Introduction
Bayesian optimization
Gaussian processes
Acquisition functions
Personalized evolutionary Bayesian optimization
Personalized Gaussian processes
Optimization of acquisition functions
Experimental studies
Test problems
\(\sigma _\varepsilon ^{2}=0\) | PBO-EA | BO | PBO-SQR | PBO-LCB | PBO-AEI | PBO-PEI | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Problem | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |||||
Ackley | 15.4 | 0.33 | 16.1 | \(\approx \) | 0.44 | 14.6 | \(\approx \) | 6.73 | 15.9 | \(\approx \) | 0.41 | 16.1 | \(\approx \) | 0.29 | 16.1 | \(\approx \) | 0.29 |
8.37 | 0.19 | 8.43 | \(\approx \) | 0.24 | 7.10 | \(\approx \) | 2.30 | 8.21 | \(\approx \) | 0.36 | 8.43 | \(\approx \) | 0.29 | 8.43 | \(\approx \) | 0.29 | |
13.1 | 0.41 | 13.3 | \(\approx \) | 0.39 | 13.0 | \(\approx \) | 1.01 | 13.1 | \(\approx \) | 0.31 | 13.4 | \(\approx \) | 0.30 | 13.4 | \(\approx \) | 0.21 | |
6.11 | 0.12 | 5.97 | \(\approx \) | 0.15 | 5.73 | \(\approx \) | 0.65 | 6.25 | \(\approx \) | 0.24 | 6.19 | \(\approx \) | 0.34 | 6.19 | \(\approx \) | 0.22 | |
12.2 | 0.22 | 12.2 | \(\approx \) | 0.23 | 12.4 | \(\approx \) | 0.23 | 12.2 | \(\approx \) | 0.23 | 12.2 | \(\approx \) | 0.35 | 12.2 | \(\approx \) | 0.34 | |
11.9 | 0.36 | 11.5 | \(\approx \) | 0.35 | 12.1 | \(\approx \) | 0.27 | 12.2 | \(\approx \) | 0.35 | 11.6 | \(\approx \) | 0.21 | 12.2 | \(\approx \) | 0.21 | |
9.37 | 0.28 | 9.34 | \(\approx \) | 0.23 | 8.78 | \(\approx \) | 0.64 | 9.55 | \(\approx \) | 0.21 | 9.54 | \(\approx \) | 0.29 | 9.43 | \(\approx \) | 0.20 | |
16.1 | 0.21 | 16.7 | \(\approx \) | 0.33 | 16.8 | \(\approx \) | 0.41 | 16.6 | \(\approx \) | 0.31 | 16.6 | \(\approx \) | 0.25 | 16.3 | \(\approx \) | 0.41 | |
3.67 | 0.23 | 3.90 | \(\approx \) | 0.38 | 3.94 | \(\approx \) | 0.41 | 3.90 | \(\approx \) | 0.22 | 4.20 | – | 0.16 | 3.91 | \(\approx \) | 0.23 | |
13.9 | 0.37 | 14.3 | \(\approx \) | 0.33 | 14.1 | \(\approx \) | 0.21 | 14.2 | \(\approx \) | 0.13 | 14.2 | \(\approx \) | 0.25 | 14.3 | \(\approx \) | 0.3 | |
Rosenbrock | 0.82 | 0.19 | 6.23 | + | 8.5 | 299 | + | 279 | 3.6e3 | + | 321 | 1.29 | \(\approx \) | 0.47 | 2.89 | + | 0.41 |
0.36 | 0.21 | 2.19 | + | 2.2 | 89.6 | + | 68.9 | 3.9e3 | + | 4.09 | 0.67 | + | 0.28 | 2.17 | + | 0.5 | |
0.25 | 0.11 | 4.31 | + | 12.9 | 71.0 | + | 63.6 | 3.8e3 | + | 4.23 | 0.39 | + | 0.21 | 1.41 | + | 0.39 | |
0.15 | 0.13 | 4.39 | \(\approx \) | 10.8 | 30.1 | + | 12.1 | 3.7e3 | + | 3.23 | 0.21 | \(\approx \) | 0.23 | 1.07 | + | 0.49 | |
0.09 | 0.12 | 2.98 | \(\approx \) | 8.93 | 20.9 | + | 15.3 | 3.9e3 | + | 4.06 | 0.14 | \(\approx \) | 0.17 | 0.56 | + | 0.23 | |
0.07 | 0.11 | 3.09 | \(\approx \) | 4.81 | 19.7 | + | 17.3 | 3.8e3 | + | 3.89 | 0.08 | \(\approx \) | 0.10 | 0.41 | + | 0.41 | |
0.04 | 0.04 | 48.3 | \(\approx \) | 101 | 18.1 | + | 16.4 | 3.8e3 | + | 4.01 | 0.04 | \(\approx \) | 0.05 | 0.12 | \(\approx \) | 0.10 | |
0.10 | 0.05 | 45.6 | \(\approx \) | 129 | 31.5 | + | 28.2 | 3.8e3 | + | 4.30 | 0.09 | \(\approx \) | 0.07 | 0.06 | \(\approx \) | 0.19 | |
0.10 | 0.04 | 945 | + | 1.4e3 | 68.4 | + | 43.1 | 3.8e3 | + | 3.91 | 0.12 | \(\approx \) | 0.06 | 0.17 | \(\approx \) | 0.19 | |
0.17 | 0.05 | 947 | + | 1.5e3 | 209 | + | 57.9 | 3.9e3 | + | 3.39 | 0.39 | \(\approx \) | 0.32 | 0.21 | \(\approx \) | 0.11 | |
Branin | 0.69 | 0.21 | 3.09 | + | 3.08 | 29.7 | + | 21.5 | 97 | + | 38.1 | 0.72 | \(\approx \) | 0.36 | 1.41 | \(\approx \) | 0.91 |
1.61 | 2.29 | 8.09 | + | 5.87 | 19.1 | + | 6.01 | 89.0 | + | 64.6 | 1.00 | \(\approx \) | 0.73 | 4.25 | + | 3.71 | |
1.76 | 3.84 | 12.9 | + | 5.41 | 19.2 | + | 3.99 | 50.9 | + | 57.4 | 4.09 | + | 4.87 | 10.8 | + | 6.70 | |
1.71 | 2.32 | 7.07 | + | 6.41 | 15.2 | + | 4.43 | 16.1 | + | 14.3 | 6.87 | + | 7.00 | 11.2 | + | 4.81 | |
0.87 | 0.43 | 3.28 | \(\approx \) | 4.39 | 21.2 | + | 20.0 | 19.8 | + | 12.1 | 2.46 | + | 2.47 | 3.78 | + | 2.49 | |
0.89 | 0.35 | 13.2 | + | 22.1 | 82.2 | + | 50.1 | 21.1 | + | 22.3 | 0.86 | \(\approx \) | 0.27 | 1.15 | \(\approx \) | 0.91 | |
