Experiment Results - The New Multimodal Optimization Technique

The New Multimodal Optimization Technique

7.3 Experiment Results

Generally speaking, we can see a huge improvement in multimodal problems and hy-brid problems when combining our technique with ACO_R, since the underlying model for ACO_R is a Gaussian Kernel that intrinsically models hills. We can see improvement in some cases when combining our technique with CMA-ES. However, for SPSO, most of the results costs more evaluations to get to same error level. This is because the underly-ing model for SPSO is not a Gaussian, but a random topology. The flyunderly-ing particles make reclustering and redefining ROIs really unstable.

Problems 1 to 5 are unimodal functions. The optimum solution of Problem 5 is on the diagonal of the search space, making it extremely easy to solve for SPSO and ACO_R. Thus there is no error line for these two algorithms on Problem 5 We can see that CMA-ES is extremely efficient on solving unimodal problems, since the underlying model is a multivariate Gaussian and it converges very fast. As expected, our methods to cost slightly more evaluations than the original algorithms on unimodal functions, since we sample more points in the beginning. Also, although our clustering methods recognize unimodals in most of the times, it is not good at handling narrow valleys, e.g. Problem 3, and starts to split into multiple clusters once falls into the valley with small gradient. This multiplication gets worse since the smaller the ROI becomes, the less gradient info there is. Thus, the clustering methods would try its best to generate “hills” from the plane with almost no gradient. This is the reason why our technique does not improve the original algorithms. it falls into the valley

Problems 6 to 12 are basic multimodal functions. We can see significant improvement in Problem 11 and Problem 12 on algorithm CMA-ES and ACO_R. Besides, ACO_R com-bined with our techniques leads with the minimum error in most of the problems. This

means that the divide-and-conquer technique works well on problems with multiple hills.

The narrow valley characteristics also appears in Problem 6, so we spend more evalua-tions than the original algorithms. One interesting thing happens in Problem 9, where our technique failed to improve CMA-ES on a Rastrigin Problem. We suspect that it is due not being able to recognize the small hills, since CMA-ES samples with a normal distribution.

Therefore, even if particles did get to the top of different hills, the MDL method would still recognize them as one normal distribution. Thus, no reclustering was performed and the population converged to the wrong hills sometimes.

Problem 13 and Problem 14 are expended problems, and they are a bit too difficult for most of the algorithms.

Problems 15 to 25 are hybrid composition functions. SPSO is suitable for these kinds of problems, since SPSO is good at exploring and takes a bit longer to converge. However, our methods enhanced the ability of ACO_R, making it compatible with SPSO, and even better sometimes.

Figure 7.1: Average error of Problem 1 to Problem 8.

Figure 7.2: Average error of Problem 9 to Problem 16.

Figure 7.3: Average error of Problem 17 to Problem 24.

Figure 7.4: Average error of Problem 25.

Figure 7.5: Median error of Problem 1 to Problem 8.

Figure 7.6: Median error of Problem 9 to Problem 16.

Figure 7.7: Median error of Problem 17 to Problem 24.

Figure 7.8: Median error of Problem 25.

FEs Problems 1 2 3 4 5 6 7 8

1e3

1st(Best) 3.18E-10 1.21E-11 1.91E-08 3.06E-11 1.50E-10 4.92E-07 7.78E-03 6.67E+00 7th 2.10E-09 2.25E-09 1.83E-05 2.42E-09 2.38E-09 1.14E-02 2.67E-02 2.00E+01 13th(Median) 3.97E-09 5.45E-09 2.20E-04 5.98E-09 5.13E-09 6.40E-01 4.27E-02 2.00E+01 19th 8.46E-09 2.52E-07 8.97E-03 7.93E-09 8.29E-09 4.85E+00 7.76E-02 2.00E+01 25th(Worst) 3.05E-05 3.22E-01 1.09E+04 6.02E-07 1.15E-08 5.80E+01 5.28E-01 2.00E+01 Mean 1.23E-06 2.03E-02 6.15E+02 5.35E-08 5.15E-09 8.37E+00 7.14E-02 1.92E+01 Std 5.97E-06 7.15E-02 2.22E+03 1.42E-07 3.46E-09 1.57E+01 9.99E-02 2.74E+00

1e4

1st(Best) 2.88E-10 1.21E-11 1.31E-10 3.06E-11 1.50E-10 5.42E-10 1.88E-03 5.20E-05 7th 1.69E-09 2.25E-09 2.22E-09 2.15E-09 1.68E-09 2.85E-09 9.86E-03 2.00E+01 13th(Median) 3.66E-09 3.67E-09 4.22E-09 4.56E-09 3.60E-09 6.24E-09 2.67E-02 2.00E+01 19th 5.86E-09 6.67E-09 8.10E-09 6.12E-09 8.22E-09 7.17E-09 4.27E-02 2.00E+01 25th(Worst) 9.92E-09 8.80E-09 9.98E-09 7.93E-09 9.77E-09 8.17E-08 1.85E-01 2.00E+01 Mean 4.15E-09 4.24E-09 5.00E-09 4.19E-09 4.74E-09 8.28E-09 3.69E-02 1.83E+01 Std 2.94E-09 2.82E-09 3.29E-09 2.32E-09 3.29E-09 1.52E-08 3.86E-02 4.79E+00

2e4

Table 7.2: Error Values Achieved of CMA-ES for Problems 1-8

FEs Problems 9 10 11 12 13 14 15 16 17

1e3

1st(Best) 3.51E-10 6.79E-09 2.75E-04 1.52E-09 2.87E-12 6.68E-03 6.13E-09 8.46E-09 8.57E-07 7th 9.28E-09 1.03E-06 2.98E-03 1.93E-06 9.89E-09 1.94E-02 2.16E+01 1.82E+01 1.17E+01 13th(Median) 9.84E-03 1.03E-03 9.62E-03 4.18E-01 2.55E-02 1.99E-02 7.91E+01 4.27E+01 3.53E+01 19th 9.95E-01 3.04E-01 1.04E-02 1.78E+00 5.73E-02 4.22E-02 1.27E+02 1.00E+02 4.63E+01 25th(Worst) 2.58E+00 1.52E+00 5.92E-01 1.80E+00 1.54E-01 1.93E-01 2.06E+02 2.40E+02 7.55E+01 Mean 4.89E-01 2.72E-01 5.06E-02 8.68E-01 4.13E-02 4.17E-02 8.68E+01 6.74E+01 3.04E+01 Std 6.50E-01 4.31E-01 1.32E-01 8.32E-01 4.49E-02 4.43E-02 7.26E+01 6.12E+01 2.30E+01

1e4

1st(Best) 3.51E-10 2.21E-10 6.47E-10 2.56E-10 2.87E-12 6.68E-03 2.60E-09 7.36E-09 5.81E-10 7th 3.39E-09 2.85E-09 5.87E-09 6.29E-09 9.89E-09 1.94E-02 8.13E-09 1.82E+01 1.97E+00 13th(Median) 9.25E-09 7.63E-09 9.34E-03 4.14E-01 2.49E-02 1.94E-02 7.91E+01 4.27E+01 1.13E+01 19th 9.95E-01 3.04E-01 9.34E-03 1.78E+00 5.55E-02 2.41E-02 1.27E+02 1.00E+02 2.62E+01 25th(Worst) 2.58E+00 1.52E+00 3.72E-01 1.78E+00 1.54E-01 1.61E-01 2.00E+02 2.40E+02 6.59E+01 Mean 4.08E-01 2.59E-01 2.09E-02 8.49E-01 3.98E-02 3.71E-02 8.49E+01 6.74E+01 1.64E+01 Std 6.14E-01 4.35E-01 7.22E-02 8.36E-01 4.50E-02 4.03E-02 7.24E+01 6.12E+01 1.59E+01

