E+029.28E+019.52E+013.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+027th1.01E+021.53E+021.01E+021.47E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+0213th1.06E+021.59E+021.11E+021.56E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+0219th1.15E+021.66E+021.19E+021.66E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+0225th1.27E+021.76E+021.75E+021.84E+023.00E+028.00E+028.00E+028.00E+028.00E+028.00E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+02
Mean1.07E+021.56E+021.14E+021.52E+023.00E+023.40E+023.20E+023.20E+023.20E+023.40E+023.00E+023.00E+023.00E+023.00E+023.00E+023.00E+02Std1.01E+011.76E+011.91E+012.18E+010.00E+001.36E+029.80E+019.80E+019.80E+011.36E+020.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
1E+0 5 5E+0
5 10D 17182019ffff
1E+0 3 1E+0
4
Table 14. Error values achieved for - (10D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 1.23E+031.28E+031.22E+031.17E+039.21E+028.82E+029.36E+029.73E+021.19E+031.31E+031.22E+031.31E+031.10E+031.01E+031.15E+031.21E+036.92E+025.93E+026.17E+021.37E+037th1.34E+031.34E+031.34E+031.37E+031.01E+031.04E+031.02E+031.07E+031.34E+031.35E+031.35E+031.36E+031.19E+031.23E+031.26E+031.29E+039.06E+021.07E+039.15E+021.55E+0313th1.37E+031.36E+031.36E+031.40E+031.05E+031.06E+031.07E+031.12E+031.36E+031.36E+031.37E+031.39E+031.26E+031.27E+031.27E+031.32E+031.01E+031.27E+031.08E+031.60E+0319th1.39E+031.38E+031.38E+031.42E+031.11E+031.09E+031.10E+031.15E+031.39E+031.38E+031.39E+031.40E+031.31E+031.30E+031.32E+031.34E+031.15E+031.42E+031.25E+031.64E+0325th1.43E+031.41E+031.41E+031.46E+031.18E+031.17E+031.15E+031.21E+031.40E+031.40E+031.41E+031.45E+031.38E+031.34E+031.40E+031.38E+031.45E+031.53E+031.49E+031.73E+03 Mean1.37E+031.36E+031.35E+031.39E+031.06E+031.06E+031.06E+031.11E+031.35E+031.36E+031.36E+031.38E+031.25E+031.24E+031.28E+031.31E+031.02E+031.18E+031.06E+031.59E+03Std4.19E+013.06E+015.26E+015.28E+016.54E+015.52E+015.32E+015.98E+014.59E+012.51E+013.90E+013.45E+017.78E+018.09E+015.76E+014.31E+012.03E+022.78E+022.33E+028.58E+011st 7.07E+025.19E+027.00E+027.28E+027.99E+027.89E+028.03E+028.16E+027.53E+025.95E+027.17E+027.33E+023.06E+022.10E+022.44E+023.09E+024.24E+024.08E+024.36E+024.24E+027th8.10E+025.74E+028.18E+029.59E+028.25E+027.99E+028.29E+028.37E+028.37E+026.44E+028.43E+028.47E+023.75E+022.42E+023.21E+023.59E+024.61E+024.15E+024.52E+024.49E+0213th8.48E+026.09E+028.68E+029.91E+028.29E+028.06E+028.34E+028.48E+028.74E+026.82E+029.03E+029.42E+024.03E+022.56E+023.55E+024.17E+024.71E+024.27E+024.64E+024.54E+0219th8.76E+026.69E+029.24E+021.05E+038.35E+028.16E+028.42E+028.56E+029.39E+027.42E+029.73E+021.02E+034.44E+022.84E+024.04E+025.66E+024.77E+024.38E+024.86E+024.68E+0225th1.01E+037.93E+021.07E+031.13E+038.50E+028.54E+028.53E+028.85E+021.04E+039.41E+021.03E+031.18E+034.71E+024.29E+024.93E+027.12E+024.98E+024.60E+025.05E+025.25E+02
Mean8.43E+026.28E+028.71E+029.93E+028.29E+028.09E+028.34E+028.49E+028.83E+027.03E+029.02E+029.34E+024.04E+022.74E+023.64E+024.74E+024.69E+024.28E+024.69E+024.58E+02Std6.68E+017.35E+019.50E+019.16E+011.06E+011.38E+011.20E+011.70E+016.80E+018.04E+018.27E+011.26E+024.56E+015.63E+016.36E+011.27E+021.57E+011.42E+011.87E+012.02E+011st 5.00E+025.00E+025.00E+025.00E+023.00E+027.63E+023.03E+027.64E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+023.97E+023.94E+023.92E+023.94E+027th5.00E+025.00E+025.00E+025.00E+027.71E+027.72E+027.75E+027.71E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.02E+024.00E+024.03E+024.03E+0213th5.00E+025.00E+025.00E+025.00E+027.79E+027.73E+027.78E+027.74E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.05E+024.04E+024.06E+024.06E+0219th5.00E+025.00E+025.00E+025.00E+027.81E+027.75E+027.82E+027.76E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.08E+024.06E+024.07E+024.07E+0225th5.00E+025.00E+025.00E+025.00E+027.85E+027.80E+027.85E+027.82E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.11E+024.09E+024.11E+024.10E+02 Mean5.00E+025.00E+025.00E+025.00E+026.84E+027.73E+027.60E+027.73E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.05E+024.03E+024.05E+024.05E+02Std2.60E-053.27E-137.80E-061.62E-051.90E+024.03E+009.32E+013.63E+003.41E-136.11E-133.41E-133.41E-131.26E-050.00E+003.60E-072.17E-064.16E+004.00E+004.58E+004.08E+001st 5.00E+025.00E+025.00E+025.00E+023.00E+027.15E+023.00E+027.45E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+023.94E+023.93E+023.91E+023.93E+027th5.00E+025.00E+025.00E+025.00E+027.25E+027.58E+027.66E+027.63E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.00E+023.99E+024.01E+024.02E+0213th5.00E+025.00E+025.00E+025.00E+027.68E+027.64E+027.71E+027.65E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.03E+024.03E+024.03E+024.05E+0219th5.00E+025.00E+025.00E+025.00E+027.73E+027.68E+027.73E+027.66E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.06E+024.05E+024.05E+024.06E+0225th5.00E+025.00E+025.00E+025.00E+027.76E+027.70E+027.77E+027.69E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.09E+024.08E+024.09E+024.08E+02
Mean5.00E+025.00E+025.00E+025.00E+026.54E+027.59E+027.50E+027.63E+025.59E+025.59E+025.59E+025.59E+022.00E+022.00E+022.00E+022.00E+024.03E+024.02E+024.03E+024.04E+02Std3.41E-133.27E-133.41E-133.41E-131.99E+021.35E+019.21E+015.67E+003.41E-136.11E-133.41E-133.41E-130.00E+000.00E+000.00E+000.00E+004.17E+004.08E+004.50E+004.00E+00
1E+0 5 5E+0
5 10D 2122242523fffff
1E+0 3 1E+0
4
Table 15. FES to reach the specified accuracy and success rate and performance for
(10D).