0.79 | 0.23 | 14.3 | + | 14.7 | 57.1 | + | 44.1 | 24.6 | + | 29.1 | 1.49 | + | 0.38 | 4.01 | + | 4.43 | |
0.68 | 0.36 | 34.6 | + | 19.6 | 103 | + | 69.9 | 56.1 | + | 44.2 | 2.98 | + | 4.37 | 12.2 | + | 11.2 | |
3.03 | 4.56 | 13.8 | + | 14.1 | 34.5 | + | 38.1 | 32.2 | + | 21.2 | 5.02 | \(\approx \) | 7.93 | 11.9 | + | 4.76 | |
27.7 | 11.4 | 64.7 | + | 65.0 | 69.7 | + | 49.6 | 45.1 | \(\approx \) | 44.7 | 14.7 | – | 11.7 | 45.7 | \(\approx \) | 36.3 | |
Harmann3 | \(-\)4.21 | 0.21 | \(-\)3.21 | + | 0.70 | \(-\)1.72 | + | 0.63 | \(-\)2.31 | + | 0.87 | \(-\)4.01 | \(\approx \) | 0.14 | \(-\)3.78 | \(\approx \) | 0.31 |
\(-\)4.19 | 0.31 | \(-\)3.38 | + | 0.81 | \(-\)2.21 | + | 0.89 | \(-\)2.31 | + | 1.04 | \(-\)4.25 | \(\approx \) | 0.19 | \(-\)4.13 | \(\approx \) | 0.21 | |
\(-\)3.64 | 0.24 | \(-\)3.00 | + | 0.67 | \(-\)1.41 | + | 0.80 | \(-\)2.41 | + | 0.65 | \(-\)3.44 | \(\approx \) | 0.34 | \(-\)3.34 | \(\approx \) | 0.41 | |
\(-\)3.79 | 0.13 | \(-\)2.95 | + | 1.1 | \(-\)2.41 | + | 0.84 | \(-\)1.91 | + | 1.02 | \(-\)3.78 | \(\approx \) | 0.17 | \(-\)3.65 | + | 0.21 | |
\(-\)3.47 | 0.19 | \(-\)2.86 | \(\approx \) | 1.91 | \(-\)1.82 | + | 0.72 | \(-\)1.54 | + | 0.94 | \(-\)3.52 | \(\approx \) | 0.14 | \(-\)3.52 | \(\approx \) | 0.22 | |
\(-\)4.78 | 0.28 | \(-\)4.54 | \(\approx \) | 0.74 | \(-\)3.36 | + | 1.01 | \(-\)1.75 | + | 0.94 | \(-\)4.92 | \(\approx \) | 0.14 | \(-\)4.79 | \(\approx \) | 0.37 | |
\(-\)3.78 | 0.23 | \(-\)3.26 | + | 0.67 | \(-\)2.21 | + | 0.36 | \(-\)1.79 | + | 1.22 | \(-\)3.84 | \(\approx \) | 0.21 | \(-\)3.57 | \(\approx \) | 0.38 | |
\(-\)4.54 | 0.40 | \(-\)3.51 | + | 0.96 | \(-\)2.84 | + | 0.67 | \(-\)1.44 | + | 1.28 | \(-\)4.55 | \(\approx \) | 0.30 | \(-\)4.51 | \(\approx \) | 0.23 | |
\(-\)3.92 | 0.16 | \(-\)2.93 | + | 0.65 | \(-\)2.16 | + | 0.92 | \(-\)0.25 | + | 0.11 | \(-\)3.85 | \(\approx \) | 0.21 | \(-\)3.84 | \(\approx \) | 0.24 | |
+/\(\approx \)/– | 20/20/0 | 30/9/1 | 29/11/0 | 5/32/3 | 14/26/0 |
\(\sigma _\varepsilon ^{2}\)=0.05 | PBO-EA | BO | PBO-SQR | PBO-LCB | PBO-AEI | PBO-PEI | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Problem | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |||||
Ackley | 16.5 | 0.25 | 16.4 | \(\approx \) | 0.35 | 15.0 | \(\approx \) | 3.83 | 16.4 | \(\approx \) | 0.41 | 16.5 | \(\approx \) | 0.43 | 16.5 | \(\approx \) | 0.29 |
8.37 | 0.26 | 8.30 | \(\approx \) | 0.29 | 8.38 | \(\approx \) | 0.66 | 8.42 | \(\approx \) | 0.28 | 8.45 | \(\approx \) | 0.33 | 8.52 | \(\approx \) | 0.21 | |
13.2 | 0.24 | 13.2 | \(\approx \) | 0.32 | 12.4 | – | 0.95 | 13.3 | \(\approx \) | 0.38 | 13.2 | \(\approx \) | 0.39 | 13.6 | + | 0.29 | |
6.14 | 0.26 | 6.14 | \(\approx \) | 0.23 | 6.16 | \(\approx \) | 0.21 | 5.89 | \(\approx \) | 0.16 | 6.26 | \(\approx \) | 0.14 | 6.24 | \(\approx \) | 0.18 | |
12.3 | 0.32 | 12.3 | \(\approx \) | 0.22 | 12.6 | \(\approx \) | 0.31 | 12.3 | \(\approx \) | 0.26 | 12.4 | \(\approx \) | 0.18 | 12.3 | \(\approx \) | 0.38 | |
12.6 | 0.32 | 12.4 | \(\approx \) | 0.29 | 12.6 | \(\approx \) | 0.33 | 12.5 | \(\approx \) | 0.25 | 12.6 | \(\approx \) | 0.18 | 12.6 | \(\approx \) | 0.13 | |
9.41 | 0.27 | 9.52 | \(\approx \) | 0.22 | 9.30 | \(\approx \) | 0.54 | 9.45 | \(\approx \) | 0.12 | 9.32 | \(\approx \) | 0.32 | 9.58 | + | 0.23 | |
16.7 | 0.37 | 16.7 | \(\approx \) | 0.22 | 16.5 | \(\approx \) | 0.61 | 16.3 | – | 0.32 | 16.6 | \(\approx \) | 0.27 | 16.5 | \(\approx \) | 0.29 | |
3.98 | 0.18 | 3.93 | \(\approx \) | 0.16 | 3.90 | \(\approx \) | 0.45 | 4.03 | \(\approx \) | 0.16 | 3.99 | \(\approx \) | 0.21 | 4.02 | \(\approx \) | 0.23 | |
14.4 | 0.35 | 14.3 | \(\approx \) | 0.36 | 4.65 | \(\approx \) | 0.22 | 14.1 | \(\approx \) | 0.22 | 14.3 | \(\approx \) | 0.22 | 14.2 | \(\approx \) | 0.33 | |
Rosenbrock | 1.06 | 0.27 | 1.65 | \(\approx \) | 1.78 | 356 | + | 476 | 3.4e3 | + | 1.0e3 | 1.75 | + | 0.29 | 3.17 | + | 0.41 |