2e4

Table 7.3: Error Values Achieved of CMA-ES for Problems 9-17

FEs Problems 18 19 20 21 22 23 24 25

1e3

1st(Best) 2.03E+02 3.00E+02 2.00E+02 5.41E+00 2.00E+02 2.54E+02 2.00E+02 2.00E+02 7th 3.00E+02 3.00E+02 3.00E+02 5.00E+02 2.00E+02 5.69E+02 2.00E+02 2.00E+02 13th(Median) 3.57E+02 3.87E+02 3.45E+02 5.00E+02 2.63E+02 5.69E+02 2.00E+02 2.00E+02 19th 4.00E+02 4.00E+02 4.00E+02 6.00E+02 3.00E+02 8.53E+02 2.00E+02 2.00E+02 25th(Worst) 5.44E+02 4.00E+02 4.32E+02 8.32E+02 3.97E+02 9.28E+02 3.07E+02 2.00E+02 Mean 3.53E+02 3.57E+02 3.39E+02 5.06E+02 2.55E+02 6.91E+02 2.04E+02 2.00E+02 Std 6.63E+01 4.62E+01 5.96E+01 1.55E+02 5.46E+01 1.73E+02 2.09E+01 1.89E-03

1e4

1st(Best) 2.03E+02 3.00E+02 2.00E+02 5.41E+00 2.00E+02 2.54E+02 2.00E+02 2.00E+02 7th 3.00E+02 3.00E+02 3.00E+02 5.00E+02 2.00E+02 5.69E+02 2.00E+02 2.00E+02 13th(Median) 3.57E+02 3.87E+02 3.45E+02 5.00E+02 2.00E+02 5.69E+02 2.00E+02 2.00E+02 19th 4.00E+02 4.00E+02 4.00E+02 6.00E+02 3.00E+02 8.53E+02 2.00E+02 2.00E+02 25th(Worst) 5.44E+02 4.00E+02 4.32E+02 8.32E+02 3.97E+02 9.28E+02 3.07E+02 2.00E+02 Mean 3.53E+02 3.57E+02 3.39E+02 5.06E+02 2.50E+02 6.91E+02 2.04E+02 2.00E+02 Std 6.63E+01 4.62E+01 5.96E+01 1.55E+02 5.59E+01 1.73E+02 2.09E+01 0.00E+00

2e4

1st(Best) 2.03E+02 3.00E+02 2.00E+02 5.41E+00 2.00E+02 2.54E+02 2.00E+02 2.00E+02 7th 3.00E+02 3.00E+02 3.00E+02 5.00E+02 2.00E+02 5.69E+02 2.00E+02 2.00E+02 13th(Median) 3.57E+02 3.87E+02 3.45E+02 5.00E+02 2.00E+02 5.69E+02 2.00E+02 2.00E+02 19th 4.00E+02 4.00E+02 4.00E+02 6.00E+02 3.00E+02 8.53E+02 2.00E+02 2.00E+02 25th(Worst) 5.44E+02 4.00E+02 4.32E+02 8.32E+02 3.97E+02 9.28E+02 3.07E+02 2.00E+02 Mean 3.53E+02 3.57E+02 3.39E+02 5.06E+02 2.50E+02 6.91E+02 2.04E+02 2.00E+02 Std 6.63E+01 4.62E+01 5.96E+01 1.55E+02 5.59E+01 1.73E+02 2.09E+01 0.00E+00

Table 7.4: Error Values Achieved of CMA-ES for Problems 18-25

FEs Problems 1 2 3 4 5 6 7 8

1e3

1st(Best) 1.58E-04 2.39E-03 2.52E+02 8.01E-04 0.00E+00 4.22E-01 1.33E-02 5.43E-01 7th 7.32E-03 1.90E-02 1.70E+03 2.38E-02 0.00E+00 7.13E+00 7.77E-02 2.00E+01 13th(Median) 1.63E-02 4.12E-02 5.47E+03 5.69E-02 0.00E+00 2.46E+01 1.15E-01 2.00E+01 19th 2.73E-02 9.44E-02 8.02E+03 1.07E-01 0.00E+00 5.67E+01 1.67E-01 2.00E+01 25th(Worst) 9.58E-02 1.57E-01 1.44E+04 3.33E-01 0.00E+00 1.36E+02 3.52E-01 2.00E+01 Mean 2.49E-02 5.81E-02 5.35E+03 7.88E-02 0.00E+00 3.45E+01 1.23E-01 1.81E+01 Std 2.67E-02 4.47E-02 3.77E+03 7.38E-02 0.00E+00 3.39E+01 7.67E-02 5.01E+00

1e4

1st(Best) 5.91E-10 5.51E-12 2.20E-02 1.37E-09 0.00E+00 3.24E-09 2.07E-07 1.63E-02 7th 3.15E-09 2.48E-09 3.03E+01 3.55E-09 0.00E+00 1.65E-03 5.82E-04 2.00E+01 13th(Median) 4.36E-09 4.50E-09 1.00E+02 5.30E-09 0.00E+00 2.06E-02 8.25E-03 2.00E+01 19th 6.60E-09 6.69E-09 1.43E+03 8.12E-09 0.00E+00 1.23E+01 9.89E-03 2.00E+01 25th(Worst) 9.35E-09 9.55E-09 4.44E+03 9.55E-09 0.00E+00 8.05E+01 4.02E-02 2.00E+01 Mean 4.71E-09 4.41E-09 1.02E+03 5.63E-09 0.00E+00 1.25E+01 9.70E-03 1.64E+01 Std 2.53E-09 2.55E-09 1.41E+03 2.67E-09 0.00E+00 2.22E+01 1.02E-02 7.08E+00

2e4

1st(Best) 5.91E-10 5.51E-12 2.91E-03 1.37E-09 0.00E+00 5.61E-10 4.85E-09 1.63E-02 7th 3.15E-09 2.48E-09 1.53E+00 3.55E-09 0.00E+00 6.32E-09 8.34E-06 2.00E+01 13th(Median) 4.36E-09 4.50E-09 6.28E+01 5.30E-09 0.00E+00 2.32E-04 7.40E-03 2.00E+01 19th 6.60E-09 6.69E-09 1.43E+03 8.12E-09 0.00E+00 1.23E+01 9.86E-03 2.00E+01 25th(Worst) 9.35E-09 9.55E-09 4.44E+03 9.55E-09 0.00E+00 8.05E+01 3.97E-02 2.00E+01 Mean 4.71E-09 4.41E-09 9.99E+02 5.63E-09 0.00E+00 1.25E+01 8.80E-03 1.58E+01 Std 2.53E-09 2.55E-09 1.42E+03 2.67E-09 0.00E+00 2.22E+01 1.04E-02 7.58E+00

Table 7.5: Error Values Achieved of SPSO 2011 for Problems 1-8

FEs Problems 9 10 11 12 13 14 15 16 17

1e3

1st(Best) 2.23E-02 1.11E-02 1.12E-01 3.20E-01 5.03E-08 1.13E-02 1.21E-01 1.28E-01 1.01E+00 7th 1.32E-01 2.14E-01 2.16E-01 2.50E+00 6.35E-06 2.01E-02 9.45E-01 8.77E+00 9.17E+00 13th(Median) 3.85E-01 5.35E-01 2.77E-01 4.86E+00 6.76E-05 4.28E-02 1.25E+01 3.97E+01 3.69E+01 19th 1.01E+00 1.13E+00 3.90E-01 5.96E+00 1.81E-04 7.73E-02 4.95E+01 5.01E+01 5.51E+01 25th(Worst) 1.45E+00 2.21E+00 9.00E-01 1.26E+01 5.69E-04 2.61E-01 8.69E+01 1.05E+02 8.17E+01 Mean 5.58E-01 7.21E-01 3.26E-01 4.70E+00 1.18E-04 6.23E-02 2.76E+01 3.73E+01 3.54E+01 Std 4.67E-01 6.31E-01 1.74E-01 3.13E+00 1.49E-04 5.50E-02 2.85E+01 3.26E+01 2.61E+01