1.0e5 5.0e5 1.0e5 5.0e5
DE 6700 7057 7320 7391 7611 7.23E+03 2.52E+02 100 % 100 % 7225 7225 VDE-1 5284 5948 6619 7216 9108 6.75E+03 9.47E+02 100 % 100 % 6753 6753 VDE-2 6713 7227 7388 7577 8421 7.43E+03 3.54E+02 100 % 100 % 7425 7425 VDE-3 4863 5988 6379 6901 7885 6.42E+03 7.43E+02 100 % 100 % 6423 6423 DE 19653 21367 22012 22817 24883 2.21E+04 1.25E+03 100 % 100 % 22077 22077 VDE-1 10374 12470 13158 14299 15156 1.31E+04 1.41E+03 100 % 100 % 13082 13082 VDE-2 17317 18804 20620 21844 28806 2.10E+04 2.84E+03 100 % 100 % 20973 20973 VDE-3 11118 12513 14578 15732 18032 1.44E+04 2.00E+03 100 % 100 % 14407 14407 DE 198379 210434 214462 221219 228329 2.15E+05 7.80E+03 0 % 100 % - 214981 VDE-1 61742 66000 68280 70015 83189 6.90E+04 4.46E+03 100 % 100 % 69047 69047 VDE-2 148189 154231 157669 162679 169465 1.58E+05 6.09E+03 0 % 100 % - 158404 VDE-3 27771 34235 37874 41353 53307 3.87E+04 6.35E+03 100 % 100 % 38705 38705
DE 22615 23851 24842 26284 28467 2.52E+04 1.59E+03 100 % 100 % 25150 25150 VDE-1 11908 13366 14252 14991 17233 1.43E+04 1.30E+03 100 % 100 % 14269 14269 VDE-2 19195 20814 23428 26538 29249 2.37E+04 3.12E+03 100 % 100 % 23718 23718 VDE-3 11634 16449 19682 23142 35650 2.02E+04 5.24E+03 100 % 100 % 20181 20181 DE 24110 24713 25871 26576 29054 2.59E+04 1.38E+03 100 % 100 % 25941 25941 VDE-1 14994 15858 16419 17028 18591 1.65E+04 9.95E+02 100 % 100 % 16521 16521 VDE-2 22239 23328 24396 26084 30038 2.50E+04 2.06E+03 100 % 100 % 24966 24966 VDE-3 13265 14127 15799 16918 20600 1.58E+04 1.86E+03 100 % 100 % 15784 15784 DE 20830 23359 25762 27061 32193 2.57E+04 2.97E+03 100 % 100 % 25695 25695 VDE-1 13214 21535 24948 31530 - 1.19E+05 1.91E+05 80 % 80 % 148598 148598 VDE-2 21596 23967 27646 34331 - 1.22E+05 1.89E+05 80 % 80 % 151964 151964 VDE-3 19707 25724 37291 44755 - 9.05E+04 1.52E+05 88 % 88 % 102892 102892
DE 17053 - - - - 4.81E+05 9.46E+04 4 % 4 % 12017053 12017053
-DE 4864 5059 5273 5722 - 6.46E+04 1.61E+05 88 % 88 % 73431 73431
VDE-1 4286 5202 5488 5781 6193 5.47E+03 4.67E+02 100 % 100 % 5471 5471 VDE-2 4358 4761 5023 5352 6363 5.13E+03 4.65E+02 100 % 100 % 5131 5131 VDE-3 4433 4819 5036 5393 - 2.49E+04 9.70E+04 96 % 96 % 25987 25987
DE - - - - - - - 0 % 0 % -
-VDE-1 - - - - - - - 0 % 0 % -
-VDE-2 - - - - - - - 0 % 0 % -
-VDE-3 - - - - - - - 0 % 0 % -
-DE 166607 241520 281317 - - 3.47E+05 1.28E+05 0 % 60 % - 577880
VDE-1 81502 114160 146990 309156 - 2.36E+05 1.57E+05 8 % 76 % 2943979 309892
VDE-2 163091 190560 324064 - - 3.49E+05 1.46E+05 0 % 56 % - 623373
VDE-3 91127 168894 - - - 3.54E+05 1.65E+05 4 % 48 % 8851540 737628
DE 246805 279548 304142 320004 357217 3.02E+05 2.90E+04 0 % 100 % - 301774 VDE-1 72749 76204 86604 96289 - 1.19E+05 1.13E+05 84 % 92 % 141862 129526 VDE-2 233437 270650 290990 311889 - 3.07E+05 6.31E+04 0 % 96 % - 319799 VDE-3 37570 56162 63623 72037 - 9.81E+04 1.19E+05 88 % 92 % 111446 106601
DE - - - - - - - 0 % 0 % -
25th Mea n Std Successrate Success Performance 13th 19th
f1
Func Strategy 1s t 7th
Table 16. FES to reach the specified accuracy and success rate and performance for
VDE-2 165024 192324 238663 - - 2.95E+05 1.32E+05 0 % 72 % - 409138
VDE-3 304732 444091 494302 - - 4.55E+05 6.00E+04 0 % 60 % - 758861
DE - - - - - - - 0 % 0 % -
Func Strategy 1s t 7th 13th 19th
The ranking of the algorithms is sorted according to the results from the functions that were reliably solved by an algorithm (extended success performance >50%). 10 functions were solved by the algorithms ( , , , , , , , , and ). Table 17 shows how many times the algorithms ranked first, second, third, and fourth on the basis of their extended success performance (5.0e5), in addition to the average ranking.
On average VDE-3 is the best but very even with VDE-1. VDE-3 is the third best and DE performs worst. Noteworthy are the good performances of DE for , VDE-1 for
, VDE-2 for and VDE-3 for and . In section 5.3 these functions are examined more in detail.
Table 17. Ranking based on success performance of the algorithms for solved 10D functions. Table illustrates how many times the algorithm ranked 1st/2nd/3rd/4th.
Strategy 1st 2nd 3rd 4th Avg. Rank
DE 1 1 3 5 3.2
VDE-1 3 5 1 1 2
VDE-2 2 0 5 3 2.9
VDE-3 4 4 1 1 1.9
Results for 30-dimensional functions 5.2.
Table 18. Error values achieved for - (30D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 1.51E+041.39E+041.36E+041.03E+045.28E+044.76E+044.99E+045.17E+043.84E+082.80E+083.03E+085.45E+087.09E+046.04E+045.53E+046.11E+047th1.91E+041.66E+041.98E+041.63E+046.83E+046.46E+047.21E+045.87E+045.07E+084.55E+084.56E+081.09E+099.40E+048.43E+048.12E+049.14E+0413th2.10E+042.12E+042.13E+041.96E+047.58E+048.30E+048.68E+046.89E+046.12E+086.39E+087.00E+081.36E+091.00E+059.13E+049.11E+041.05E+0519th2.33E+042.32E+042.37E+042.36E+048.43E+049.14E+049.22E+047.72E+047.69E+087.35E+088.08E+081.46E+091.13E+051.05E+051.05E+051.21E+0525th2.66E+043.22E+043.24E+043.59E+041.07E+059.71E+041.12E+058.21E+049.25E+081.05E+091.02E+091.73E+091.31E+051.31E+051.27E+051.54E+05 Mean2.10E+042.02E+042.18E+042.05E+047.68E+047.82E+048.33E+046.75E+046.30E+086.12E+086.51E+081.28E+091.02E+059.33E+049.36E+041.07E+05Std2.95E+034.06E+034.51E+036.36E+031.34E+041.44E+041.54E+049.41E+031.68E+082.08E+082.02E+082.81E+081.56E+041.60E+041.71E+042.26E+041st 2.61E+001.73E-031.32E+001.37E-031.08E+042.35E+035.31E+032.30E+034.62E+075.13E+068.02E+062.39E+081.78E+047.34E+031.14E+043.53E+047th4.13E+001.86E-022.55E+004.12E-011.60E+046.37E+031.03E+044.00E+036.95E+071.34E+071.55E+073.10E+083.28E+041.45E+041.69E+044.66E+0413th5.26E+003.97E-023.12E+001.28E+002.09E+041.05E+041.45E+046.21E+038.88E+071.93E+072.13E+073.63E+083.69E+041.72E+042.43E+045.80E+0419th5.79E+002.55E-014.46E+003.77E+002.85E+041.30E+042.12E+047.16E+031.15E+083.89E+076.42E+074.33E+084.59E+042.73E+043.65E+046.58E+0425th7.15E+008.62E+016.87E+008.23E+003.84E+043.37E+044.84E+042.18E+042.06E+089.25E+072.73E+086.63E+085.12E+044.50E+045.81E+048.00E+04