0.43 | 0.17 | 3.35 | + | 9.50 | 53.4 | + | 67.6 | 3.5e3 | + | 787 | 0.91 | + | 0.21 | 2.06 | + | 0.35 | |
0.24 | 0.14 | 0.61 | \(\approx \) | 2.05 | 99.5 | + | 123 | 3.6e3 | + | 687 | 0.47 | + | 0.12 | 1.54 | + | 0.32 | |
0.14 | 0.11 | 0.53 | + | 1.42 | 42.8 | + | 47.1 | 3.5e3 | + | 9.1e2 | 0.26 | + | 0.08 | 1.04 | + | 0.24 | |
0.09 | 0.06 | 0.08 | \(\approx \) | 0.14 | 20.2 | + | 9.31 | 3.8e3 | + | 4.01 | 0.11 | \(\approx \) | 0.06 | 0.68 | + | 0.26 | |
0.05 | 0.05 | 0.64 | + | 1.83 | 29.4 | + | 40.7 | 3.9e3 | + | 4.10 | 0.06 | \(\approx \) | 0.08 | 0.36 | + | 0.16 | |
0.06 | 0.08 | 0.62 | \(\approx \) | 1.06 | 68.1 | + | 113 | 3.8e3 | + | 4.01 | 0.03 | \(\approx \) | 0.05 | 0.12 | \(\approx \) | 0.04 | |
0.06 | 0.05 | 1.69 | \(\approx \) | 1.86 | 89.9 | + | 173 | 3.9e3 | + | 4.01 | 0.08 | \(\approx \) | 0.05 | 0.06 | \(\approx \) | 0.07 | |
0.20 | 0.19 | 6.23 | + | 10.3 | 250 | + | 198 | 3.8e3 | + | 4.11 | 0.15 | \(\approx \) | 0.04 | 0.19 | \(\approx \) | 0.17 | |
0.24 | 0.21 | 7.74 | + | 14.3 | 411 | + | 191 | 3.8e3 | + | 4.02 | 0.25 | \(\approx \) | 0.11 | 0.76 | \(\approx \) | 1.23 | |
Branin | 0.61 | 0.21 | 1.26 | \(\approx \) | 0.82 | 23.6 | + | 12.3 | 8.40 | + | 4.32 | 0.88 | \(\approx \) | 0.51 | 1.78 | + | 1.22 |
0.63 | 0.22 | 7.08 | + | 5.71 | 21.2 | + | 6.39 | 4.74 | + | 5.11 | 3.21 | \(\approx \) | 3.26 | 2.98 | + | 4.23 | |
2.88 | 4.46 | 12.1 | + | 6.04 | 14.9 | + | 4.91 | 36.4 | + | 22.8 | 8.09 | + | 6.75 | 6.01 | + | 4.41 | |
3.37 | 4.68 | 6.19 | \(\approx \) | 6.33 | 15.6 | + | 10.1 | 63.6 | + | 43.6 | 9.72 | + | 6.39 | 13.4 | + | 5.40 | |
2.81 | 4.20 | 4.24 | \(\approx \) | 4.98 | 41.0 | + | 26.5 | 85.4 | + | 93.4 | 4.38 | + | 3.37 | 4.25 | + | 2.31 | |
1.20 | 0.81 | 9.00 | \(\approx \) | 13.1 | 49.0 | + | 44.0 | 175 | + | 117 | 0.87 | \(\approx \) | 0.25 | 1.56 | \(\approx \) | 1.58 | |
0.95 | 0.38 | 9.01 | + | 7.04 | 49.5 | + | 33.1 | 66.3 | + | 55.7 | 2.21 | + | 1.29 | 2.70 | \(\approx \) | 5.21 | |
1.09 | 1.35 | 29.6 | + | 17.5 | 159 | + | 91.8 | 247 | + | 125 | 9.47 | \(\approx \) | 9.86 | 7.34 | + | 9.42 | |
3.08 | 4.63 | 7.44 | + | 5.55 | 84 | + | 79.5 | 51.0 | + | 61.6 | 13.3 | + | 12.8 | 13.4 | + | 6.51 | |
28.2 | 9.70 | 46.5 | + | 33.1 | 151 | + | 149 | 168 | + | 159 | 40.7 | \(\approx \) | 28.3 | 48.6 | \(\approx \) | 27.6 | |
Hartmann3 | \(-\)3.86 | 0.35 | \(-\)3.42 | + | 0.35 | \(-\)2.15 | + | 0.90 | \(-\)2.49 | + | 0.84 | \(-\)4.01 | \(\approx \) | 0.17 | \(-\)3.93 | \(\approx \) | 0.35 |
\(-\)4.15 | 0.30 | \(-\)3.60 | + | 0.25 | \(-\)2.09 | + | 0.64 | \(-\)2.28 | + | 1.14 | \(-\)4.13 | \(\approx \) | 0.28 | \(-\)4.16 | \(\approx \) | 0.22 | |
\(-\)3.69 | 0.09 | \(-\)3.12 | \(\approx \) | 0.41 | \(-\)2.02 | + | 0.90 | \(-\)2.07 | + | 0.83 | \(-\)3.47 | \(\approx \) | 0.38 | \(-\)3.25 | + | 0.50 | |
\(-\)3.88 | 0.12 | \(-\)3.15 | \(\approx \) | 0.52 | \(-\)1.95 | + | 1.16 | \(-\)2.12 | + | 0.92 | \(-\)3.73 | \(\approx \) | 0.24 | \(-\)3.81 | \(\approx \) | 0.17 | |
\(-\)3.61 | 0.21 | \(-\)2.77 | + | 0.93 | \(-\)2.12 | + | 0.85 | \(-\)1.87 | + | 1.04 | \(-\)3.61 | \(\approx \) | 0.17 | \(-\)3.49 | \(\approx \) | 0.33 | |
\(-\)4.80 | 0.23 | \(-\)3.64 | + | 1.24 | \(-\)2.65 | + | 0.97 | \(-\)2.01 | + | 1.31 | \(-\)5.10 | – | 0.12 | \(-\)4.87 | \(\approx \) | 0.18 | |
\(-\)3.86 | 0.41 | \(-\)3.10 | \(\approx \) | 1.12 | \(-\)2.59 | + | 0.73 | \(-\)1.83 | + | 1.06 | \(-\)3.86 | \(\approx \) | 0.27 | \(-\)3.75 | \(\approx \) | 0.29 | |
\(-\)4.63 | 0.23 | \(-\)3.35 | + | 1.21 | \(-\)2.87 | + | 0.86 | \(-\)1.31 | + | 1.31 | \(-\)4.48 | \(\approx \) | 0.31 | \(-\)4.23 | \(\approx \) | 0.42 | |
\(-\)3.84 | 0.18 | \(-\)3.18 | \(\approx \) | 1.21 | \(-\)2.41 | + | 0.84 | \(-\)0.28 | + | 0.06 | \(-\)3.84 | \(\approx \) | 0.25 | \(-\)3.79 | \(\approx \) | 0.18 | |
+/\(\approx \)/– | 16/24/0 | 30/9/1 | 30/9/1 | 11/28/1 | 16/24/0 |
\(\sigma _\varepsilon ^{2}\)=0.1 | PBO-EA | BO | PBO-SQR | PBO-LCB | PBO-AEI | PBO-PEI | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Problem | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |||||