1e4

1st(Best) 2.06E-10 5.39E-10 8.48E-03 4.22E-09 3.41E-10 4.34E-09 8.03E-11 5.53E-09 1.49E-09 7th 4.25E-09 2.92E-09 3.73E-02 1.56E-02 1.51E-09 5.32E-05 3.42E-09 8.60E-04 4.01E-08 13th(Median) 8.83E-09 8.04E-09 3.73E-02 4.14E-01 3.34E-09 7.79E-03 6.40E-09 1.01E-01 7.96E-05 19th 3.63E-04 2.74E-07 3.73E-02 4.69E+00 5.89E-09 1.94E-02 9.10E-09 1.98E+00 1.00E+01 25th(Worst) 9.95E-01 9.95E-01 4.00E-02 4.69E+00 9.43E-09 1.94E-02 5.19E+01 5.90E+01 2.06E+01 Mean 6.85E-02 1.23E-01 3.08E-02 1.60E+00 3.94E-09 9.83E-03 4.18E+00 6.09E+00 4.66E+00 Std 2.21E-01 3.22E-01 1.21E-02 1.99E+00 2.71E-09 9.34E-03 1.41E+01 1.51E+01 7.20E+00

2e4

1st(Best) 2.06E-10 5.39E-10 7.75E-05 4.22E-09 3.41E-10 1.04E-09 8.03E-11 5.53E-09 1.61E-10 7th 3.37E-09 2.55E-09 3.73E-02 4.54E-04 1.51E-09 4.33E-07 3.42E-09 9.21E-08 3.89E-09 13th(Median) 8.02E-09 7.02E-09 3.73E-02 3.44E-01 3.34E-09 4.19E-04 6.22E-09 2.95E-02 8.55E-09 19th 1.08E-04 9.38E-09 3.73E-02 4.69E+00 5.89E-09 1.94E-02 8.58E-09 7.03E-01 1.00E+01 25th(Worst) 9.95E-01 9.95E-01 3.76E-02 4.69E+00 9.43E-09 1.94E-02 5.19E+01 5.90E+01 2.06E+01 Mean 6.58E-02 1.19E-01 3.02E-02 1.55E+00 3.94E-09 7.91E-03 4.18E+00 5.90E+00 4.06E+00 Std 2.21E-01 3.23E-01 1.27E-02 2.01E+00 2.71E-09 9.41E-03 1.41E+01 1.52E+01 6.59E+00

Table 7.6: Error Values Achieved of SPSO 2011 for Problems 9-17

FEs Problems 18 19 20 21 22 23 24 25

1e3

1st(Best) 4.66E+01 2.01E+02 2.84E-01 1.19E+02 3.96E+01 4.14E+00 2.00E+02 2.00E+02 7th 2.37E+02 2.54E+02 3.76E+01 2.04E+02 2.03E+02 2.54E+02 2.00E+02 2.00E+02 13th(Median) 3.02E+02 3.03E+02 2.28E+02 3.11E+02 2.27E+02 4.57E+02 2.00E+02 2.01E+02 19th 3.12E+02 3.33E+02 3.44E+02 4.69E+02 3.00E+02 5.69E+02 2.00E+02 2.01E+02 25th(Worst) 4.00E+02 4.00E+02 4.01E+02 5.01E+02 5.00E+02 5.69E+02 2.01E+02 2.08E+02 Mean 2.75E+02 3.05E+02 2.09E+02 3.30E+02 2.45E+02 4.08E+02 2.00E+02 2.01E+02 Std 8.36E+01 6.41E+01 1.50E+02 1.28E+02 9.15E+01 1.59E+02 2.86E-01 2.10E+00

1e4

1st(Best) 8.64E-03 1.03E+02 1.83E-09 6.78E-05 2.01E-02 4.63E-05 2.00E+02 2.00E+02 7th 2.00E+02 2.00E+02 2.65E-05 8.63E-01 1.15E+02 5.53E+00 2.00E+02 2.00E+02 13th(Median) 3.00E+02 3.00E+02 2.01E+02 2.03E+01 2.00E+02 4.57E+02 2.00E+02 2.00E+02 19th 3.00E+02 3.00E+02 3.00E+02 2.76E+02 2.00E+02 4.57E+02 2.00E+02 2.00E+02 25th(Worst) 4.00E+02 4.00E+02 4.00E+02 5.00E+02 5.00E+02 5.69E+02 2.00E+02 2.00E+02 Mean 2.17E+02 2.64E+02 1.83E+02 1.57E+02 1.76E+02 3.06E+02 2.00E+02 2.00E+02 Std 1.17E+02 7.38E+01 1.43E+02 1.89E+02 1.01E+02 2.44E+02 0.00E+00 0.00E+00

2e4

1st(Best) 5.02E-06 1.03E+02 1.83E-09 9.85E-09 6.16E-04 3.23E-09 2.00E+02 2.00E+02 7th 2.80E+01 2.00E+02 2.65E-05 5.29E-03 5.36E+01 1.15E+00 2.00E+02 2.00E+02 13th(Median) 3.00E+02 3.00E+02 2.00E+02 4.44E+00 2.00E+02 4.57E+02 2.00E+02 2.00E+02 19th 3.00E+02 3.00E+02 3.00E+02 2.76E+02 2.00E+02 4.57E+02 2.00E+02 2.00E+02 25th(Worst) 4.00E+02 4.00E+02 4.00E+02 5.00E+02 5.00E+02 5.69E+02 2.00E+02 2.00E+02 Mean 1.94E+02 2.59E+02 1.72E+02 1.24E+02 1.55E+02 2.70E+02 2.00E+02 2.00E+02 Std 1.32E+02 6.64E+01 1.39E+02 1.98E+02 1.17E+02 2.52E+02 0.00E+00 0.00E+00

Table 7.7: Error Values Achieved of SPSO 2011 for Problems 18-25

FEs Problems 1 2 3 4 5 6 7 8

1e3

1st(Best) 1.98E-10 1.22E-10 5.42E+00 7.55E-10 0.00E+00 1.24E-03 1.54E-03 7.90E-05 7th 2.56E-09 8.57E-10 4.04E+02 3.91E-09 0.00E+00 1.25E-01 1.06E-02 2.17E+00 13th(Median) 4.20E-09 2.77E-09 2.12E+03 6.45E-09 0.00E+00 4.87E+00 2.37E-02 2.00E+01 19th 5.64E-09 5.16E-09 5.43E+03 8.43E-09 0.00E+00 5.91E+01 3.71E-02 2.00E+01 25th(Worst) 7.97E-09 9.58E-09 1.40E+04 9.91E-09 0.00E+00 1.28E+02 1.13E-01 2.01E+01 Mean 4.12E-09 3.60E-09 3.60E+03 5.92E-09 0.00E+00 3.22E+01 3.06E-02 1.38E+01 Std 2.24E-09 3.07E-09 3.96E+03 3.02E-09 0.00E+00 4.36E+01 2.91E-02 9.07E+00

1e4

1st(Best) 1.98E-10 1.22E-10 1.13E-03 7.55E-10 0.00E+00 1.70E-10 4.99E-11 7.53E-10 7th 2.56E-09 8.57E-10 5.61E-01 3.91E-09 0.00E+00 3.21E-09 3.83E-09 5.80E-09 13th(Median) 4.20E-09 2.77E-09 1.25E+01 6.45E-09 0.00E+00 4.78E-09 7.40E-03 2.00E+01 19th 5.64E-09 5.16E-09 4.89E+01 8.43E-09 0.00E+00 6.87E-09 7.40E-03 2.00E+01 25th(Worst) 7.97E-09 9.58E-09 5.25E+02 9.91E-09 0.00E+00 1.24E-05 1.06E-01 2.00E+01 Mean 4.12E-09 3.60E-09 5.58E+01 5.92E-09 0.00E+00 8.35E-07 1.41E-02 1.28E+01 Std 2.24E-09 3.07E-09 1.11E+02 3.02E-09 0.00E+00 2.87E-06 2.73E-02 9.60E+00