Mean5.05E+005.92E+003.45E+002.22E+002.22E+041.06E+041.78E+046.45E+039.67E+073.03E+075.50E+073.81E+083.74E+042.06E+042.82E+045.71E+04Std1.05E+001.81E+011.32E+002.29E+007.95E+036.45E+031.18E+043.86E+033.75E+072.41E+076.15E+071.06E+088.79E+039.05E+031.28E+041.84E+011st 0.00E+000.00E+000.00E+000.00E+001.22E-013.30E-063.83E-036.21E-033.65E+051.80E+051.32E+053.59E+061.57E+012.72E+001.67E+021.66E+027th0.00E+000.00E+000.00E+000.00E+009.22E-018.39E-052.14E-022.15E-027.38E+053.03E+053.46E+058.67E+061.59E+021.87E+012.53E+022.59E+0213th0.00E+000.00E+000.00E+000.00E+002.10E+002.56E-045.09E-023.34E-029.58E+054.34E+054.37E+059.96E+062.47E+024.95E+014.02E+021.24E+0419th0.00E+000.00E+000.00E+000.00E+002.66E+001.18E-031.04E-019.19E-021.72E+066.13E+056.13E+051.68E+075.17E+021.19E+026.54E+027.21E+0225th0.00E+000.00E+000.00E+000.00E+007.50E+003.44E-027.02E+006.43E-012.61E+061.22E+069.54E+051.12E+081.85E+031.08E+033.78E+032.99E+04 Mean0.00E+000.00E+000.00E+000.00E+002.16E+002.18E-034.49E-011.13E-011.19E+064.77E+054.81E+052.06E+074.26E+021.52E+026.68E+023.10E+03Std0.00E+000.00E+000.00E+000.00E+001.73E+006.73E-031.41E+001.74E-016.11E+052.35E+052.22E+052.63E+074.16E+022.56E+027.95E+027.56E+031st 0.00E+000.00E+000.00E+000.00E+007.82E-110.00E+000.00E+000.00E+007.03E+044.25E+043.07E+031.63E+033.19E-048.75E-061.74E-021.60E-047th0.00E+000.00E+000.00E+000.00E+006.07E-100.00E+000.00E+000.00E+001.71E+056.81E+045.76E+041.13E+045.87E-022.11E-041.33E-014.65E-0313th0.00E+000.00E+000.00E+000.00E+001.93E-090.00E+000.00E+000.00E+002.14E+051.02E+056.86E+042.55E+041.77E-012.50E-032.53E+001.46E-0219th0.00E+000.00E+000.00E+000.00E+009.90E-090.00E+000.00E+001.99E-122.76E+051.23E+059.98E+044.92E+043.64E-012.44E-021.23E+018.82E-0225th0.00E+000.00E+000.00E+000.00E+007.98E-080.00E+003.71E-104.27E-098.08E+051.90E+051.44E+057.07E+041.20E+013.85E-018.07E+023.21E+01
Mean0.00E+000.00E+000.00E+000.00E+001.07E-080.00E+001.55E-111.75E-102.51E+051.01E+057.45E+042.89E+041.27E+003.38E-024.36E+011.47E+00Std0.00E+000.00E+000.00E+000.00E+002.03E-080.00E+007.26E-118.36E-101.50E+053.83E+043.44E+042.05E+042.82E+007.98E-021.58E+026.29E+001st 0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+003.69E+041.59E+041.30E+033.46E+001.01E-071.38E-102.05E-051.10E-087th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+007.15E+042.99E+042.02E+045.59E+011.60E-055.18E-099.45E-049.68E-0813th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+009.90E+044.18E+043.02E+041.07E+026.20E-059.35E-083.92E-032.07E-0619th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+001.53E+055.27E+043.42E+042.95E+022.61E-048.13E-065.44E-021.73E-0525th0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+003.95E+051.21E+054.57E+042.82E+033.30E-021.26E-033.54E+021.02E-03 Mean0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+001.20E+054.72E+042.84E+043.08E+022.05E-037.14E-051.44E+015.80E-05Std0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+000.00E+007.43E+042.50E+041.17E+045.54E+026.75E-032.53E-046.93E+011.99E-04
5E+0
5 5 3E+0 5 1E+0
30D f1
1E+0 4
f2f4f3
1E+0 3
Table 19. Error values achieved for - (30D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 2.00E+042.76E+041.85E+042.98E+043.54E+091.34E+106.80E+091.08E+104.85E+035.11E+035.00E+035.20E+032.11E+012.10E+012.11E+012.11E+017th2.44E+043.02E+042.56E+043.26E+041.08E+102.46E+101.02E+101.78E+105.71E+036.40E+036.57E+036.07E+032.12E+012.12E+012.12E+012.11E+0113th2.73E+043.11E+042.78E+043.46E+041.36E+102.65E+101.47E+102.41E+107.03E+037.10E+037.30E+036.35E+032.12E+012.12E+012.12E+012.12E+0119th2.84E+043.18E+042.90E+043.62E+042.04E+103.36E+101.68E+102.67E+107.92E+037.50E+037.75E+037.12E+032.13E+012.13E+012.13E+012.12E+0125th3.24E+043.74E+043.30E+043.92E+043.37E+105.89E+102.34E+104.06E+109.48E+038.72E+031.03E+048.41E+032.13E+012.13E+012.13E+012.13E+01 Mean2.65E+043.13E+042.71E+043.43E+041.61E+102.84E+101.44E+102.40E+106.97E+037.02E+037.22E+036.57E+032.12E+012.12E+012.12E+012.12E+01Std2.97E+032.18E+033.35E+032.40E+037.63E+099.76E+094.40E+097.58E+091.27E+038.59E+021.18E+037.92E+026.25E-026.06E-025.25E-026.07E-021st 5.74E+038.70E+035.39E+031.17E+042.46E+062.15E+071.83E+064.86E+062.02E+034.23E+022.79E+031.07E+022.09E+012.09E+012.10E+012.10E+017th7.42E+039.86E+036.78E+031.39E+047.20E+065.28E+071.15E+071.68E+073.60E+037.72E+023.74E+033.31E+022.11E+012.11E+012.11E+012.11E+0113th8.50E+031.13E+047.16E+031.54E+041.03E+079.87E+072.38E+073.26E+073.92E+031.01E+034.13E+035.57E+022.11E+012.11E+012.11E+012.11E+0119th9.55E+031.26E+048.23E+031.72E+042.13E+071.78E+084.58E+079.34E+074.76E+031.35E+034.61E+031.10E+032.11E+012.11E+012.11E+012.11E+0125th1.20E+041.76E+041.20E+042.09E+041.56E+086.22E+081.45E+081.03E+095.14E+032.94E+035.73E+032.21E+032.12E+012.12E+012.12E+012.12E+01
Mean8.49E+031.17E+047.92E+031.58E+042.24E+071.46E+083.40E+071.31E+084.01E+031.18E+034.14E+037.54E+022.11E+012.11E+012.11E+012.11E+01Std1.53E+032.39E+031.86E+032.51E+033.20E+071.46E+083.35E+072.30E+087.72E+025.99E+026.79E+025.47E+026.10E-026.67E-024.44E-024.68E-021st 1.24E+021.57E+024.31E+023.47E+021.56E+011.97E+012.09E+011.87E+013.59E+014.83E-041.13E+011.74E-072.09E+012.09E+012.09E+012.08E+017th7.37E+023.10E+027.60E+026.25E+022.32E+012.21E+012.79E+012.12E+016.75E+012.92E-036.09E+014.72E-062.10E+012.10E+012.10E+012.09E+0113th1.11E+034.66E+021.25E+038.29E+022.61E+012.29E+012.94E+012.28E+019.69E+014.80E-033.77E+024.06E-052.10E+012.10E+012.10E+012.10E+0119th1.42E+036.74E+021.60E+039.54E+021.13E+022.40E+011.10E+022.44E+011.30E+021.16E-029.50E+021.17E-032.10E+012.10E+012.10E+012.10E+0125th2.19E+031.43E+033.55E+031.45E+032.08E+021.52E+021.56E+039.98E+012.04E+023.57E-021.87E+031.09E-012.11E+012.11E+012.11E+012.11E+01 Mean1.08E+035.82E+021.30E+038.46E+027.44E+013.88E+011.50E+022.99E+011.04E+028.04E-035.97E+028.82E-032.10E+012.10E+012.10E+012.10E+01Std5.16E+023.58E+026.57E+022.89E+026.56E+013.49E+013.27E+022.10E+014.66E+017.96E-036.04E+022.23E-024.08E-025.25E-024.56E-026.23E-021st 2.78E+003.82E-026.61E+013.82E-012.69E-042.18E-074.90E-049.00E-075.61E-010.00E+003.64E-010.00E+002.09E+012.08E+012.09E+012.08E+017th1.24E+022.60E-013.74E+026.70E-012.38E-016.64E-053.99E+002.11E-049.79E-010.00E+001.33E+000.00E+002.09E+012.09E+012.09E+012.09E+0113th2.19E+024.64E-016.50E+021.03E+002.73E+007.98E-047.93E+001.94E-031.01E+000.00E+003.96E+000.00E+002.09E+012.10E+012.10E+012.09E+0119th4.97E+029.29E-011.11E+032.44E+006.49E+003.89E-031.09E+011.21E-021.06E+000.00E+005.19E+010.00E+002.10E+012.10E+012.10E+012.10E+0125th1.45E+037.44E+001.41E+035.48E+001.90E+012.95E+007.33E+017.52E-011.10E+007.40E-033.15E+022.95E-022.10E+012.10E+012.10E+012.10E+01
Mean3.49E+021.16E+006.88E+021.59E+004.85E+001.23E-011.24E+013.89E-029.74E-018.88E-044.88E+014.33E-032.10E+012.10E+012.10E+012.09E+01Std3.36E+021.71E+004.12E+021.29E+005.40E+005.78E-011.83E+011.47E-011.32E-012.40E-037.83E+019.24E-033.60E-025.27E-023.61E-025.49E-021st 2.51E-012.21E-069.72E+002.44E-046.42E-110.00E+005.91E-120.00E+005.09E-040.00E+005.52E-050.00E+002.08E+012.07E+012.08E+012.08E+017th3.05E+012.05E-041.08E+027.86E-048.47E-090.00E+001.96E-060.00E+002.29E-030.00E+001.43E-010.00E+002.09E+012.09E+012.09E+012.09E+0113th8.11E+015.41E-043.34E+021.12E-031.42E-050.00E+005.93E-030.00E+005.23E-030.00E+001.35E+000.00E+002.09E+012.09E+012.10E+012.09E+0119th2.30E+021.47E-037.46E+023.80E-037.32E-050.00E+001.58E-010.00E+001.40E-020.00E+009.42E+000.00E+002.09E+012.10E+012.10E+012.09E+0125th1.38E+031.55E-021.11E+031.83E-023.99E+000.00E+004.01E+000.00E+003.49E-027.40E-034.54E+012.95E-022.10E+012.10E+012.10E+012.10E+01