Ackley | 16.4 | 0.31 | 13.6 | \(\approx \) | 4.65 | 15.2 | \(\approx \) | 0.45 | 16.3 | \(\approx \) | 0.33 | 15.4 | \(\approx \) | 0.34 | 14.0 | \(\approx \) | 1.54 |
8.23 | 0.28 | 8.19 | \(\approx \) | 0.33 | 8.56 | \(\approx \) | 0.30 | 8.63 | \(\approx \) | 0.24 | 8.64 | \(\approx \) | 0.37 | 7.13 | \(\approx \) | 1.58 | |
13.2 | 0.25 | 13.3 | \(\approx \) | 0.74 | 13.2 | \(\approx \) | 0.16 | 13.3 | \(\approx \) | 0.30 | 13.3 | \(\approx \) | 0.45 | 12.1 | \(\approx \) | 14.7 | |
6.27 | 0.24 | 6.01 | \(\approx \) | 0.33 | 6.13 | \(\approx \) | 0.14 | 6.12 | \(\approx \) | 0.21 | 6.25 | \(\approx \) | 0.19 | 6.03 | \(\approx \) | 16.5 | |
12.1 | 0.23 | 12.2 | \(\approx \) | 0.31 | 12.4 | \(\approx \) | 0.28 | 11.7 | \(\approx \) | 0.32 | 12.6 | \(\approx \) | 0.28 | 26.7 | + | 6.41 | |
12.1 | 0.28 | 12.6 | + | 0.28 | 12.4 | \(\approx \) | 0.30 | 12.4 | \(\approx \) | 0.29 | 12.4 | \(\approx \) | 0.33 | 17.9 | + | 9.13 | |
9.28 | 0.23 | 9.10 | \(\approx \) | 0.51 | 9.48 | \(\approx \) | 0.08 | 9.55 | \(\approx \) | 0.20 | 9.60 | \(\approx \) | 0.15 | 12.4 | + | 7.67 | |
16.4 | 0.34 | 16.6 | \(\approx \) | 0.43 | 16.5 | \(\approx \) | 0.26 | 16.6 | \(\approx \) | 0.36 | 16.3 | \(\approx \) | 0.28 | 22.4 | + | 10.4 | |
4.01 | 0.19 | 3.83 | \(\approx \) | 0.67 | 4.07 | \(\approx \) | 0.24 | 4.01 | \(\approx \) | 0.14 | 4.14 | \(\approx \) | 0.19 | 6.63 | + | 1.26 | |
14.5 | 0.34 | 14.3 | \(\approx \) | 0.62 | 14.5 | \(\approx \) | 0.18 | 14.1 | \(\approx \) | 0.16 | 14.2 | \(\approx \) | 0.19 | 14.0 | \(\approx \) | 89.9 | |
Rosenbrock | 1.03 | 0.41 | 390 | + | 469 | 2.2e3 | + | 1.2e3 | 2.14 | + | 0.41 | 3.39 | + | 0.70 | 4.25 | + | 3.71 |
0.44 | 0.23 | 95.4 | + | 102 | 1.8e3 | + | 1.7e3 | 1.28 | + | 0.39 | 2.21 | + | 0.37 | 11.8 | + | 6.72 | |
0.24 | 0.15 | 89.7 | + | 61.6 | 1.6e3 | + | 1.9e3 | 0.72 | + | 0.19 | 1.47 | + | 0.29 | 11.5 | + | 4.83 | |
0.14 | 0.10 | 23.2 | + | 23.2 | 1.3e3 | + | 1.7e3 | 0.36 | + | 0.14 | 0.99 | + | 0.23 | 3.36 | + | 2.49 | |
0.08 | 0.06 | 40.3 | + | 61.6 | 1.3e3 | + | 1.7e3 | 0.12 | \(\approx \) | 0.09 | 0.81 | + | 0.38 | 1.15 | \(\approx \) | 0.97 | |
0.11 | 0.08 | 10.0 | + | 14.1 | 1.2e3 | + | 1.7e3 | 0.07 | \(\approx \) | 0.08 | 0.42 | + | 0.20 | 4.25 | + | 4.95 | |
0.05 | 0.05 | 7.01 | + | 7.22 | 1.4e3 | + | 1.7e3 | 0.03 | \(\approx \) | 0.03 | 0.14 | + | 0.11 | 12.7 | + | 13.6 | |
0.07 | 0.04 | 83.8 | + | 112 | 1.3e3 | + | 1.7e3 | 0.09 | \(\approx \) | 0.05 | 0.10 | \(\approx \) | 0.14 | 13.1 | + | 5.05 | |
0.16 | 0.09 | 381 | + | 227 | 1.3e3 | + | 1.7e3 | 0.22 | \(\approx \) | 0.15 | 0.14 | \(\approx \) | 0.12 | 47.9 | + | 37.8 | |
0.28 | 0.13 | 547 | + | 218 | 1.3e3 | + | 914 | 0.39 | \(\approx \) | 0.29 | 0.79 | + | 0.71 | 1.90 | + | 0.37 | |
Branin | 2.23 | 3.23 | 24 | + | 19.71 | 7.98 | + | 4.46 | 1.11 | \(\approx \) | 0.53 | 1.64 | \(\approx \) | 1.33 | 1.40 | \(\approx \) | 0.9 |
1.87 | 2.60 | 15.4 | + | 5.50 | 16.5 | + | 17.7 | 3.07 | \(\approx \) | 3.90 | 6.10 | + | 4.02 | 4.25 | + | 3.7 | |
0.80 | 0.46 | 17.0 | + | 7.68 | 35.3 | + | 29.4 | 7.63 | + | 6.63 | 12.2 | + | 5.19 | 11.8 | + | 6.77 | |
2.89 | 5.23 | 18.5 | + | 16.9 | 81.8 | + | 68.1 | 9.15 | + | 7.20 | 13.3 | + | 5.14 | 11.6 | + | 4.88 | |
2.16 | 4.79 | 20.6 | + | 19.0 | 64.4 | + | 33.6 | 2.33 | + | 1.17 | 2.84 | + | 1.21 | 3.36 | + | 2.49 | |
1.07 | 0.48 | 100 | + | 54.5 | 167 | + | 72.7 | 1.05 | \(\approx \) | 0.92 | 1.21 | \(\approx \) | 0.71 | 1.15 | \(\approx \) | 0.98 | |
1.92 | 2.69 | 41.8 | + | 29.9 | 50.9 | + | 45.6 | 4.19 | + | 6.08 | 3.60 | + | 2.56 | 4.25 | + | 4.97 | |
6.43 | 14.5 | 104 | + | 87.4 | 245 | + | 97.1 | 8.42 | + | 10.4 | 14.8 | + | 11.0 | 11.7 | + | 11.3 | |
3.78 | 4.56 | 68.2 | + | 83.1 | 65.8 | + | 71.8 | 9.08 | + | 7.71 | 13.9 | + | 8.74 | 12.7 | + | 5.01 | |
32.1 | 14.5 | 169 | + | 124 | 157 | + | 98.9 | 32.6 | \(\approx \) | 15.1 | 47.1 | + | 18.2 | 47.1 | \(\approx \) | 37.1 | |