2e4

1st(Best) 1.98E-10 1.22E-10 2.00E-04 7.55E-10 0.00E+00 1.70E-10 4.99E-11 7.53E-10 7th 2.56E-09 8.57E-10 9.48E-03 3.91E-09 0.00E+00 2.59E-09 3.83E-09 5.80E-09 13th(Median) 4.20E-09 2.77E-09 8.40E-02 6.45E-09 0.00E+00 4.53E-09 7.40E-03 2.00E+01 19th 5.64E-09 5.16E-09 5.74E-01 8.43E-09 0.00E+00 6.49E-09 7.40E-03 2.00E+01 25th(Worst) 7.97E-09 9.58E-09 2.59E+01 9.91E-09 0.00E+00 9.01E-09 1.06E-01 2.00E+01 Mean 4.12E-09 3.60E-09 2.05E+00 5.92E-09 0.00E+00 4.66E-09 1.41E-02 1.28E+01 Std 2.24E-09 3.07E-09 5.68E+00 3.02E-09 0.00E+00 2.60E-09 2.73E-02 9.60E+00

Table 7.8: Error Values Achieved of ACORfor Problems 1-8

FEs Problems 9 10 11 12 13 14 15 16 17

1e3

1st(Best) 9.89E-10 2.66E-09 1.10E-03 4.26E-10 3.69E-13 1.10E-03 1.93E-09 2.84E-10 3.27E-07 7th 6.54E-09 5.12E-04 1.15E-02 9.17E-09 8.24E-09 1.94E-02 1.02E-07 3.96E+01 5.44E+00 13th(Median) 1.44E-06 9.95E-01 1.97E-02 1.78E+00 1.98E-02 1.94E-02 1.66E-06 1.00E+02 2.95E+01 19th 9.95E-01 9.95E-01 3.91E-02 1.78E+00 4.02E-02 1.95E-02 5.74E+01 1.08E+02 5.62E+01 25th(Worst) 1.99E+00 4.97E+00 3.21E-01 4.69E+00 1.43E-01 7.45E-02 3.03E+02 3.76E+02 1.95E+02 Mean 4.78E-01 6.91E-01 3.98E-02 1.49E+00 3.65E-02 1.99E-02 5.04E+01 9.84E+01 4.84E+01 Std 6.37E-01 9.96E-01 6.22E-02 1.60E+00 4.58E-02 1.20E-02 8.38E+01 1.10E+02 5.41E+01

1e4

1st(Best) 9.89E-10 5.60E-10 9.32E-10 4.26E-10 3.69E-13 1.00E-09 4.32E-10 2.84E-10 1.33E-09 7th 6.01E-09 5.39E-09 9.34E-03 9.17E-09 5.57E-09 1.94E-02 3.76E-09 3.96E+01 8.01E-09 13th(Median) 8.23E-09 9.95E-01 9.34E-03 1.78E+00 1.97E-02 1.94E-02 8.18E-09 1.00E+02 1.71E+01 19th 9.95E-01 9.95E-01 3.73E-02 1.78E+00 1.97E-02 1.94E-02 5.74E+01 1.08E+02 3.01E+01 25th(Worst) 1.99E+00 4.97E+00 3.73E-02 4.69E+00 1.22E-01 7.44E-02 3.00E+02 3.76E+02 1.31E+02 Mean 4.78E-01 6.77E-01 1.97E-02 1.49E+00 2.51E-02 1.70E-02 5.03E+01 9.83E+01 2.88E+01 Std 6.37E-01 1.00E+00 1.45E-02 1.60E+00 3.62E-02 1.43E-02 8.34E+01 1.10E+02 3.73E+01

2e4

1st(Best) 9.89E-10 5.60E-10 9.32E-10 4.26E-10 3.69E-13 1.00E-09 4.32E-10 2.84E-10 1.33E-09 7th 6.01E-09 5.39E-09 9.34E-03 9.17E-09 5.57E-09 1.94E-02 3.76E-09 3.96E+01 8.01E-09 13th(Median) 8.23E-09 9.95E-01 9.34E-03 1.78E+00 1.97E-02 1.94E-02 8.18E-09 1.00E+02 1.11E+01 19th 9.95E-01 9.95E-01 3.73E-02 1.78E+00 1.97E-02 1.94E-02 5.74E+01 1.08E+02 2.61E+01 25th(Worst) 1.99E+00 4.97E+00 3.73E-02 4.69E+00 1.22E-01 7.44E-02 3.00E+02 3.76E+02 9.70E+01 Mean 4.78E-01 6.77E-01 1.97E-02 1.49E+00 2.51E-02 1.70E-02 5.03E+01 9.83E+01 2.24E+01 Std 6.37E-01 1.00E+00 1.45E-02 1.60E+00 3.62E-02 1.43E-02 8.34E+01 1.10E+02 2.81E+01

Table 7.9: Error Values Achieved of ACO_Rfor Problems 9-17

FEs Problems 18 19 20 21 22 23 24 25

1e3

1st(Best) 1.13E+01 2.00E+02 3.00E+02 2.47E+01 2.00E+02 2.54E+02 2.00E+02 2.00E+02 7th 3.00E+02 3.00E+02 4.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 13th(Median) 4.00E+02 4.00E+02 4.00E+02 3.24E+02 3.00E+02 5.69E+02 2.00E+02 2.00E+02 19th 4.00E+02 4.00E+02 4.00E+02 5.00E+02 5.00E+02 5.69E+02 2.00E+02 2.00E+02 25th(Worst) 5.00E+02 5.00E+02 5.00E+02 6.00E+02 5.00E+02 9.28E+02 6.60E+02 6.72E+02 Mean 3.31E+02 3.44E+02 3.89E+02 3.50E+02 3.21E+02 5.00E+02 2.26E+02 2.46E+02 Std 1.40E+02 8.96E+01 5.74E+01 1.63E+02 1.29E+02 1.96E+02 9.68E+01 1.26E+02

1e4

1st(Best) 5.52E-10 2.00E+02 1.00E+02 1.09E-09 7.07E-10 2.54E+02 2.00E+02 1.79E+02 7th 3.00E+02 3.00E+02 4.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 13th(Median) 4.00E+02 4.00E+02 4.00E+02 3.00E+02 3.00E+02 5.69E+02 2.00E+02 2.00E+02 19th 4.00E+02 4.00E+02 4.00E+02 5.00E+02 5.00E+02 5.69E+02 2.00E+02 2.00E+02 25th(Worst) 5.00E+02 5.00E+02 5.00E+02 6.00E+02 5.00E+02 9.28E+02 6.58E+02 6.34E+02 Mean 3.28E+02 3.44E+02 3.80E+02 3.42E+02 3.12E+02 4.98E+02 2.26E+02 2.29E+02 Std 1.46E+02 8.98E+01 8.00E+01 1.73E+02 1.42E+02 1.93E+02 9.64E+01 1.02E+02

2e4

1st(Best) 5.52E-10 2.00E+02 1.00E+02 1.09E-09 7.07E-10 2.54E+02 2.00E+02 1.55E+02 7th 3.00E+02 3.00E+02 4.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 13th(Median) 4.00E+02 4.00E+02 4.00E+02 3.00E+02 3.00E+02 5.69E+02 2.00E+02 2.00E+02 19th 4.00E+02 4.00E+02 4.00E+02 5.00E+02 5.00E+02 5.69E+02 2.00E+02 2.00E+02 25th(Worst) 5.00E+02 5.00E+02 5.00E+02 6.00E+02 5.00E+02 9.28E+02 6.57E+02 6.34E+02 Mean 3.28E+02 3.44E+02 3.80E+02 3.42E+02 3.12E+02 4.98E+02 2.26E+02 2.28E+02 Std 1.46E+02 8.98E+01 8.00E+01 1.73E+02 1.42E+02 1.93E+02 9.63E+01 1.02E+02

Table 7.10: Error Values Achieved of ACO_Rfor Problems 18-25

FEs Problems 1 2 3 4 5 6 7 8

1e3

1st(Best) 4.58E-10 3.31E-10 7.41E-08 1.02E-09 1.74E-03 4.24E-04 1.05E-02 3.47E+00 7th 3.31E-09 7.56E-09 8.42E+00 8.10E-09 3.31E-02 4.82E-01 6.56E-02 2.00E+01 13th(Median) 5.15E-09 1.35E-07 9.81E+01 5.87E-08 2.04E+00 2.38E+00 1.15E-01 2.00E+01 19th 8.97E-09 1.23E-05 6.35E+02 8.03E-05 1.53E+01 8.56E+00 1.65E-01 2.00E+01 25th(Worst) 4.23E-04 6.99E-01 1.00E+04 3.47E-01 2.23E+02 1.65E+02 5.16E-01 2.00E+01 Mean 1.69E-05 5.46E-02 1.14E+03 2.11E-02 2.12E+01 1.54E+01 1.50E-01 1.87E+01 Std 8.29E-05 1.76E-01 2.47E+03 7.25E-02 4.90E+01 3.61E+01 1.37E-01 3.54E+00