Mean1.73E+022.23E-034.33E+023.11E-036.38E-010.00E+006.86E-010.00E+009.78E-038.88E-047.49E+004.33E-032.09E+012.09E+012.09E+012.09E+01Std2.73E+023.95E-033.40E+024.14E-031.46E+000.00E+001.45E+000.00E+009.11E-032.40E-031.14E+019.24E-034.11E-025.87E-024.11E-024.73E-02 f6f8f7
1E+0 3 5E+0
5 5 3E+0 5 1E+0
30D f5
1E+0 4
Table 20. Error values achieved for - (30D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 2.84E+023.11E+023.33E+023.18E+024.03E+023.49E+024.05E+023.46E+024.07E+014.23E+013.97E+014.20E+011.29E+069.36E+051.19E+061.30E+067th3.55E+023.65E+023.67E+023.52E+024.55E+024.35E+024.57E+024.02E+024.40E+014.41E+014.47E+014.50E+011.44E+061.56E+061.45E+061.50E+0613th3.71E+023.79E+023.82E+023.68E+024.99E+024.60E+024.83E+024.50E+024.55E+014.50E+014.54E+014.58E+011.61E+061.67E+061.56E+061.69E+0619th3.88E+023.98E+024.01E+023.95E+025.22E+024.91E+025.41E+024.71E+024.65E+014.58E+014.63E+014.72E+011.74E+061.82E+061.66E+061.75E+0625th4.29E+024.20E+024.26E+024.61E+025.60E+025.78E+025.97E+025.57E+024.83E+014.71E+014.87E+014.81E+012.02E+062.14E+061.88E+062.03E+06 Mean3.69E+023.76E+023.82E+023.72E+024.87E+024.61E+024.96E+024.46E+024.52E+014.48E+014.54E+014.58E+011.60E+061.67E+061.55E+061.66E+06Std3.30E+012.71E+012.35E+013.47E+014.32E+015.41E+015.70E+015.01E+011.66E+001.20E+001.82E+001.56E+001.77E+052.22E+051.65E+051.91E+051st 8.79E+018.76E+016.84E+016.45E+012.31E+021.94E+022.31E+021.15E+023.89E+014.07E+013.13E+014.07E+017.55E+056.44E+056.52E+056.21E+057th9.56E+019.57E+018.60E+018.44E+012.64E+022.29E+022.63E+022.36E+024.17E+014.23E+014.16E+014.20E+019.01E+058.71E+059.54E+058.36E+0513th1.02E+021.01E+029.10E+018.86E+012.78E+022.45E+022.77E+022.51E+024.31E+014.29E+014.21E+014.27E+019.81E+059.10E+059.81E+059.03E+0519th1.05E+021.11E+021.03E+021.01E+022.93E+022.56E+022.95E+022.59E+024.38E+014.39E+014.36E+014.35E+011.09E+061.04E+061.04E+069.91E+0525th1.15E+021.19E+021.66E+021.41E+023.32E+022.76E+023.19E+022.80E+024.50E+014.49E+014.53E+014.52E+011.26E+061.35E+061.15E+061.15E+06
Mean1.01E+021.03E+029.83E+019.36E+012.78E+022.42E+022.77E+022.42E+024.26E+014.30E+014.13E+014.27E+019.87E+059.60E+059.81E+059.04E+05Std6.86E+008.98E+002.29E+011.79E+012.13E+011.97E+012.26E+013.12E+011.55E+001.12E+003.70E+001.02E+001.23E+051.68E+051.02E+051.28E+051st 1.78E-080.00E+001.00E-100.00E+002.50E+012.39E+013.38E+012.31E+011.42E+011.10E+011.42E+013.77E+011.38E+051.08E+054.74E+042.11E+037th4.11E-080.00E+002.82E-100.00E+004.77E+013.68E+015.67E+015.87E+013.91E+013.90E+012.01E+014.01E+012.20E+052.29E+052.66E+057.69E+0313th7.93E-084.83E-124.40E-101.14E-126.82E+015.49E+012.05E+021.81E+024.05E+014.07E+013.05E+014.09E+012.64E+053.37E+054.90E+053.62E+0419th1.04E-076.51E-111.23E-091.03E-112.01E+021.92E+022.18E+021.91E+024.12E+014.10E+013.97E+014.12E+013.38E+054.41E+055.78E+053.69E+0525th6.20E-073.06E-087.52E-092.21E-092.28E+022.24E+022.37E+022.03E+024.25E+014.18E+014.21E+014.19E+016.32E+056.08E+056.80E+055.05E+05 Mean1.09E-072.83E-091.25E-091.47E-101.22E+021.13E+021.55E+021.33E+023.90E+013.87E+012.91E+014.06E+012.94E+053.25E+054.31E+051.52E+05Std1.23E-078.05E-091.71E-094.48E-107.98E+018.06E+018.05E+016.92E+015.64E+006.04E+001.03E+019.44E-011.17E+051.37E+051.94E+051.80E+051st 0.00E+000.00E+000.00E+000.00E+002.29E+012.39E+012.69E+012.19E+011.17E+019.48E+006.94E+003.60E+011.42E+031.44E+032.28E+021.90E-027th0.00E+000.00E+000.00E+000.00E+003.08E+013.28E+014.48E+014.08E+012.28E+011.58E+011.84E+013.88E+014.84E+034.36E+034.26E+031.21E+0213th0.00E+000.00E+000.00E+000.00E+004.08E+014.28E+019.09E+016.37E+013.89E+013.85E+012.23E+014.03E+017.50E+038.20E+032.37E+051.19E+0319th0.00E+000.00E+000.00E+000.00E+005.37E+011.69E+022.00E+021.73E+023.99E+014.00E+013.79E+014.09E+011.16E+042.87E+043.79E+053.26E+0325th0.00E+000.00E+000.00E+000.00E+009.89E+011.91E+022.17E+021.91E+024.11E+014.10E+014.07E+014.12E+013.71E+041.79E+055.50E+053.36E+05
Mean0.00E+000.00E+000.00E+000.00E+004.38E+018.19E+011.18E+021.03E+023.23E+013.09E+012.58E+013.96E+019.11E+032.44E+042.12E+053.54E+04Std0.00E+000.00E+000.00E+000.00E+001.77E+016.81E+017.68E+016.75E+011.07E+011.24E+011.10E+011.44E+007.10E+033.79E+041.95E+059.28E+041st 0.00E+000.00E+000.00E+000.00E+002.29E+012.39E+012.69E+012.09E+011.17E+019.48E+006.94E+003.60E+012.92E+012.21E+012.98E+012.22E-097th0.00E+000.00E+000.00E+000.00E+003.08E+012.89E+014.28E+013.65E+012.13E+011.26E+011.84E+013.78E+017.27E+024.95E+021.28E+033.11E+0013th0.00E+000.00E+000.00E+000.00E+004.08E+013.68E+017.50E+015.87E+013.69E+013.81E+012.23E+013.92E+012.24E+031.13E+037.60E+034.31E+0219th0.00E+000.00E+000.00E+000.00E+005.37E+014.28E+011.89E+021.73E+023.90E+013.95E+013.60E+014.02E+013.33E+032.28E+031.88E+052.20E+0325th0.00E+000.00E+000.00E+000.00E+009.89E+011.81E+022.17E+021.91E+024.06E+014.10E+014.04E+014.08E+017.71E+031.03E+044.60E+053.36E+05
Mean0.00E+000.00E+000.00E+000.00E+004.38E+015.02E+011.08E+029.13E+013.02E+012.79E+012.51E+013.89E+012.47E+032.03E+031.18E+051.45E+04Std0.00E+000.00E+000.00E+000.00E+001.77E+014.32E+017.33E+016.71E+011.09E+011.29E+011.04E+011.35E+002.06E+032.38E+031.71E+056.56E+04 f10f12f11
1E+0 3 5E+0
5 5 3E+0 5 1E+0
30D f9
1E+0 4
Table 21. Error values achieved for - (30D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 6.99E+015.48E+016.25E+014.60E+011.38E+011.38E+011.39E+011.38E+018.93E+029.31E+028.36E+028.92E+026.23E+026.11E+026.30E+027.69E+027th1.54E+021.00E+021.63E+029.83E+011.41E+011.41E+011.41E+011.40E+011.03E+031.08E+031.05E+031.12E+037.48E+028.11E+027.64E+028.41E+0213th2.23E+021.60E+022.31E+021.62E+021.42E+011.42E+011.42E+011.42E+011.05E+031.13E+031.08E+031.13E+037.95E+029.17E+028.12E+029.10E+0219th3.34E+022.16E+022.73E+022.25E+021.43E+011.42E+011.42E+011.43E+011.10E+031.15E+031.11E+031.16E+038.40E+029.63E+028.63E+029.54E+0225th5.42E+024.35E+024.16E+023.21E+021.44E+011.44E+011.43E+011.44E+011.21E+031.23E+031.16E+031.22E+039.54E+021.02E+031.02E+031.09E+03 Mean2.47E+021.74E+022.25E+021.72E+021.42E+011.42E+011.42E+011.42E+011.05E+031.11E+031.07E+031.12E+037.93E+028.84E+028.18E+029.10E+02Std1.22E+029.36E+019.16E+017.83E+011.28E-011.40E-011.19E-011.84E-017.26E+018.00E+016.61E+016.50E+017.75E+011.04E+028.53E+017.71E+011st 1.51E+011.65E+011.84E+011.73E+011.35E+011.37E+011.36E+011.34E+015.97E+026.10E+026.08E+025.66E+023.90E+023.61E+024.02E+023.29E+027th2.16E+011.82E+012.19E+011.91E+011.38E+011.38E+011.38E+011.38E+016.73E+026.51E+026.34E+026.37E+024.17E+024.23E+024.38E+023.86E+0213th2.30E+011.93E+012.23E+011.96E+011.39E+011.39E+011.39E+011.38E+016.89E+026.78E+026.80E+026.97E+024.54E+024.43E+024.59E+024.02E+0219th2.43E+012.11E+012.39E+012.03E+011.39E+011.39E+011.40E+011.39E+017.47E+027.47E+027.06E+027.45E+024.84E+024.79E+024.74E+024.45E+0225th2.86E+012.32E+013.61E+012.19E+011.41E+011.41E+011.41E+011.41E+018.54E+028.12E+028.54E+028.10E+025.52E+025.42E+025.36E+025.15E+02