Hartmann3 | \(-\)3.89 | 0.24 | \(-\)2.89 | \(\approx \) | 1.01 | \(-\)2.78 | + | 0.78 | \(-\)3.76 | \(\approx \) | 0.28 | \(-\)3.83 | \(\approx \) | 0.37 | \(-\)3.91 | \(\approx \) | 0.33 |
\(-\)4.23 | 0.28 | \(-\)2.23 | \(\approx \) | 1.01 | \(-\)1.84 | + | 0.81 | \(-\)3.76 | \(\approx \) | 0.33 | \(-\)3.71 | \(\approx \) | 0.48 | \(-\)4.19 | \(\approx \) | 0.24 | |
\(-\)3.48 | 0.32 | \(-\)1.28 | + | 0.43 | \(-\)1.98 | + | 1.08 | \(-\)3.48 | \(\approx \) | 0.33 | \(-\)3.49 | \(\approx \) | 0.12 | \(-\)3.29 | \(\approx \) | 0.37 | |
\(-\)3.86 | 0.18 | \(-\)1.69 | + | 0.89 | \(-\)2.15 | + | 1.09 | \(-\)3.74 | \(\approx \) | 0.24 | \(-\)3.79 | \(\approx \) | 0.39 | \(-\)3.70 | + | 0.29 | |
\(-\)3.47 | 0.31 | \(-\)1.53 | + | 0.45 | \(-\)1.49 | + | 0.87 | \(-\)3.59 | \(\approx \) | 0.20 | \(-\)3.37 | \(\approx \) | 0.29 | \(-\)3.56 | \(\approx \) | 0.25 | |
\(-\)4.78 | 0.33 | \(-\)3.24 | + | 0.53 | \(-\)2.21 | + | 1.34 | \(-\)4.65 | \(\approx \) | 0.54 | \(-\)4.89 | \(\approx \) | 0.43 | \(-\)4.80 | \(\approx \) | 0.38 | |
\(-\)3.82 | 0.23 | \(-\)2.61 | \(\approx \) | 0.80 | \(-\)1.56 | + | 1.13 | \(-\)3.82 | \(\approx \) | 0.33 | \(-\)3.85 | \(\approx \) | 0.21 | \(-\)3.71 | \(\approx \) | 0.39 | |
\(-\)4.59 | 0.15 | \(-\)2.89 | \(\approx \) | 0.74 | \(-\)1.75 | + | 1.39 | \(-\)4.43 | \(\approx \) | 0.46 | \(-\)4.61 | \(\approx \) | 0.16 | \(-\)4.60 | \(\approx \) | 0.26 | |
\(-\)3.82 | 0.26 | \(-\)2.12 | \(\approx \) | 0.75 | \(-\)0.27 | + | 0.13 | \(-\)3.82 | \(\approx \) | 0.24 | \(-\)3.66 | \(\approx \) | 0.26 | \(-\)3.84 | \(\approx \) | 0.28 | |
1/0/–1 | 27/13/0 | 30/10/0 | 10/30/0 | 16/24/0 | 22/18/0 |
\(\sigma _\varepsilon ^{2}\)=0.5 | PBO-EA | BO | PBO-SQR | PBO-LCB | PBO-AEI | PBO-PEI | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Problem | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |||||
Ackley | 16.6 | 0.31 | 16.6 | \(\approx \) | 0.53 | 10.4 | – | 6.50 | 16.7 | \(\approx \) | 0.36 | 16.7 | \(\approx \) | 0.30 | 16.5 | \(\approx \) | 0.32 |
8.35 | 0.23 | 8.54 | \(\approx \) | 0.19 | 7.37 | \(\approx \) | 2.87 | 8.54 | \(\approx \) | 0.22 | 8.54 | + | 0.23 | 8.63 | \(\approx \) | 0.23 | |
13.2 | 0.28 | 13.1 | \(\approx \) | 0.22 | 12.6 | \(\approx \) | 1.21 | 13.5 | \(\approx \) | 0.21 | 13.4 | \(\approx \) | 0.34 | 13.3 | \(\approx \) | 0.33 | |
6.28 | 0.19 | 6.29 | \(\approx \) | 0.18 | 6.11 | \(\approx \) | 0.33 | 6.17 | \(\approx \) | 0.23 | 6.33 | \(\approx \) | 0.17 | 6.27 | \(\approx \) | 0.19 | |
12.1 | 0.23 | 12.4 | \(\approx \) | 0.22 | 12.6 | \(\approx \) | 0.23 | 12.5 | \(\approx \) | 0.25 | 12.6 | \(\approx \) | 0.20 | 12.5 | \(\approx \) | 0.23 | |
12.5 | 0.21 | 12.3 | \(\approx \) | 0.22 | 12.6 | \(\approx \) | 0.41 | 12.5 | \(\approx \) | 0.32 | 12.7 | \(\approx \) | 0.30 | 14.3 | + | 0.23 | |
9.43 | 0.23 | 9.62 | \(\approx \) | 0.24 | 9.53 | \(\approx \) | 0.59 | 9.58 | \(\approx \) | 0.22 | 9.51 | \(\approx \) | 0.23 | 9.63 | \(\approx \) | 0.17 | |
16.2 | 0.21 | 16.9 | + | 0.22 | 16.3 | \(\approx \) | 0.61 | 16.4 | \(\approx \) | 0.42 | 16.6 | \(\approx \) | 0.26 | 16.2 | \(\approx \) | 0.32 | |
3.87 | 0.28 | 3.88 | \(\approx \) | 0.27 | 4.08 | \(\approx \) | 0.58 | 4.02 | + | 0.18 | 4.01 | \(\approx \) | 0.13 | 4.02 | \(\approx \) | 0.21 | |
14.6 | 0.21 | 14.6 | \(\approx \) | 0.19 | 14.6 | \(\approx \) | 0.50 | 14.6 | \(\approx \) | 0.23 | 14.5 | \(\approx \) | 0.27 | 14.5 | \(\approx \) | 0.23 | |
Rosenbrock | 56.9 | 303 | 5.31 | – | 6.42 | 88.2 | \(\approx \) | 134 | 1.2e3 | + | 642 | 55.2 | \(\approx \) | 103 | 62.1 | \(\approx \) | 124 |
9.54 | 16.9 | 9.39 | \(\approx \) | 9.47 | 363 | \(\approx \) | 894 | 946 | + | 986 | 32.0 | + | 48.1 | 141 | + | 179 | |
16.0 | 116 | 12.6 | \(\approx \) | 26.7 | 214 | + | 431 | 343 | + | 483 | 15.5 | \(\approx \) | 26.7 | 10.7 | \(\approx \) | 10.6 | |
4.27 | 73.2 | 3.81 | \(\approx \) | 3.56 | 13.4 | + | 13.2 | 212 | + | 315 | 46.1 | + | 112 | 6.24 | + | 10.0 | |
11.5 | 30.7 | 3.73 | – | 4.30 | 651 | + | 999 | 8.77 | – | 7.02 | 9.32 | \(\approx \) | 16.5 | 18.4 | + | 26.7 | |