1e4

1st(Best) 4.58E-10 5.67E-11 8.17E-10 4.34E-10 5.32E-09 5.57E-10 4.97E-09 3.43E-04 7th 2.96E-09 2.94E-09 6.06E-09 1.57E-09 9.13E-09 3.11E-09 7.40E-03 1.51E+01 13th(Median) 5.12E-09 4.45E-09 1.45E-01 3.70E-09 5.30E-06 7.09E-09 7.88E-03 2.00E+01 19th 8.08E-09 8.03E-09 5.46E+01 8.10E-09 9.55E-04 9.51E-09 3.07E-02 2.00E+01 25th(Worst) 9.44E-09 9.72E-09 3.20E+03 9.74E-09 4.12E-01 1.98E+00 2.75E-01 2.00E+01 Mean 5.18E-09 5.02E-09 1.83E+02 4.51E-09 2.18E-02 1.04E-01 3.50E-02 1.64E+01 Std 3.04E-09 3.18E-09 6.25E+02 3.20E-09 8.27E-02 3.99E-01 6.04E-02 6.46E+00

2e4

1st(Best) 4.58E-10 5.67E-11 8.17E-10 4.34E-10 2.75E-09 1.37E-10 4.97E-09 8.61E-09 7th 2.96E-09 2.94E-09 5.50E-09 1.57E-09 7.74E-09 2.09E-09 7.40E-03 1.12E+01 13th(Median) 5.12E-09 4.45E-09 1.07E-02 3.70E-09 8.64E-09 4.13E-09 7.40E-03 2.00E+01 19th 8.08E-09 8.03E-09 1.61E+01 8.10E-09 2.40E-08 7.45E-09 2.95E-02 2.00E+01 25th(Worst) 9.44E-09 9.72E-09 5.90E+02 9.74E-09 4.12E-01 9.75E-09 2.75E-01 2.00E+01 Mean 5.18E-09 5.02E-09 6.17E+01 4.51E-09 2.17E-02 4.95E-09 3.40E-02 1.52E+01 Std 3.04E-09 3.18E-09 1.45E+02 3.20E-09 8.27E-02 3.08E-09 6.06E-02 7.05E+00

Table 7.11: Error Values Achieved of CMA-ES + Bandit for Problems 1-8

FEs Problems 9 10 11 12 13 14 15 16 17

1e3

1st(Best) 3.78E-07 2.63E-08 4.15E-02 1.08E-07 2.03E-06 1.94E-02 1.18E+00 1.47E+00 1.25E-02 7th 2.67E-01 3.62E-03 1.24E-01 3.36E-01 9.93E-03 7.96E-02 2.09E+01 2.42E+01 1.57E+01 13th(Median) 9.95E-01 6.22E-02 1.80E-01 6.57E-01 4.09E-02 1.57E-01 5.21E+01 4.05E+01 4.37E+01 19th 1.12E+00 9.95E-01 2.44E-01 1.64E+00 5.61E-02 3.23E-01 1.00E+02 5.21E+01 6.05E+01 25th(Worst) 2.18E+00 1.38E+00 4.26E-01 5.64E+00 1.81E-01 6.68E-01 1.93E+02 9.92E+01 1.11E+02 Mean 8.65E-01 4.26E-01 1.91E-01 1.12E+00 4.76E-02 2.39E-01 6.54E+01 4.12E+01 4.19E+01 Std 6.13E-01 5.12E-01 9.02E-02 1.26E+00 4.72E-02 1.94E-01 5.64E+01 2.14E+01 3.18E+01

1e4

1st(Best) 9.38E-10 7.88E-10 8.80E-12 1.03E-10 1.20E-11 3.20E-07 1.89E-09 3.97E-09 1.99E-10 7th 5.64E-09 3.32E-09 3.41E-09 4.13E-09 4.77E-09 1.94E-02 4.60E-09 6.64E-03 5.38E-09 13th(Median) 9.95E-01 6.75E-09 5.79E-09 7.38E-09 1.97E-02 1.94E-02 1.24E+01 3.66E+01 1.24E+01 19th 9.95E-01 4.42E-01 2.04E-07 9.95E-09 4.65E-02 2.20E-02 8.46E+01 3.96E+01 1.40E+01 25th(Worst) 1.99E+00 9.95E-01 3.68E-02 1.78E+00 1.76E-01 1.63E-01 1.74E+02 5.39E+01 3.48E+01 Mean 5.71E-01 2.60E-01 2.65E-03 1.54E-01 3.84E-02 3.68E-02 4.15E+01 2.46E+01 1.06E+01 Std 5.59E-01 4.22E-01 7.62E-03 3.70E-01 4.95E-02 4.35E-02 4.80E+01 1.92E+01 9.83E+00

2e4

1st(Best) 9.38E-10 7.88E-10 8.80E-12 1.03E-10 1.20E-11 1.36E-09 1.89E-09 1.24E-09 1.99E-10 7th 5.64E-09 3.32E-09 1.90E-09 4.13E-09 4.77E-09 1.94E-02 4.60E-09 8.48E-09 5.38E-09 13th(Median) 9.95E-01 6.75E-09 4.63E-09 7.38E-09 1.97E-02 1.94E-02 1.24E+01 3.66E+01 7.95E+00 19th 9.95E-01 4.42E-01 8.78E-09 9.95E-09 4.65E-02 1.96E-02 8.46E+01 3.96E+01 1.10E+01 25th(Worst) 1.99E+00 9.95E-01 3.68E-02 1.78E+00 1.76E-01 1.56E-01 1.74E+02 5.39E+01 3.48E+01 Mean 5.71E-01 2.60E-01 2.22E-03 1.54E-01 3.84E-02 3.26E-02 4.15E+01 2.46E+01 8.87E+00 Std 5.59E-01 4.22E-01 7.50E-03 3.70E-01 4.95E-02 3.62E-02 4.80E+01 1.92E+01 9.54E+00

Table 7.12: Error Values Achieved of CMA-ES + Bandit for Problems 9-17

FEs Problems 18 19 20 21 22 23 24 25

1e3

1st(Best) 1.38E+02 1.03E+02 5.36E+00 1.62E+02 2.00E+02 1.72E+02 2.00E+02 1.43E+02 7th 2.86E+02 2.93E+02 2.69E+02 2.00E+02 2.04E+02 2.54E+02 2.00E+02 2.44E+02 13th(Median) 4.00E+02 3.27E+02 3.64E+02 2.00E+02 2.27E+02 2.54E+02 2.00E+02 3.92E+02 19th 4.00E+02 3.80E+02 4.00E+02 2.22E+02 3.65E+02 2.54E+02 2.10E+02 4.51E+02 25th(Worst) 4.47E+02 5.00E+02 4.34E+02 5.91E+02 5.00E+02 4.57E+02 2.88E+02 5.33E+02 Mean 3.44E+02 3.22E+02 3.22E+02 2.60E+02 2.79E+02 2.62E+02 2.08E+02 3.55E+02 Std 8.87E+01 8.31E+01 1.05E+02 1.19E+02 8.96E+01 4.60E+01 1.80E+01 1.21E+02