Mean2.27E+011.95E+012.38E+011.95E+011.39E+011.39E+011.39E+011.38E+017.02E+026.96E+026.83E+026.89E+024.53E+024.51E+024.58E+024.13E+02Std2.75E+001.76E+004.37E+001.01E+001.14E-011.16E-011.25E-011.37E-016.46E+015.87E+015.82E+016.69E+014.10E+014.39E+013.54E+014.84E+011st 1.83E+001.93E+001.88E+001.54E+001.31E+011.34E+011.26E+011.29E+015.05E+024.77E+024.85E+024.54E+022.90E+022.24E+022.58E+022.29E+027th2.59E+002.81E+003.17E+002.36E+001.35E+011.34E+011.33E+011.34E+015.12E+024.81E+024.89E+024.70E+023.06E+022.64E+022.74E+022.45E+0213th3.22E+002.96E+004.62E+002.72E+001.36E+011.35E+011.35E+011.35E+015.17E+024.82E+024.94E+024.74E+023.17E+022.70E+022.85E+022.54E+0219th4.50E+003.83E+007.58E+003.92E+001.37E+011.36E+011.36E+011.36E+015.22E+024.84E+025.00E+024.77E+023.24E+022.78E+023.07E+022.60E+0225th7.18E+001.56E+011.72E+011.62E+011.37E+011.38E+011.38E+011.37E+016.35E+024.90E+025.65E+025.66E+023.70E+022.96E+023.19E+022.75E+02 Mean3.59E+003.61E+006.02E+005.14E+001.36E+011.35E+011.35E+011.35E+015.38E+024.83E+024.98E+024.79E+023.17E+022.68E+022.88E+022.53E+02Std1.38E+002.53E+004.25E+004.96E+001.59E-011.24E-012.71E-011.91E-014.61E+012.83E+001.58E+012.34E+011.71E+011.45E+011.91E+011.18E+011st 1.80E+001.92E+001.84E+001.48E+001.29E+011.29E+011.26E+011.18E+014.63E+024.02E+024.53E+024.00E+022.56E+022.10E+022.48E+022.10E+027th2.53E+002.66E+002.89E+002.29E+001.33E+011.32E+011.31E+011.30E+014.78E+024.03E+024.59E+024.00E+022.79E+022.29E+022.58E+022.26E+0213th3.22E+002.91E+003.22E+002.56E+001.34E+011.33E+011.33E+011.32E+014.80E+024.03E+024.62E+024.00E+022.86E+022.39E+022.61E+022.32E+0219th4.50E+003.44E+004.70E+003.92E+001.35E+011.34E+011.35E+011.34E+014.83E+024.05E+024.68E+024.00E+022.89E+022.47E+022.73E+022.37E+0225th7.18E+004.09E+001.50E+011.51E+011.36E+011.37E+011.38E+011.37E+015.92E+024.06E+024.88E+025.00E+023.53E+022.61E+022.89E+022.44E+02
Mean3.54E+002.98E+004.36E+004.73E+001.34E+011.33E+011.32E+011.31E+014.97E+024.04E+024.64E+024.08E+022.85E+022.38E+022.66E+022.31E+02Std1.39E+006.18E-012.76E+004.29E+001.60E-011.67E-012.98E-014.19E-013.99E+011.16E+007.14E+002.71E+011.74E+011.25E+011.24E+017.85E+001st 1.80E+001.91E+001.84E+001.47E+001.26E+011.29E+011.26E+011.11E+013.51E+024.00E+023.98E+024.00E+022.41E+022.04E+022.04E+021.94E+027th2.53E+002.66E+002.89E+002.25E+001.32E+011.31E+011.31E+011.25E+014.47E+024.00E+024.22E+024.00E+022.60E+022.21E+022.49E+022.14E+0213th3.22E+002.91E+003.22E+002.54E+001.33E+011.33E+011.33E+011.30E+014.53E+024.00E+024.31E+024.00E+022.70E+022.29E+022.54E+022.23E+0219th4.50E+003.44E+004.70E+003.23E+001.34E+011.34E+011.34E+011.33E+014.61E+024.00E+024.36E+024.00E+022.81E+022.36E+022.60E+022.27E+0225th7.18E+004.09E+008.46E+001.50E+011.36E+011.35E+011.38E+011.35E+015.73E+024.00E+024.42E+025.00E+023.49E+022.50E+022.79E+022.41E+02
Mean3.54E+002.96E+004.05E+003.69E+001.33E+011.32E+011.32E+011.28E+014.66E+024.00E+024.28E+024.08E+022.72E+022.29E+022.54E+022.20E+02Std1.39E+006.38E-011.88E+003.16E+002.41E-011.63E-012.84E-016.61E-014.90E+015.39E-031.12E+012.71E+012.06E+011.11E+011.51E+011.17E+01 f14f16f15
1E+0 3 5E+0
5 5 3E+0 5 1E+0
30D f13
1E+0 4
Table 22. Error values achieved for - (30D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 7.44E+025.99E+025.39E+027.63E+021.24E+031.17E+031.17E+031.18E+031.20E+031.08E+031.21E+031.17E+031.19E+031.20E+031.20E+031.20E+037th8.39E+028.36E+028.60E+029.55E+021.28E+031.24E+031.27E+031.31E+031.27E+031.26E+031.26E+031.31E+031.25E+031.26E+031.27E+031.29E+0313th8.88E+029.44E+029.24E+021.01E+031.30E+031.27E+031.30E+031.33E+031.28E+031.28E+031.29E+031.32E+031.28E+031.29E+031.29E+031.31E+0319th9.69E+021.02E+039.64E+021.08E+031.33E+031.30E+031.33E+031.34E+031.31E+031.30E+031.31E+031.34E+031.30E+031.30E+031.31E+031.34E+0325th1.06E+031.19E+031.06E+031.15E+031.37E+031.34E+031.36E+031.39E+031.34E+031.33E+031.35E+031.38E+031.36E+031.33E+031.35E+031.35E+03 Mean9.05E+029.21E+028.97E+021.00E+031.30E+031.27E+031.29E+031.31E+031.29E+031.27E+031.29E+031.32E+031.27E+031.28E+031.28E+031.30E+03Std8.84E+011.33E+021.07E+029.31E+013.35E+013.99E+015.10E+014.29E+013.32E+015.04E+013.34E+014.13E+014.15E+013.43E+014.15E+014.30E+011st 4.32E+023.54E+024.16E+024.10E+021.07E+039.95E+021.04E+031.05E+031.04E+039.68E+021.02E+031.06E+031.04E+039.69E+021.04E+031.03E+037th4.86E+023.86E+024.75E+024.68E+021.09E+031.01E+031.08E+031.08E+031.07E+031.01E+031.07E+031.07E+031.09E+031.01E+031.07E+031.06E+0313th5.21E+024.12E+025.27E+024.92E+021.11E+031.03E+031.09E+031.09E+031.10E+031.03E+031.08E+031.09E+031.10E+031.04E+031.09E+031.09E+0319th5.37E+024.22E+025.48E+025.36E+021.12E+031.05E+031.11E+031.12E+031.12E+031.05E+031.10E+031.11E+031.11E+031.07E+031.10E+031.11E+0325th5.89E+025.14E+026.70E+025.95E+021.14E+031.10E+031.13E+031.17E+031.14E+031.13E+031.12E+031.14E+031.14E+031.11E+031.12E+031.15E+03
Mean5.15E+024.14E+025.17E+025.01E+021.11E+031.03E+031.09E+031.10E+031.10E+031.04E+031.09E+031.09E+031.10E+031.04E+031.08E+031.09E+03Std3.84E+013.82E+015.87E+015.46E+011.99E+012.83E+012.51E+013.33E+012.74E+013.74E+012.23E+012.68E+012.01E+013.73E+012.07E+013.02E+011st 3.35E+022.50E+023.02E+022.58E+029.30E+029.07E+029.17E+029.10E+029.29E+029.07E+029.18E+029.11E+029.28E+029.07E+029.18E+029.10E+027th3.46E+022.67E+023.24E+022.75E+029.35E+029.07E+029.20E+029.11E+029.33E+029.07E+029.22E+029.12E+029.33E+029.07E+029.20E+029.12E+0213th3.59E+022.81E+023.35E+022.92E+029.38E+029.08E+029.24E+029.12E+029.37E+029.07E+029.26E+029.14E+029.35E+029.07E+029.24E+029.13E+0219th3.69E+022.89E+023.49E+023.01E+029.45E+029.08E+029.32E+029.14E+029.39E+029.08E+029.35E+029.15E+029.38E+029.08E+029.28E+029.14E+0225th3.89E+023.05E+024.05E+023.15E+029.61E+029.08E+029.48E+029.17E+029.52E+029.08E+029.44E+029.22E+029.48E+029.08E+029.42E+029.25E+02 Mean3.59E+022.78E+023.37E+022.88E+029.40E+029.08E+029.27E+029.13E+029.37E+029.07E+029.27E+029.14E+029.36E+029.07E+029.25E+029.14E+02Std1.60E+011.41E+012.15E+011.59E+017.24E+003.83E-017.67E+001.94E+005.31E+002.44E-017.23E+002.77E+004.65E+002.48E-016.42E+002.87E+001st 2.93E+022.22E+022.72E+022.45E+029.11E+029.06E+029.07E+029.06E+029.11E+029.05E+029.08E+029.06E+029.10E+029.05E+029.08E+029.06E+027th3.04E+022.45E+022.92E+022.53E+029.12E+029.06E+029.08E+029.06E+029.12E+029.06E+029.08E+029.06E+029.12E+029.06E+029.08E+029.06E+0213th3.14E+022.59E+023.08E+022.59E+029.13E+029.06E+029.09E+029.06E+029.13E+029.06E+029.08E+029.06E+029.13E+029.06E+029.09E+029.06E+0219th3.32E+022.63E+023.24E+022.64E+029.13E+029.06E+029.09E+029.06E+029.14E+029.06E+029.10E+029.07E+029.14E+029.06E+029.09E+029.07E+0225th3.41E+022.74E+024.04E+022.80E+029.15E+029.07E+029.12E+029.07E+029.16E+029.06E+029.12E+029.07E+029.16E+029.07E+029.13E+029.07E+02