21.7 | 169 | 23.1 | \(\approx \) | 56.4 | 997 | + | 998 | 63.0 | + | 95.8 | 45.6 | + | 72.7 | 74.1 | + | 121 | |
2.72 | 2.65 | 12.1 | + | 16.2 | 63.2 | + | 121 | 587 | + | 767 | 123 | + | 203 | 7.43 | + | 19.1 | |
23.9 | 531 | 30.0 | \(\approx \) | 67.2 | 9.19 | – | 10.1 | 812 | + | 756 | 54.6 | \(\approx \) | 145 | 95.2 | + | 234 | |
20.1 | 19.9 | 12.4 | \(\approx \) | 24.5 | 18.5 | \(\approx \) | 27.6 | 2.3e3 | + | 1.2e3 | 51.2 | + | 98.9 | 298 | + | 389 | |
256 | 387 | 305 | + | 876 | 201 | \(\approx \) | 564 | 2.0e3 | + | 1.0e3 | 795 | + | 1.1e3 | 402 | + | 813 | |
Branin | 7.21 | 5.56 | 32.3 | + | 53.1 | 59.6 | + | 67.3 | 34.2 | + | 37.2 | 5.10 | \(\approx \) | 4.34 | 11.1 | + | 10.5 |
12.1 | 24.2 | 11.8 | \(\approx \) | 12.5 | 39.9 | + | 57.1 | 58.4 | + | 63.4 | 23.6 | + | 20.4 | 21.7 | \(\approx \) | 20.4 | |
22.6 | 21.0 | 10.9 | – | 14.4 | 80.3 | + | 116 | 87.7 | + | 67.8 | 32.5 | + | 18.5 | 27.7 | + | 19.6 | |
22.1 | 23.2 | 5.87 | – | 3.87 | 33.8 | + | 19.0 | 141 | + | 97.8 | 36.7 | \(\approx \) | 38.00 | 17.5 | \(\approx \) | 12.2 | |
21.4 | 20.9 | 15.4 | \(\approx \) | 18.9 | 25.2 | \(\approx \) | 17.4 | 153 | + | 96.9 | 20.0 | \(\approx \) | 36.8 | 27.3 | \(\approx \) | 48.4 | |
29.8 | 28.4 | 26.3 | \(\approx \) | 32.3 | 87.0 | + | 130 | 235 | + | 164 | 15.6 | – | 24.1 | 56.9 | + | 90.2 | |
8.91 | 9.00 | 24.8 | + | 41.7 | 54.3 | + | 55.8 | 75.4 | + | 62.0 | 35.1 | + | 53.7 | 32.9 | + | 57.3 | |
12.6 | 10.2 | 8.79 | \(\approx -\) | 8.28 | 105 | + | 141 | 214 | + | 118 | 73.5 | + | 81.2 | 32.0 | + | 23.0 | |
35.3 | 29.1 | 19.2 | – | 24.0 | 55.6 | + | 55.2 | 104 | + | 72.92 | 16.9 | – | 16.0 | 32.1 | \(\approx \) | 59.1 | |
38.6 | 31.1 | 16.3 | – | 19.4 | 19.6 | – | 13.1 | 119 | + | 44.1 | 65.0 | + | 50.4 | 45.0 | + | 28.6 | |
Harmann3 | \(-\)3.19 | 0.67 | \(-\)1.53 | + | 0.96 | \(-\)1.97 | + | 0.55 | \(-\)2.11 | + | 0.81 | \(-\)2.98 | \(\approx \) | 0.52 | \(-\)3.11 | \(\approx \) | 0.68 |
\(-\)3.74 | 0.75 | \(-\)1.63 | + | 1.14 | \(-\)1.44 | + | 0.49 | \(-\)2.24 | + | 1.28 | \(-\)3.37 | \(\approx \) | 0.65 | \(-\)3.69 | \(\approx \) | 0.69 | |
\(-\)3.22 | 0.31 | \(-\)1.51 | + | 1.12 | \(-\)1.56 | + | 0.27 | \(-\)2.03 | + | 1.01 | \(-\)3.03 | \(\approx \) | 0.66 | \(-\)3.23 | \(\approx \) | 0.23 | |
\(-\)3.21 | 0.58 | \(-\)2.37 | + | 1.18 | \(-\)1.57 | + | 0.43 | \(-\)1.99 | + | 0.73 | \(-\)3.57 | \(\approx \) | 0.41 | \(-\)3.12 | \(\approx \) | 0.64 | |
\(-\)2.99 | 0.42 | \(-\)2.19 | + | 1.13 | \(-\)1.29 | + | 0.55 | \(-\)1.89 | + | 0.87 | \(-\)3.36 | \(\approx \) | 0.33 | \(-\)2.86 | \(\approx \) | 0.49 | |
\(-\)3.45 | 1.23 | \(-\)3.17 | \(\approx \) | 1.55 | \(-\)2.34 | + | 0.56 | \(-\)2.39 | + | 1.18 | \(-\)4.12 | \(\approx \) | 0.50 | \(-\)3.80 | \(\approx \) | 0.81 | |
\(-\)3.43 | 0.76 | \(-\)2.35 | + | 1.01 | \(-\)1.69 | + | 0.41 | \(-\)1.54 | + | 0.91 | \(-\)3.81 | \(\approx \) | 0.37 | \(-\)2.85 | + | 0.73 | |
\(-\)3.89 | 0.81 | \(-\)2.65 | + | 1.49 | \(-\)1.82 | + | 0.48 | \(-\)2.31 | + | 1.40 | \(-\)4.34 | \(\approx \) | 0.61 | \(-\)4.23 | \(\approx \) | 0.77 | |
\(-\)2.80 | 0.92 | \(-\)2.43 | \(\approx \) | 1.26 | \(-\)1.87 | + | 0.44 | \(-\)1.09 | + | 0.75 | \(-\)3.13 | \(\approx \) | 0.73 | \(-\)3.19 | \(\approx \) | 1.03 | |
\(-\)2.45 | 1.12 | \(-\)2.31 | \(\approx \) | 1.41 | \(-\)2.04 | \(\approx \) | 0.45 | \(-\)1.22 | + | 1.29 | \(-\)3.43 | – | \(-\)3.69 | – | 0.72 | ||
1/0/–1 | 12/22/6 | 22/15/3 | 30/9/1 | 12/26/2 | 16/23/1 |
\(\sigma _\varepsilon ^{2}\)=1 | PBO-EA | BO | PBO-SQR | PBO-LCB | PBO-AEI | PBO-PEI | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Problem | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | Std | |||||
Ackley | 14.8 | 0.31 | 16.1 | \(\approx \) | 0.34 | 15.4 | \(\approx \) | 2.51 | 12.3 | \(\approx \) | 0.47 | 16.5 | \(\approx \) | 0.52 | 16.2 | \(\approx \) | 0.29 |
8.32 | 0.18 | 8.36 | \(\approx \) | 0.24 | 7.07 | \(\approx \) | 2.21 | 8.49 | \(\approx \) | 0.28 | 8.62 | \(\approx \) | 0.24 | 8.45 | \(\approx \) | 0.21 | |