1e4

1st(Best) 2.65E-09 1.00E+02 9.58E-09 1.62E+02 2.00E+02 1.72E+02 2.00E+02 1.02E+02 7th 2.84E+02 2.00E+02 2.07E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 13th(Median) 4.00E+02 3.00E+02 3.64E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 19th 4.00E+02 3.04E+02 4.00E+02 2.00E+02 3.00E+02 2.54E+02 2.00E+02 2.00E+02 25th(Worst) 4.13E+02 5.00E+02 4.08E+02 5.12E+02 5.00E+02 4.57E+02 2.00E+02 5.00E+02 Mean 3.19E+02 2.70E+02 3.06E+02 2.52E+02 2.57E+02 2.62E+02 2.00E+02 2.26E+02 Std 1.22E+02 1.13E+02 1.21E+02 1.07E+02 8.80E+01 4.60E+01 0.00E+00 1.04E+02

2e4

1st(Best) 2.65E-09 1.00E+02 9.58E-09 1.62E+02 2.00E+02 1.72E+02 2.00E+02 1.00E+02 7th 2.84E+02 2.00E+02 2.07E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 13th(Median) 4.00E+02 3.00E+02 3.64E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 19th 4.00E+02 3.04E+02 4.00E+02 2.00E+02 3.00E+02 2.54E+02 2.00E+02 2.00E+02 25th(Worst) 4.13E+02 5.00E+02 4.08E+02 5.12E+02 5.00E+02 4.57E+02 2.00E+02 5.00E+02 Mean 3.19E+02 2.70E+02 3.06E+02 2.52E+02 2.57E+02 2.62E+02 2.00E+02 2.26E+02 Std 1.22E+02 1.13E+02 1.21E+02 1.07E+02 8.80E+01 4.60E+01 0.00E+00 1.04E+02

Table 7.13: Error Values Achieved of CMA-ES + Bandit for Problems 18-25

FEs Problems 1 2 3 4 5 6 7 8

1e3

1st(Best) 7.98E-04 7.39E-03 1.86E+02 2.39E-01 3.19E+00 4.23E+00 5.13E-02 9.52E+00 7th 2.14E-02 3.98E-01 1.11E+03 6.61E-01 1.17E+01 2.04E+01 1.48E-01 1.52E+01 13th(Median) 6.17E-02 6.48E-01 2.87E+03 1.76E+00 2.28E+01 3.64E+01 3.36E-01 2.00E+01 19th 2.45E-01 1.28E+00 4.41E+03 2.94E+00 5.84E+01 5.46E+01 5.66E-01 2.00E+01 25th(Worst) 4.02E+00 1.12E+01 3.55E+04 2.60E+01 3.63E+02 3.62E+02 2.33E+00 2.00E+01 Mean 4.19E-01 1.67E+00 4.92E+03 3.61E+00 5.03E+01 5.71E+01 5.30E-01 1.78E+01 Std 9.03E-01 2.76E+00 7.34E+03 5.84E+00 7.40E+01 7.47E+01 5.84E-01 3.22E+00

1e4

1st(Best) 1.57E-09 4.33E-10 3.00E-01 1.38E-09 2.99E-09 1.06E-02 2.69E-09 6.62E+00 7th 3.41E-09 3.63E-09 3.51E+01 2.65E-09 2.27E-04 1.98E+00 8.31E-03 1.03E+01 13th(Median) 5.11E-09 6.74E-09 3.60E+02 7.13E-09 1.93E+00 9.10E+00 1.89E-02 1.97E+01 19th 7.30E-09 8.82E-09 1.07E+03 3.27E-08 9.33E+00 3.56E+01 7.85E-02 2.00E+01 25th(Worst) 9.80E-09 3.42E-03 6.22E+03 2.71E-01 8.27E+01 9.26E+01 5.66E-01 2.00E+01 Mean 5.30E-09 1.72E-04 8.80E+02 1.09E-02 1.03E+01 2.20E+01 7.13E-02 1.56E+01 Std 2.55E-09 6.85E-04 1.41E+03 5.31E-02 1.95E+01 2.60E+01 1.22E-01 5.21E+00

2e4

1st(Best) 1.57E-09 4.33E-10 2.11E-03 1.38E-09 2.99E-09 8.17E-05 2.69E-09 4.48E-02 7th 3.41E-09 3.63E-09 8.80E+00 2.62E-09 9.48E-09 6.10E-01 8.01E-03 9.52E+00 13th(Median) 5.11E-09 6.25E-09 1.73E+02 3.68E-09 4.82E-01 9.10E+00 1.16E-02 1.84E+01 19th 7.30E-09 8.62E-09 9.37E+02 7.81E-09 3.07E+00 2.65E+01 5.13E-02 2.00E+01 25th(Worst) 9.80E-09 8.73E-04 2.59E+03 2.31E-05 5.54E+01 9.26E+01 5.66E-01 2.00E+01 Mean 5.30E-09 3.49E-05 5.09E+02 1.19E-06 6.43E+00 2.06E+01 6.13E-02 1.45E+01 Std 2.55E-09 1.71E-04 6.84E+02 4.65E-06 1.29E+01 2.60E+01 1.22E-01 6.72E+00

Table 7.14: Error Values Achieved of SPSO 2011 + Bandit for Problems 1-8

FEs Problems 9 10 11 12 13 14 15 16 17

1e3

1st(Best) 6.52E-02 1.14E-01 1.67E-01 4.57E-01 3.31E-08 1.97E-02 6.06E+00 1.42E+00 1.36E+01 7th 3.88E-01 6.74E-01 2.94E-01 2.90E+00 2.14E-04 5.65E-02 6.76E+01 3.77E+01 3.11E+01 13th(Median) 1.01E+00 1.15E+00 3.65E-01 6.82E+00 1.76E-03 8.41E-02 7.64E+01 5.02E+01 5.16E+01 19th 1.17E+00 1.72E+00 4.47E-01 1.52E+01 1.01E-02 1.56E-01 1.03E+02 8.47E+01 7.47E+01 25th(Worst) 2.72E+00 2.66E+00 7.17E-01 8.08E+01 8.94E-02 2.72E-01 2.25E+02 1.16E+02 1.11E+02 Mean 9.29E-01 1.22E+00 3.94E-01 1.49E+01 9.96E-03 1.07E-01 8.59E+01 5.78E+01 5.22E+01 Std 6.68E-01 6.73E-01 1.30E-01 2.06E+01 1.99E-02 7.28E-02 4.74E+01 3.15E+01 2.55E+01

1e4

1st(Best) 1.33E-09 2.59E-09 1.40E-02 1.10E-09 2.61E-10 3.68E-04 4.42E-04 1.46E-04 6.81E-06 7th 4.86E-09 4.87E-06 3.13E-02 6.63E-06 2.62E-09 1.69E-02 3.09E-02 1.54E+00 1.98E+01 13th(Median) 1.64E-03 1.90E-01 4.65E-02 2.36E-04 9.27E-09 1.94E-02 6.06E+00 3.76E+01 3.08E+01 19th 3.88E-01 9.95E-01 6.82E-02 5.92E-02 2.62E-06 2.60E-02 5.74E+01 4.22E+01 4.01E+01 25th(Worst) 9.95E-01 2.66E+00 1.87E-01 6.85E-01 1.31E-02 7.47E-02 1.01E+02 9.10E+01 6.82E+01 Mean 2.61E-01 5.50E-01 6.17E-02 1.08E-01 8.91E-04 2.36E-02 2.76E+01 3.34E+01 3.08E+01 Std 3.64E-01 7.68E-01 4.72E-02 1.97E-01 2.86E-03 1.73E-02 3.29E+01 2.70E+01 1.59E+01

2e4

1st(Best) 1.33E-09 2.55E-09 1.99E-04 9.82E-10 9.09E-13 9.98E-06 4.34E-09 4.08E-06 6.81E-06 7th 3.48E-09 9.28E-09 7.87E-04 4.42E-09 1.64E-09 1.13E-02 1.00E-02 2.06E-01 1.58E+01 13th(Median) 1.64E-03 1.90E-01 9.39E-03 7.70E-09 7.91E-09 1.94E-02 6.06E+00 3.76E+01 2.47E+01 19th 3.88E-01 9.95E-01 1.01E-02 2.04E-05 2.62E-06 1.95E-02 5.74E+01 4.22E+01 3.89E+01 25th(Worst) 9.95E-01 2.66E+00 6.82E-02 4.15E-01 1.31E-02 3.04E-02 1.01E+02 9.10E+01 6.82E+01 Mean 2.57E-01 5.50E-01 9.07E-03 5.22E-02 8.90E-04 1.63E-02 2.66E+01 3.29E+01 2.66E+01 Std 3.67E-01 7.68E-01 1.34E-02 1.34E-01 2.86E-03 8.27E-03 3.16E+01 2.75E+01 1.69E+01