Mean3.17E+022.54E+023.11E+022.60E+029.13E+029.06E+029.09E+029.06E+029.13E+029.06E+029.09E+029.06E+029.13E+029.06E+029.09E+029.06E+02Std1.48E+011.30E+012.55E+018.07E+001.29E+002.10E-011.24E+002.32E-011.45E+003.28E-011.31E+002.87E-011.42E+002.28E-011.45E+002.59E-011st 2.53E+022.21E+022.67E+022.14E+029.08E+029.03E+029.07E+029.04E+029.08E+029.03E+029.07E+029.04E+029.08E+029.03E+029.07E+029.04E+027th2.93E+022.42E+022.90E+022.40E+029.08E+029.05E+029.07E+029.05E+029.08E+029.04E+029.07E+029.05E+029.08E+029.04E+029.07E+029.05E+0213th3.00E+022.43E+022.96E+022.48E+029.09E+029.06E+029.07E+029.06E+029.08E+029.04E+029.07E+029.06E+029.09E+029.05E+029.07E+029.06E+0219th3.08E+022.53E+023.05E+022.54E+029.09E+029.06E+029.07E+029.06E+029.09E+029.06E+029.07E+029.06E+029.09E+029.06E+029.07E+029.06E+0225th3.34E+022.63E+023.89E+022.66E+029.10E+029.06E+029.08E+029.06E+029.10E+029.06E+029.08E+029.06E+029.09E+029.06E+029.08E+029.07E+02
Mean2.99E+022.44E+022.98E+022.47E+029.08E+029.05E+029.07E+029.06E+029.09E+029.05E+029.07E+029.06E+029.09E+029.05E+029.07E+029.06E+02Std1.63E+011.14E+012.21E+011.18E+015.40E-018.88E-012.19E-017.88E-014.94E-011.07E+003.17E-015.39E-015.06E-011.03E+002.63E-016.46E-01 f18f20f19
1E+0 3 5E+0
5 5 3E+0 5 1E+0
30D f17
1E+0 4
Table 23. Error values achieved for - (30D).
DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-3DEVDE-1VDE-2VDE-31st 1.38E+031.38E+031.38E+031.34E+031.40E+031.37E+031.32E+031.44E+031.35E+031.38E+031.37E+031.34E+031.42E+031.41E+031.42E+031.43E+031.61E+031.60E+031.59E+031.68E+037th1.41E+031.40E+031.43E+031.44E+031.51E+031.52E+031.51E+031.55E+031.40E+031.41E+031.41E+031.41E+031.45E+031.45E+031.45E+031.48E+031.67E+031.68E+031.67E+031.76E+0313th1.43E+031.43E+031.44E+031.46E+031.56E+031.56E+031.57E+031.59E+031.43E+031.43E+031.44E+031.45E+031.48E+031.47E+031.46E+031.51E+031.72E+031.73E+031.70E+031.79E+0319th1.44E+031.45E+031.47E+031.49E+031.60E+031.62E+031.61E+031.66E+031.45E+031.45E+031.46E+031.46E+031.49E+031.48E+031.48E+031.52E+031.78E+031.74E+031.73E+031.82E+0325th1.49E+031.48E+031.49E+031.52E+031.64E+031.74E+031.80E+031.77E+031.51E+031.48E+031.50E+031.52E+031.54E+031.52E+031.52E+031.53E+031.83E+031.83E+031.79E+031.87E+03 Mean1.43E+031.43E+031.45E+031.46E+031.55E+031.56E+031.55E+031.60E+031.43E+031.43E+031.43E+031.45E+031.47E+031.47E+031.47E+031.50E+031.73E+031.72E+031.70E+031.79E+03Std2.55E+012.95E+012.91E+013.93E+016.96E+018.29E+011.12E+028.89E+014.36E+012.63E+013.39E+014.48E+013.07E+012.56E+012.57E+013.06E+016.32E+015.68E+015.29E+014.81E+011st 1.22E+039.94E+021.20E+031.19E+031.13E+031.09E+031.16E+031.16E+031.22E+031.10E+031.18E+031.16E+031.18E+031.01E+031.17E+031.16E+034.57E+023.22E+024.64E+025.09E+027th1.24E+031.13E+031.23E+031.22E+031.25E+031.15E+031.21E+031.21E+031.23E+031.15E+031.22E+031.24E+031.22E+031.10E+031.20E+031.20E+036.36E+024.24E+025.20E+025.50E+0213th1.25E+031.17E+031.23E+031.26E+031.26E+031.17E+031.27E+031.24E+031.23E+031.18E+031.24E+031.27E+031.24E+031.12E+031.21E+031.26E+037.18E+024.77E+026.77E+026.77E+0219th1.26E+031.21E+031.27E+031.26E+031.29E+031.19E+031.29E+031.27E+031.27E+031.22E+031.25E+031.28E+031.26E+031.16E+031.23E+031.27E+038.22E+025.15E+027.36E+027.83E+0225th1.29E+031.25E+031.28E+031.28E+031.36E+031.29E+031.33E+031.35E+031.28E+031.25E+031.28E+031.31E+031.31E+031.19E+031.26E+031.31E+031.19E+036.55E+021.07E+031.04E+03
Mean1.25E+031.16E+031.24E+031.24E+031.26E+031.17E+031.25E+031.24E+031.25E+031.18E+031.24E+031.26E+031.24E+031.12E+031.21E+031.24E+037.41E+024.80E+026.71E+026.90E+02Std1.53E+016.30E+012.32E+012.76E+014.81E+014.09E+014.86E+014.35E+012.18E+014.42E+012.24E+013.48E+012.97E+014.78E+012.48E+014.20E+011.55E+028.73E+011.55E+021.53E+021st 8.21E+025.00E+026.28E+025.11E+029.90E+028.92E+029.78E+029.02E+028.94E+025.67E+027.13E+026.11E+026.19E+022.00E+024.51E+022.19E+022.82E+022.20E+022.67E+022.12E+027th8.63E+025.00E+026.79E+025.21E+021.02E+039.06E+021.02E+039.17E+029.72E+025.89E+027.91E+026.37E+027.14E+022.01E+024.72E+022.55E+023.45E+022.24E+023.18E+022.16E+0213th8.81E+025.01E+027.11E+025.31E+021.04E+039.17E+021.03E+039.22E+029.98E+025.97E+028.17E+026.51E+027.67E+022.01E+025.24E+022.59E+023.96E+022.26E+023.66E+022.18E+0219th9.15E+025.01E+027.62E+025.37E+021.05E+039.24E+021.05E+039.42E+021.00E+036.08E+028.40E+026.60E+027.88E+022.02E+025.91E+022.79E+024.29E+022.30E+023.94E+022.21E+0225th9.51E+025.02E+029.16E+026.06E+021.07E+039.50E+021.09E+039.58E+021.03E+036.37E+029.42E+027.24E+028.30E+022.03E+027.76E+023.87E+024.50E+022.42E+024.69E+022.46E+02 Mean8.88E+025.01E+027.29E+025.36E+021.03E+039.16E+021.03E+039.27E+029.85E+025.97E+028.15E+026.51E+027.55E+022.01E+025.52E+022.69E+023.84E+022.27E+023.61E+022.20E+02Std3.32E+013.70E-016.78E+012.29E+012.04E+011.30E+012.27E+011.49E+013.17E+011.57E+014.80E+012.23E+015.04E+015.76E-019.25E+013.36E+014.78E+015.40E+005.46E+016.98E+001st 5.41E+025.00E+025.08E+025.00E+029.44E+028.75E+029.29E+028.70E+026.46E+025.34E+025.50E+025.34E+022.60E+022.00E+022.14E+022.00E+022.39E+022.10E+022.25E+022.09E+027th5.56E+025.00E+025.10E+025.00E+029.56E+028.80E+029.52E+028.78E+026.80E+025.34E+025.60E+025.49E+022.84E+022.00E+022.20E+022.00E+022.58E+022.10E+022.30E+022.09E+0213th5.72E+025.00E+025.13E+025.00E+029.64E+028.82E+029.62E+028.81E+027.12E+025.34E+025.66E+025.60E+022.97E+022.00E+022.26E+022.00E+022.78E+022.11E+022.38E+022.09E+0219th5.86E+025.00E+025.17E+025.00E+029.78E+028.84E+029.68E+028.85E+027.39E+025.34E+025.74E+025.61E+023.35E+022.00E+022.36E+022.00E+022.93E+022.11E+022.95E+022.09E+0225th6.22E+025.00E+025.33E+025.00E+029.91E+028.98E+029.81E+028.88E+027.76E+025.47E+026.19E+025.73E+024.14E+022.00E+022.73E+022.00E+023.26E+022.12E+023.66E+022.09E+02