12.1 | 0.27 | 13.4 | \(\approx \) | 0.29 | 13.4 | \(\approx \) | 1.39 | 13.4 | \(\approx \) | 0.22 | 13.7 | \(\approx \) | 0.25 | 13.43 | \(\approx \) | 0.42 | |
6.07 | 0.28 | 6.01 | \(\approx \) | 0.21 | 5.21 | \(\approx \) | 1.32 | 6.11 | \(\approx \) | 0.20 | 6.28 | \(\approx \) | 0.26 | 6.23 | \(\approx \) | 0.11 | |
11.2 | 0.23 | 12.2 | \(\approx \) | 0.42 | 9.24 | – | 3.12 | 12.7 | \(\approx \) | 0.28 | 12.9 | \(\approx \) | 0.32 | 12.6 | \(\approx \) | 0.28 | |
12.1 | 0.34 | 13.1 | \(\approx \) | 0.29 | 9.87 | – | 2.31 | 12.3 | \(\approx \) | 0.22 | 12.7 | \(\approx \) | 0.31 | 12.7 | \(\approx \) | 0.19 | |
9.43 | 0.12 | 9.52 | \(\approx \) | 0.12 | 7.93 | \(\approx \) | 1.92 | 9.69 | \(\approx \) | 0.15 | 9.50 | \(\approx \) | 0.18 | 9.54 | \(\approx \) | 0.37 | |
16.5 | 0.29 | 16.5 | \(\approx \) | 0.56 | 13.9 | \(\approx \) | 4.81 | 16.6 | \(\approx \) | 0.25 | 16.6 | \(\approx \) | 0.28 | 16.6 | \(\approx \) | 0.16 | |
4.15 | 0.20 | 4.10 | \(\approx \) | 0.15 | 3.22 | \(\approx \) | 1.03 | 4.05 | \(\approx \) | 0.20 | 3.99 | \(\approx \) | 0.27 | 4.10 | \(\approx \) | 0.26 | |
14.5 | 0.23 | 14.5 | \(\approx \) | 0.35 | 12.6 | \(\approx \) | 3.15 | 14.5 | \(\approx \) | 0.25 | 14.4 | \(\approx \) | 0.22 | 14.5 | \(\approx \) | 0.24 | |
Rosenbrock | 116 | 96.1 | 956 | + | 768 | 967 | + | 786 | 1.2e3 | + | 865 | 701 | + | 568 | 445 | + | 498 |
193 | 278 | 181 | \(\approx \) | 176 | 1.3e3 | + | 1.1e3 | 854 | + | 753 | 488 | \(\approx \) | 832 | 490 | \(\approx \) | 293 | |
121 | 109 | 334 | + | 423 | 821 | + | 1.2e3 | 542 | + | 454 | 434 | + | 601 | 205 | \(\approx \) | 184 | |
119 | 109 | 131 | \(\approx \) | 176 | 1.2e3 | + | 1.5e3 | 323 | + | 276 | 212 | + | 123 | 124 | \(\approx \) | 123 | |
123 | 112 | 189 | + | 214 | 1.2e3 | + | 1.3e3 | 476 | + | 346 | 287 | + | 251 | 119 | \(\approx \) | 125 | |
143 | 287 | 149 | \(\approx \) | 207 | 423 | + | 687 | 621 | + | 625 | 276 | + | 213 | 241 | + | 179 | |
117 | 187 | 178 | + | 256 | 397 | + | 798 | 897 | + | 798 | 214 | + | 423 | 345 | + | 313 | |
176 | 512 | 134 | \(\approx \) | 176 | 578 | + | 821 | 1.4e3 | + | 821 | 501 | + | 423 | 156 | \(\approx \) | 157 | |
556 | 780 | 512 | \(\approx \) | 1.1e3 | 387 | – | 865 | 2.5e3 | + | 787 | 921 | + | 723 | 1.0e3 | + | 613 | |
612 | 761 | 474 | – | 1.0e3 | 1.0e3 | + | 1.1e3 | 2.1e3 | + | 1.0e3 | 1.0e3 | + | 765 | 1.1e3 | + | 731 | |
Branin | 31.9 | 19.6 | 81.1 | + | 61.7 | 90.4 | + | 59.5 | 75.1 | + | 54.4 | 48.1 | \(\approx \) | 33.4 | 28.1 | \(\approx \) | 21.8 |
36.3 | 34.4 | 109 | + | 123 | 131 | + | 91.1 | 121 | + | 63.3 | 31.2 | \(\approx \) | 13.4 | 34.4 | \(\approx \) | 17.4 | |
37.1 | 24.8 | 93.0 | + | 75.4 | 92.4 | + | 87.1 | 183 | + | 67.12 | 32.2 | \(\approx \) | 17.9 | 34.8 | \(\approx \) | 28.9 | |
73.0 | 95.1 | 129 | \(\approx \) | 126 | 190 | + | 156 | 221 | + | 102 | 51.5 | \(\approx \) | 27.6 | 43.5 | – | 42.7 | |
82.3 | 91.3 | 126 | \(\approx \) | 81.6 | 134 | + | 137 | 212 | + | 110 | 87.2 | \(\approx \) | 77.0 | 50.8 | \(\approx \) | 54.9 | |
102 | 95.3 | 71.4 | \(\approx \) | 81.4 | 180 | + | 188 | 328 | + | 126 | 88.3 | \(\approx \) | 70.5 | 86.7 | \(\approx \) | 53.0 | |
68.1 | 45.1 | 52.8 | \(\approx \) | 33.1 | 96.9 | + | 91.4 | 126 | + | 58.5 | 68.8 | \(\approx \) | 64.5 | 81.1 | + | 44.92 | |
102 | 82 | 76.0 | – | 58.3 | 141 | + | 154 | 283 | + | 95.1 | 87.5 | \(\approx \) | 73.2 | 135 | + | 98.3 | |
45.4 | 36.4 | 31.2 | \(\approx \) | 20.3 | 84.7 | + | 82.2 | 132 | + | 70.5 | 80.4 | + | 72.1 | 91.0 | + | 70.3 | |
91 | 88 | 93.2 | \(\approx \) | 70.3 | 193 | + | 128 | 250 | + | 139 | 90.3 | \(\approx \) | 53.6 | 146 | + | 115 | |
Hartmann3 | \(-\)3.23 | 0.56 | \(-\)1.25 | + | 1.13 | \(-\)1.53 | + | 0.59 | \(-\)1.97 | + | 0.75 | \(-\)3.53 | \(\approx \) | 0.68 | \(-\)2.63 | + | 0.84 |
\(-\)3.61 | 0.95 | \(-\)1.36 | + | 1.44 | \(-\)1.73 | + | 0.45 | \(-\)1.95 | + | 1.00 | \(-\)3.60 | \(\approx \) | 0.52 | \(-\)2.81 | + | 0.62 | |