Table 7.15: Error Values Achieved of SPSO 2011 + Bandit for Problems 9-17

FEs Problems 18 19 20 21 22 23 24 25

1e3

1st(Best) 1.91E+01 2.03E+02 3.38E+00 1.90E+02 1.96E+02 2.20E+02 1.48E+02 1.75E+02 7th 2.39E+02 3.14E+02 2.60E+02 2.10E+02 2.35E+02 2.54E+02 2.05E+02 3.14E+02 13th(Median) 3.02E+02 3.55E+02 3.40E+02 2.44E+02 2.59E+02 2.54E+02 2.16E+02 3.79E+02 19th 3.52E+02 3.83E+02 3.77E+02 2.94E+02 3.39E+02 2.54E+02 2.46E+02 5.12E+02 25th(Worst) 4.52E+02 4.33E+02 4.62E+02 5.03E+02 5.01E+02 4.57E+02 3.40E+02 6.25E+02 Mean 2.93E+02 3.44E+02 2.98E+02 2.78E+02 2.97E+02 3.01E+02 2.30E+02 4.06E+02 Std 9.63E+01 6.29E+01 1.25E+02 8.89E+01 8.10E+01 8.79E+01 3.93E+01 1.29E+02

1e4

1st(Best) 1.35E-01 1.01E+02 3.55E-02 2.11E+00 8.59E+00 5.92E-01 1.48E+02 1.00E+02 7th 6.13E+00 2.00E+02 1.01E+02 1.43E+02 2.00E+02 2.54E+02 2.00E+02 1.57E+02 13th(Median) 1.00E+02 3.00E+02 2.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.78E+02 19th 2.09E+02 3.31E+02 2.66E+02 2.00E+02 2.02E+02 2.54E+02 2.00E+02 2.00E+02 25th(Worst) 4.24E+02 4.04E+02 4.00E+02 2.94E+02 3.08E+02 4.57E+02 2.10E+02 5.00E+02 Mean 1.19E+02 2.70E+02 1.83E+02 1.70E+02 1.94E+02 2.81E+02 1.98E+02 2.00E+02 Std 1.30E+02 8.58E+01 1.23E+02 6.57E+01 6.48E+01 1.20E+02 1.04E+01 8.14E+01

2e4

1st(Best) 1.27E-07 1.00E+02 3.19E-02 1.76E+00 8.59E+00 3.24E-06 1.48E+02 1.00E+02 7th 1.66E-01 2.00E+02 1.00E+02 1.27E+02 2.00E+02 2.54E+02 2.00E+02 1.48E+02 13th(Median) 1.00E+02 3.00E+02 2.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.75E+02 19th 2.09E+02 3.31E+02 2.66E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.00E+02 25th(Worst) 4.24E+02 4.00E+02 4.00E+02 2.94E+02 3.08E+02 4.57E+02 2.10E+02 5.00E+02 Mean 1.13E+02 2.69E+02 1.76E+02 1.65E+02 1.93E+02 2.72E+02 1.98E+02 1.88E+02 Std 1.34E+02 8.63E+01 1.26E+02 6.76E+01 6.50E+01 1.31E+02 1.04E+01 8.40E+01

Table 7.16: Error Values Achieved of SPSO 2011 + Bandit for Problems 18-25

FEs Problems 1 2 3 4 5 6 7 8

1e3

1st(Best) 7.84E-12 1.37E-09 8.01E+00 1.69E-10 8.69E-06 5.18E-02 5.68E-03 2.46E-02 7th 2.34E-09 3.25E-09 2.10E+02 2.30E-09 1.09E-04 4.38E-01 1.68E-02 5.11E+00 13th(Median) 5.28E-09 4.78E-09 7.31E+02 5.32E-09 3.14E-04 3.87E+00 2.07E-02 1.51E+01 19th 7.48E-09 7.55E-09 1.87E+03 6.83E-09 8.50E-04 2.90E+01 4.18E-02 2.00E+01 25th(Worst) 9.86E-09 9.98E-09 1.42E+04 9.60E-09 2.06E+01 6.27E+01 1.28E-01 2.00E+01 Mean 5.07E-09 5.34E-09 1.65E+03 4.88E-09 1.50E+00 1.55E+01 3.58E-02 1.29E+01 Std 2.90E-09 2.79E-09 2.86E+03 2.70E-09 5.11E+00 1.96E+01 3.12E-02 7.17E+00

1e4

1st(Best) 7.84E-12 1.37E-09 1.78E-03 1.69E-10 7.69E-10 4.12E-09 5.77E-10 6.89E-08 7th 2.34E-09 3.25E-09 5.69E-02 2.30E-09 4.37E-09 4.19E-05 3.26E-09 4.45E-02 13th(Median) 5.28E-09 4.78E-09 1.54E+00 5.32E-09 6.39E-09 3.24E-04 7.40E-03 3.40E+00 19th 7.48E-09 7.55E-09 1.90E+01 6.83E-09 8.56E-09 4.00E-03 7.40E-03 1.94E+01 25th(Worst) 9.86E-09 9.98E-09 6.44E+02 9.60E-09 1.69E+01 6.29E+00 2.71E-02 2.00E+01 Mean 5.07E-09 5.34E-09 7.36E+01 4.88E-09 6.75E-01 2.56E-01 5.92E-03 7.63E+00 Std 2.90E-09 2.79E-09 1.78E+02 2.70E-09 3.31E+00 1.23E+00 7.04E-03 8.78E+00

2e4

1st(Best) 7.84E-12 1.37E-09 3.12E-06 1.69E-10 7.69E-10 3.64E-09 5.77E-10 3.15E-09 7th 2.34E-09 3.25E-09 8.29E-04 2.30E-09 4.37E-09 9.46E-09 2.80E-09 2.02E-04 13th(Median) 5.28E-09 4.78E-09 6.72E-03 5.32E-09 6.39E-09 3.09E-08 7.40E-03 1.37E-02 19th 7.48E-09 7.55E-09 3.98E+00 6.83E-09 8.56E-09 9.56E-07 7.40E-03 1.80E+01 25th(Worst) 9.86E-09 9.98E-09 4.33E+02 9.60E-09 1.69E+01 1.12E-04 2.71E-02 2.00E+01 Mean 5.07E-09 5.34E-09 3.55E+01 4.88E-09 6.75E-01 5.26E-06 5.92E-03 6.44E+00 Std 2.90E-09 2.79E-09 9.71E+01 2.70E-09 3.31E+00 2.18E-05 7.04E-03 8.88E+00

Table 7.17: Error Values Achieved of ACO_R+ Bandit for Problems 1-8

FEs Problems 9 10 11 12 13 14 15 16 17

1e3

1st(Best) 2.61E-08 6.32E-08 7.08E-02 5.76E-03 2.34E-09 1.63E-02 2.16E-01 5.15E-03 1.82E-01 7th 1.08E-03 2.97E-02 1.87E-01 1.19E-01 9.86E-06 1.94E-02 9.00E+00 1.68E+00 1.56E+00 13th(Median) 4.55E-02 2.56E-01 2.85E-01 4.88E-01 3.00E-03 1.95E-02 3.71E+01 5.28E+00 3.01E+01 19th 2.44E-01 9.96E-01 3.94E-01 1.78E+00 2.83E-02 1.97E-02 6.58E+01 3.96E+01 4.21E+01 25th(Worst) 1.77E+00 2.08E+00 6.55E-01 4.11E+00 1.22E-01 1.58E-01 2.05E+02 5.64E+01 7.78E+01 Mean 2.63E-01 5.62E-01 3.01E-01 9.25E-01 1.98E-02 2.72E-02 4.82E+01 1.70E+01 2.69E+01 Std 4.42E-01 5.94E-01 1.50E-01 9.74E-01 2.87E-02 2.89E-02 4.82E+01 1.95E+01 2.34E+01