Mean5.75E+025.00E+025.15E+025.00E+029.66E+028.82E+029.59E+028.81E+027.12E+025.35E+025.69E+025.56E+023.11E+022.00E+022.29E+022.00E+022.78E+022.11E+022.59E+022.09E+02Std2.36E+012.47E-085.82E+007.28E-031.17E+015.50E+001.35E+014.84E+003.66E+012.51E+001.50E+019.91E+003.39E+014.93E-081.26E+018.47E-032.33E+016.25E-014.26E+015.42E-021st 5.05E+025.00E+025.00E+025.00E+029.09E+028.67E+029.05E+028.60E+025.34E+025.34E+025.34E+025.34E+022.05E+022.00E+022.01E+022.00E+022.25E+022.09E+022.13E+022.09E+027th5.07E+025.00E+025.01E+025.00E+029.32E+028.72E+029.19E+028.72E+025.50E+025.34E+025.34E+025.34E+022.10E+022.00E+022.01E+022.00E+022.30E+022.09E+022.16E+022.09E+0213th5.10E+025.00E+025.01E+025.00E+029.39E+028.76E+029.24E+028.75E+025.70E+025.34E+025.34E+025.34E+022.14E+022.00E+022.02E+022.00E+022.34E+022.09E+022.19E+022.09E+0219th5.14E+025.00E+025.01E+025.00E+029.46E+028.79E+029.37E+028.77E+025.88E+025.34E+025.34E+025.34E+022.18E+022.00E+022.02E+022.00E+022.38E+022.09E+022.20E+022.09E+0225th5.23E+025.00E+025.02E+025.00E+029.57E+028.84E+029.59E+028.79E+026.13E+025.34E+025.34E+025.34E+022.43E+022.00E+022.04E+022.00E+022.78E+022.10E+023.18E+022.09E+02
Mean5.11E+025.00E+025.01E+025.00E+029.38E+028.76E+029.28E+028.73E+025.69E+025.34E+025.34E+025.34E+022.16E+022.00E+022.02E+022.00E+022.37E+022.09E+022.31E+022.09E+02Std4.86E+001.14E-134.00E-011.70E-061.19E+014.83E+001.24E+015.00E+002.23E+014.55E-134.22E-138.59E-051.01E+011.19E-128.11E-011.83E-061.17E+011.74E-013.00E+014.50E-02 f22f24f25f23
1E+0 3 5E+0
5 5 3E+0 5 1E+0
30D f21
1E+0 4
Table 24. FES to reach the specified accuracy and success rate and performance for
(30D).
3.0e5 5.0e5 3.0e5 5.0e5
DE 26472 27452 27616 28154 28714 2.78E+04 4.99E+02 100 % 100 % 27781 27781 VDE-1 13955 16192 17311 20032 35161 1.92E+04 4.91E+03 100 % 100 % 19236 19236 VDE-2 24739 25529 26405 26773 30767 2.64E+04 1.19E+03 100 % 100 % 26424 26424 VDE-3 15786 18601 22398 25107 29367 2.22E+04 3.98E+03 100 % 100 % 22243 22243 DE 216416 229360 233489 251649 269462 2.40E+05 1.48E+04 100 % 100 % 239684 239684 VDE-1 108084 123154 134039 137428 157964 1.32E+05 1.16E+04 100 % 100 % 132137 132137 VDE-2 148451 172322 176471 190957 218857 1.81E+05 1.60E+04 100 % 100 % 181486 181486 VDE-3 157207 172891 181850 192061 227741 1.84E+05 1.77E+04 100 % 100 % 184253 184253
DE - - - - - - - 0 % 0 % -
-VDE-1 - - - - - - - 0 % 0 % -
-VDE-2 - - - - - - - 0 % 0 % -
-VDE-3 - - - - - - - 0 % 0 % -
-DE 457973 - - - - 4.98E+05 8.24E+03 0 % 4 % - 12457973
VDE-1 333315 403615 449553 - - 4.47E+05 5.15E+04 0 % 68 % - 657309
VDE-2 - - - - - - - 0 % 0 % -
-DE 273385 348740 398937 438005 - 3.94E+05 6.96E+04 12 % 84 % 3280646 468663 VDE-1 229497 272059 285691 293913 353038 2.83E+05 2.36E+04 84 % 100 % 336829 282936 VDE-2 258379 407828 491068 - - 4.41E+05 7.09E+04 4 % 52 % 11035796 848907 VDE-3 244551 275131 290412 301025 330673 2.88E+05 1.78E+04 72 % 100 % 399946 287961
DE 433097 466096 487504 - - 4.80E+05 2.10E+04 0 % 56 % - 857736
VDE-1 83739 92824 96234 100762 113870 9.69E+04 6.40E+03 100 % 100 % 96861 96861
VDE-2 385601 - - - - 4.80E+05 3.65E+04 0 % 24 % - 2001147
VDE-3 61926 71044 76162 93306 - 1.61E+05 1.70E+05 80 % 80 % 201871 201871
DE - - - - - - - 0 % 0 % -
-VDE-1 - - - - - - - 0 % 0 % -
-VDE-2 - - - - - - - 0 % 0 % -
-VDE-3 - - - - - - - 0 % 0 % -
-DE 61591 64253 65296 66444 70831 6.54E+04 2.09E+03 100 % 100 % 65397 65397 VDE-1 54100 58425 60787 63971 76712 6.14E+04 4.88E+03 100 % 100 % 61377 61377 VDE-2 46969 50719 51694 53679 62044 5.23E+04 3.29E+03 100 % 100 % 52285 52285 VDE-3 43652 45961 49230 51330 64702 4.94E+04 4.21E+03 100 % 100 % 49358 49358
DE - - - - - - - 0 % 0 % -
25th Mea n Std Successrate Success Performance 13th 19th
f1
Func Strategy 1s t 7th
Table 25. FES to reach the specified accuracy and success rate and performance for
Func Strategy 1s t 7th 13th 19th
In 30D only 6 functions were reliably solved by the algorithms ( , , , , , and ).
Table 26 shows how the algorithms ranked. In 10D, VDE-1 and VDE-3 rank very close but in 30D VDE-1 performs better having the best performance for 5/6 functions.
Table 26. Ranking based on success performance of the algorithms for solved 30D functions. Table illustrates how many times the algorithm ranked 1st/2nd/3rd/4th.
In order to see the overall performance of the algorithms, Table 27 combines the 10D and 30D results. The table clearly shows that VDE-1 was the overall best with VDE-3 a close second. VDE-3 ranked third while DE performed the worst.
Table 27. Ranking based on success performance of the algorithms for all solved functions (10D and 30D). Table illustrates how many times the algorithm ranked 1st/2nd/3rd/4th.
Convergence and value (FEMA, CREMA, c) distribution graphs 5.3.
In this section more detailed figures and graphs are provided for the convergence properties of all algorithms for selected functions. The semi-log convergence graphs show the error value achieved against the number of function evaluations (FES).
Convergence graphs include the error values of the best, median and the worst run for each algorithm. The runs are sorted by the number of function evaluations to reach the specified accuracy or the error value at MAXFES if the specified accuracy is not
Strategy 1st 2nd 3rd 4th Avg. Rank
DE 0 0 3 3 3.5
VDE-1 5 0 1 0 1.33
VDE-2 0 2 1 3 3.17
VDE-3 1 4 1 0 2
Strategy 1st 2nd 3rd 4th Avg. Rank
DE 1 1 6 8 3.31
VDE-1 8 5 2 1 1.75
VDE-2 2 2 6 6 3.00
VDE-3 5 8 2 1 1.94
reached. Because the functions show great variation, the scale of FES shown in the figures has been adjusted accordingly to present the data in a suitable scale. The data in the graphs has been gathered from every 100th function evaluations.
In addition to the convergence graphs, figures of the control parameter behavior for the best, median and the worst run are provided for the VDE-algorithms. VDE-1 graphs show the value of for each run and VDE-2 graphs show the value of . Also provided is the maximum, mean and the minimum value of and during the optimization. The max, mean and min values have been gathered until the median run reaches an accuracy of 1.0E-08 or the maximum function evaluation (MAXFES) value.
For VDE-3, the graphs show both the and values in addition to the value of the variance factor c which is calculated by these EMA values. Also the max, mean and min value of c during the optimization is provided for VDE-3. For other VDE algorithms the value of c is not provided in the graphs since its behavior is linear with the control parameter which is being adapted.
The functions have been selected according to the achieved results. The performance of one algorithm has been exceedingly better, or the function in question is otherwise interesting. The selected functions are (10D), (30D), (10D), (10D), (30D),
(30D), (30D), (10D), (10D) and (10D).
First to be analyzed is which is a unimodal and non-separable function. A 3D landscape illustration of the 2D version of the function is shown in Figure 6.