\(-\)2.41 | 0.92 | \(-\)1.15 | + | 1.22 | \(-\)1.28 | + | 0.61 | \(-\)2.19 | \(\approx \) | 0.95 | \(-\)2.99 | \(\approx \) | 0.45 | \(-\)2.42 | \(\approx \) | 0.87 | |
\(-\)2.97 | 0.76 | \(-\)1.17 | + | 1.14 | \(-\)1.32 | + | 0.55 | \(-\)2.24 | \(\approx \) | 1.21 | \(-\)3.01 | \(\approx \) | 0.79 | \(-\)2.84 | \(\approx \) | 0.76 | |
\(-\)2.99 | 0.68 | \(-\)0.89 | + | 0.76 | \(-\)1.32 | + | 0.91 | \(-\)1.36 | + | 1.09 | \(-\)2.65 | \(\approx \) | 0.72 | \(-\)2.34 | \(\approx \) | 0.74 | |
\(-\)4.56 | 1.32 | \(-\)1.23 | + | 1.16 | \(-\)2.31 | + | 0.78 | \(-\)2.32 | + | 1.28 | \(-\)3.44 | \(\approx \) | 1.08 | \(-\)3.37 | \(\approx \) | 1.29 | |
\(-\)2.59 | 0.84 | \(-\)1.15 | + | 0.99 | \(-\)1.79 | \(\approx \) | 0.63 | \(-\)2.27 | \(\approx \) | 1.31 | \(-\)2.68 | \(\approx \) | 0.81 | \(-\)2.79 | \(\approx \) | 0.83 | |
\(-\)3.85 | 1.00 | \(-\)1.28 | + | 0.99 | \(-\)2.40 | + | 0.66 | \(-\)2.61 | + | 1.32 | \(-\)3.70 | \(\approx \) | 0.66 | \(-\)3.77 | \(\approx \) | 0.85 | |
\(-\)2.77 | 0.99 | \(-\)1.26 | + | 1.02 | \(-\)1.54 | + | 0.47 | \(-\)1.38 | + | 1.05 | \(-\)3.03 | \(\approx \) | 0.92 | \(-\)2.68 | \(\approx \) | 0.70 | |
\(-\)2.77 | 1.13 | \(-\)1.19 | + | 0.94 | \(-\)2.02 | \(\approx \) | 0.32 | \(-\)2.02 | \(\approx \) | 1.56 | \(-\)2.63 | \(\approx \) | 0.81 | \(-\)3.19 | \(\approx \) | 0.81 | |
+/\(\approx \)/– | 17/21/2 | 27/10/3 | 26/14/0 | 10/30/0 | 11/28/1 |
Experimental settings
Experimental results
Algorithm | Problems | DTLZ1 | DTLZ2 | DTLZ3 | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
p | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | |
Mean | 0.54 | 0.42 | 0.45 | 0.45 | 0.46 | 0.43 | 0.63 | 0.38 | 0.35 | 0.30 | 0.23 | 0.25 | 0.49 | 0.43 | 0.42 | 0.41 | 0.42 | 0.47 | |
PGP-MOEA | Std | 0.13 | 0.29 | 0.29 | 0.24 | 0.27 | 0.28 | 0.06 | 0.25 | 0.23 | 0.22 | 0.21 | 0.22 | 0.09 | 0.22 | 0.14 | 0.15 | 0.17 | 0.20 |
0.16 | 0.22 | 0.25 | 0.21 | 0.23 | 0.19 | 0.29 | 0.16 | 0.15 | 0.17 | 0.18 | 0.18 | 0.17 | 0.11 | 0.12 | 0.12 | 0.18 | 0.14 | ||
Mean | + | + | + | + | + | + | + | + | + | + | \(\approx \) | + | + | + | + | + | + | + | |
GP-MOEA | Std | 0.08 | 0.20 | 0.25 | 0.20 | 0.23 | 0.22 | 0.06 | 0.05 | 0.04 | 0.08 | 0.07 | 0.07 | 0.10 | 0.10 | 0.16 | 0.16 | 0.17 | 0.13 |
Algorithm | Problems | DTLZ4 | DTLZ5 | DTLZ6 | |||||||||||||||
p | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | |
Mean | 0.41 | 0.40 | 0.40 | 0.37 | 0.38 | 0.38 | 0.56 | 0.59 | 0.57 | 0.55 | 0.58 | 0.57 | 0.19 | 0.19 | 0.18 | 0.19 | 0.21 | 0.23 | |
PGP-MOEA | Std | 0.14 | 0.16 | 0.17 | 0.15 | 0.17 | 0.17 | 0.12 | 0.10 | 0.11 | 0.10 | 0.08 | 0.10 | 0.03 | 0.03 | 0.03 | 0.03 | 0.05 | 0.03 |
0.41 | 0.42 | 0.39 | 0.37 | 0.39 | 0.36 | 0.16 | 0.14 | 0.11 | 0.14 | 0.09 | 0.08 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | ||
Mean | \(\approx \) | \(\approx \) | \(\approx \) | \(\approx \) | \(\approx \) | \(\approx \) | + | + | + | + | + | + | + | + | + | + | + | + | |
GP-MOEA | Std | 0.11 | 0.13 | 0.12 | 0.12 | 0.13 | 0.13 | 0.15 | 0.13 | 0.10 | 0.17 | 0.08 | 0.08 | 0.04 | 0.02 | 0.02 | 0.03 | 0.03 | 0.03 |
Algorithm | Problems | DTLZ7 | DTLZ1a | DTLZ3a | |||||||||||||||
p | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | \(p_1\) | \(p_2\) | \(p_3\) | \(p_4\) | \(p_5\) | \(p_6\) | |
Mean | 0.02 | 0.01 | 0.02 | 0.02 | 0.02 | 0.02 | 0.51 | 0.44 | 0.41 | 0.56 | 0.51 | 0.47 | 0.46 | 0.32 | 0.38 | 0.41 | 0.40 | 0.38 | |
PGP-MOEA | Std | 0.01 | 0.00 | 0.01 | 0.01 | 0.01 | 0.01 | 0.33 | 0.28 | 0.27 | 0.30 | 0.30 | 0.24 | 0.35 | 0.29 | 0.31 | 0.35 | 0.33 | 0.30 |
0.02 | 0.01 | 0.02 | 0.02 | 0.01 | 0.02 | 0.28 | 0.26 | 0.23 | 0.35 | 0.27 | 0.28 | 0.18 | 0.13 | 0.12 | 0.16 | 0.12 | 0.10 | ||
Mean | \(\approx \) | \(\approx \) | \(\approx \) | \(\approx \) | \(\approx \) | \(\approx \) | + | + | \(\approx \) | + | + | \(\approx \) | + | \(\approx \) | + | + | + | + | |
GP-MOEA | Std | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | 0.01 | 0.19 | 0.24 | 0.22 | 0.28 | 0.26 | 0.26 | 0.15 | 0.10 | 0.13 | 0.13 | 0.12 | 0.09 |