1e4

1st(Best) 3.11E-10 9.06E-10 5.85E-11 2.97E-10 5.29E-11 2.21E-09 1.82E-11 8.43E-10 8.60E-10 7th 4.33E-09 3.59E-09 2.44E-09 4.23E-09 2.43E-09 1.94E-02 3.65E-09 3.47E-09 5.90E-09 13th(Median) 5.79E-09 6.17E-09 9.41E-09 9.21E-09 4.17E-09 1.94E-02 6.85E-09 6.43E-09 7.79E-09 19th 7.00E-09 9.82E-09 9.53E-06 4.14E-01 1.44E-02 1.94E-02 8.22E-09 8.17E-09 1.11E+01 25th(Worst) 9.95E-01 9.95E-01 9.34E-03 4.14E-01 1.97E-02 1.94E-02 1.00E+02 3.96E+01 1.85E+01 Mean 3.98E-02 1.99E-01 7.72E-04 1.82E-01 6.50E-03 1.71E-02 1.41E+01 9.38E+00 4.51E+00 Std 1.95E-01 3.98E-01 2.53E-03 2.05E-01 8.44E-03 6.31E-03 3.33E+01 1.67E+01 6.77E+00

2e4

1st(Best) 3.11E-10 9.06E-10 5.85E-11 2.97E-10 5.29E-11 2.21E-09 1.82E-11 8.43E-10 8.60E-10 7th 4.33E-09 3.59E-09 2.44E-09 4.23E-09 2.43E-09 1.94E-02 3.65E-09 3.47E-09 5.90E-09 13th(Median) 5.79E-09 6.17E-09 9.41E-09 9.21E-09 3.55E-09 1.94E-02 6.85E-09 6.43E-09 7.79E-09 19th 7.00E-09 9.82E-09 9.53E-06 4.14E-01 9.92E-03 1.94E-02 8.22E-09 8.17E-09 1.11E+01 25th(Worst) 9.95E-01 9.95E-01 9.34E-03 4.14E-01 1.97E-02 1.94E-02 1.00E+02 3.96E+01 1.85E+01 Mean 3.98E-02 1.99E-01 7.72E-04 1.82E-01 6.32E-03 1.63E-02 1.41E+01 9.38E+00 4.18E+00 Std 1.95E-01 3.98E-01 2.53E-03 2.05E-01 8.32E-03 7.12E-03 3.33E+01 1.67E+01 6.47E+00

Table 7.18: Error Values Achieved of ACO_R+ Bandit for Problems 9-17

FEs Problems 18 19 20 21 22 23 24 25

1e3

1st(Best) 8.07E-01 1.04E+02 8.80E-01 9.54E+01 5.06E+01 1.53E+02 2.00E+02 1.44E+02 7th 1.02E+02 2.09E+02 1.23E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.19E+02 13th(Median) 2.08E+02 2.66E+02 2.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 2.64E+02 19th 3.00E+02 3.01E+02 3.00E+02 2.00E+02 2.04E+02 2.54E+02 2.00E+02 4.10E+02 25th(Worst) 3.11E+02 4.00E+02 4.00E+02 5.00E+02 3.01E+02 5.69E+02 2.09E+02 5.29E+02 Mean 1.92E+02 2.62E+02 2.03E+02 2.04E+02 2.07E+02 2.79E+02 2.00E+02 3.09E+02 Std 1.02E+02 7.29E+01 1.13E+02 9.54E+01 4.57E+01 8.41E+01 1.85E+00 1.23E+02

1e4

1st(Best) 1.81E-09 1.00E+02 4.24E-09 2.47E-10 2.62E-09 3.10E-09 2.00E+02 1.00E+02 7th 8.66E-09 1.00E+02 1.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.00E+02 13th(Median) 1.00E+02 2.00E+02 1.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.11E+02 19th 3.00E+02 3.00E+02 3.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.42E+02 25th(Worst) 3.00E+02 4.00E+02 4.00E+02 5.00E+02 3.00E+02 5.69E+02 2.00E+02 5.00E+02 Mean 1.28E+02 1.92E+02 1.60E+02 1.69E+02 1.80E+02 2.52E+02 2.00E+02 1.49E+02 Std 1.25E+02 9.77E+01 1.30E+02 1.04E+02 8.49E+01 1.23E+02 0.00E+00 1.05E+02

2e4

1st(Best) 1.81E-09 1.00E+02 4.24E-09 2.47E-10 2.62E-09 3.10E-09 2.00E+02 1.00E+02 7th 8.66E-09 1.00E+02 1.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.00E+02 13th(Median) 1.00E+02 2.00E+02 1.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.00E+02 19th 3.00E+02 3.00E+02 3.00E+02 2.00E+02 2.00E+02 2.54E+02 2.00E+02 1.00E+02 25th(Worst) 3.00E+02 4.00E+02 4.00E+02 5.00E+02 3.00E+02 5.69E+02 2.00E+02 5.00E+02 Mean 1.28E+02 1.92E+02 1.60E+02 1.68E+02 1.80E+02 2.52E+02 2.00E+02 1.34E+02 Std 1.25E+02 9.77E+01 1.30E+02 1.05E+02 8.49E+01 1.23E+02 0.00E+00 1.08E+02

Table 7.19: Error Values Achieved of ACO_R+ Bandit for Problems 18-25

Chapter 8 Conclusion

In this thesis, a new multimodal optimization technique is proposed that aims to break down the multimodal problem into unimodal problems. This technique is composed of three main components: unimodal detection, region of interest optimization and resource allocation. We use the three techniques to dynamically isolate potential unimodals and enlarge a specific interesting region on the subspace for algorithms to search more thor-oughly. We also manage the ratio between exploration and exploitation according to the remaining resources, so that the optimization process can be more efficient.

First, we proposed a unimodal detection technique to find potential fitness hills in a real-valued multimodal problem by hierarchical clustering and Minimum Description Length. It depicts the underlying fitness hills better than K-Means. Also unlike K-Means which needs to collaborate with other methods to determine the number of clusters, our technique intrinsically detects the number of potential unimodals.

Second, we also proposed a method to separate the search space into smaller subspaces in order to enhance performance. We use linear projection matrix to project the search space to a subspace with well-defined boundaries for feasible solutions. Therefore, the algorithm only needs to search within a hypercube, constrained by [0, 1] in all dimensions.

This technique is preferable for algorithms that requires a box-shape boundaries. It also creates ROIs on the original search space that isolates the unimodals. We also use the (1+1)-ES to optimize the projection matrix so that the ROIs have minimal-overlapping, thus enhance the searching efficiency. By cracking down the original search space into

multiple non-overlapping subspace, algorithms now only have to search a smaller portion of the landscape which ideally consists of only one unimodal.

Furthermore, we proposed a new multi-armed bandit technique that optimize the re-source allocation. Unlike traditional multi-armed bandit techniques that maximize the highest reward mean, the new multi-armed technique aims to maximize the probability of acquiring the best rank. We also explained how to maintain a more stabilize cluster shapes during recluster, yet still conserve the flexibility to split when needed. We believe that in the beginning, when there are abundant of resources left, we should invest more resources in exploration. That means we should update each clusters more equally for ex-ploration. Later, as the algorithms update the particles, we should be able to merge clusters to eliminate redundant search, or split a cluster and invest more particles to search in a cer-tain region. When there are few evaluations left, we should concentrate on exploiting the current best hill.

However, there are still many issues that can be improved. Our experiment results show that there are still stability issues. We need to investigate more on how to delete un-necessary arms. Also, this technique tends to cost more evaluations on problems that need to move for a long distance after converging to a relatively narrow valley. The proposed clustering technique for underlying unimodals identification can further be improved. The current hierarchical clustering technique can be extended to a complete tree that contains all clusters. Moreover, since this technique is composed of many relatively complex meth-ods, speeding up computational time is also a crucial issue for all hyperparameters tuning techniques.

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