Figure 6. :Shifted Schwefel’s Problem 1.2. 3-D map for 2-D function. (Suganthan et al. 2005)
Convergence of the algorithms for is illustrated in Figure 7. Each algorithm converges reasonably steady towards the global optimum. The best run is achieved by VDE-1 and the worst by VDE-2. Overall, VDE-2 was the best. In VDE-3 it is interesting to see that the worst run actually has the best error value in the early part of the optimization but its convergence slows down later.
Figure 7. Convergence graphs for (10D).
As a reminder, the scale of the following graphs for the VDE-algorithms is the same as in the convergence graphs so that the reader can compare them as need be. The EMA median and in the worst run the value stays relatively near the starting point of 0.9 and falls later to the ≈0.7 values. Distribution graph also shows the same behaviour and the
values are steadily falling towards the lower bound of the value range which is
≈0.58 with the settings used for the test.
1E-08
0 10000 20000 30000 40000 50000
Errorvalue
FES
Convergence DE Worst runMedian run Best run
0 10000 20000 30000 40000 50000
Errorvalue
FES
Convergence VDE-1 Worst runMedian run Best run
0 10000 20000 30000 40000 50000
Errorvalue
FES
Convergence VDE-2 Worst runMedian run Best run
0 10000 20000 30000 40000 50000
Errorvalue
FES
Convergence VDE-3 Worst runMedian run Best run
Figure 8. VDE-1: values for best, median and worst runs, and max, mean and min value during optimization for (10D).
For VDE-2 (Figure 9) the most interesting run is the worst run where the value stays below 0.9 for the most part of the optimization while the best and median runs values are above 0.9 and also result in better convergence speed compared to DE. Noteworthy is also the fact that the minimum value of does not fall near the lower bound of the value range which was ≈0.62 during the optimization.
Figure 9. VDE-2: values for best, median and worst runs, and max, mean and min value during optimization for (10D).
For VDE-3 (Figure 10) the parameter values of the best, median and the worst are quite identical. falls to range of 0.6-0.8 and varies in range 0.8-1.0. Convergence
0 10000 20000 30000 40000 50000 FEMA
FES
VDE-1: Value of FEMA Worst runMedian run
Best run
0 10000 20000 30000 40000 50000 FEMA
FES
VDE-1: Distribution of FEMA maxmean
min
0 10000 20000 30000 40000 50000 CREMA
FES
VDE-2: Value of CREMA Worst runMedian run
Best run
0 10000 20000 30000 40000 50000 CREMA
FES
VDE-2: Distribution of CREMA
max mean min
parameter values tend to be nearer to the lower bound than the upper bound. Overall, the behavior of the control parameters is individually similar in VDE-1 and VDE-2.
Figure 10. VDE-3: , and c values for best, median and worst runs, and c max, mean and min value during optimization for (10D).
Next are shown the convergence graphs for the 30D version of . Again VDE-1 has clearly the best convergence speed compared to the other algorithms. But also VDE-2 is converging faster and its performance is much better than that of DE. VDE-2 shows
0 10000 20000 30000 40000 50000
Value
0 10000 20000 30000 40000 50000
Value
0 10000 20000 30000 40000 50000
Value
0 10000 20000 30000 40000 50000
Value
Figure 11. Convergence graphs for (30D).
Values of for VDE-1 are again near 0.7, as in 10D, and consistent with each run.
The high fluctuation of the value is explained by the fact that the population is small (NP=20) and the new F values are generated at each generation.
Figure 12. VDE-1: values for best, median and worst runs, and max, mean and min value during optimization for (30D).
1E-08
Convergence DE Worst runMedian run Best run
Convergence VDE-1 Worst runMedian run Best run
Convergence VDE-2 Worst runMedian run Best run
Convergence VDE-3 Worst runMedian run Best run
VDE-1: Distribution of FEMA maxmean min
For VDE-2 the behaviour of is very consistent for each run. In the best run the value rises to ≈0.95 the fastest and in the worst run the slowest. The distribution graph shows that for every run the value of has stabilized >0.9. VDE-2 performed better than DE in this function so a minor change in the CR can be quite effective.
Maximum CR value with variance factor limit 1.6 and NP=20 is ≈0.99.
Figure 13. VDE-2: values for best, median and worst runs, and max, mean and min value during optimization for (30D).
While the performance of VDE-1 and VDE-3 was alike in 10D, in 30D VDE-3 no longer performs that well. In all the runs the control parameter values are close together.
is mostly >0.9 and <0.8. VDE-3 was set up with a lower variance factor boundary, 1.2, than VDE-1, 1.25, and this allows lower values which seem to slow down convergence. The c distribution graph shows that the average value of the variance factor c was a little greater than 1.3. Again the values fluctuate greatly due to the small NP which is set to 20.
VDE-2: Distribution of CREMA
max mean min
Figure 14. VDE-3: , and c values for best, median and worst runs, and c max, mean and min value during optimization for (30D).
Now for the graphs of (10D) (Figure 15) which is also a unimodal non-separable function. The difference between the other unimodal 10D functions in the benchmark was that was run with a larger population: NP=50. The convergence graphs in Figure 16 show great differences in convergence speeds. VDE-3 solves the function in
Figure 15. : Shifted Rotated High Conditioned Elliptic Function. 3-D map for 2-D function. (Suganthan et al. 2005)
Figure 16. Convergence graphs for (10D).
1E-08
Convergence DE Worst runMedian run Best run
Convergence VDE-1 Worst runMedian run Best run
Convergence VDE-2 Worst runMedian run Best run
Convergence VDE-3 Worst runMedian run Best run
As in the previous function, VDE-1 values in Figure x again fall very fast to <0.7 with the exception of the worst run in which the value is higher until ≈50000 FES. In the convergence graph it is also shown that the convergence was a bit slower until that point. The larger NP size allows a minimum value of ≈0.57 for F and the value distribution is slightly lower compared to the values for .
Figure 17. VDE-1: values for best, median and worst runs, and max, mean and min value during optimization for (10D).
VDE-2 values again are consistently >0.9 which provide better performance compared to DE. The allowed maximum CR value with variance factor limit 1.6 and NP=50 is ≈0.98.
Figure 18. VDE-2: values for best, median and worst runs, and max, mean and min value during optimization for (10D).
0.5
VDE-1: Value of FEMA Worst runMedian run
Best run
VDE-1: Distribution of FEMA maxmean
min
VDE-2: Distribution of CREMA maxmean min
The reason for the good VDE-3 performance seems to be that lower F values are allowed. values again fall rapidly near 0.6 while values stay in range 0.9-1.0. Compared to , lower F values are more effective for and confirm the fact that the optimal control parameter values vary with the function in question. High CR values are better at least for the unimodal functions.
Figure 19. VDE-3: , and c values for best, median and worst runs, and c max, mean and min value during optimization for (10D).
Overall, in unimodal non-separable functions in VDE-1 and VDE-3 values of F systematically approached the lower boundary of the allowed value range and thus speeded the convergence. In VDE-2 and VDE-3 the CR values, in contrast, approached the higher boundary of the value range. Next some of the multimodal functions are
was used for this function which caused some failed runs. DE had 100% success rate in 10D for and it was the only function in which DE had the best performance. For the VDE-algorithms a larger population size fixed the success rate in the expense of speed but NP was kept low for the actual test firstly for comparison and secondly in order to analyze what might be the problem that causes the premature convergence or stagnation.
Figure 20. : Shifted Rosenbrock’s Function. 3-D map for 2-D function. (Suganthan et al. 2005)
The convergence graphs (Figure 21) show that VDE-1 and VDE-3 converge much faster in the beginning and this seems to cause premature convergence.
Figure 21. Convergence graphs for (10D).
For VDE-1 the behavior of is again similar with the unimodal functions and the average approaches value of ≈0.7. There is no major difference with the best, median and the worst runs. The high variance in the worst run indicates that the population is clustering in the local optimum. As the selection operation selects a trial vector of an equal error value to the next generation, the current F value updates the value for the majority of the function evaluations.
Figure 22. VDE-1: values for best, median and worst runs, and max, mean and min value during optimization for (10D).
1E-08
0 20000 40000 60000 80000 100000
Errorvalue
FES
Convergence DE Worst runMedian run Best run
0 20000 40000 60000 80000 100000
Errorvalue
FES
Convergence VDE-1 Worst runMedian run Best run
0 20000 40000 60000 80000 100000
Errorvalue
FES
Convergence VDE-2 Worst runMedian run Best run
0 20000 40000 60000 80000 100000
Errorvalue
FES
Convergence VDE-3 Worst runMedian run Best run
0 20000 40000 60000 80000 100000 FEMA
FES
VDE-1: Value of FEMA Worst runMedian run
Best run
0 20000 40000 60000 80000 100000 FEMA
FES
VDE-1: Distribution of FEMA maxmean
min
values for VDE-2 had more variance compared to the unimodal functions. In the beginning successful trial vectors are found very quickly and the some of the CR values are approaching the lower bound. After the algorithm has found a local optimum (≈10000 FES), the values start to rise again and average values are above 0.9. The performance of the best and median run was very even with DE.
Figure 23. VDE-2: values for best, median and worst runs, and max, mean and min value during optimization for (10D).
VDE-3 had the most unstable convergence of the algorithms. VDE-3 has very fast
VDE-3 had the most unstable convergence of the algorithms. VDE-3 has very fast