This section describes the sets of experiments on the proposed method to reveal the change in performance when different splitting ratios of the two subpopulations are
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Figure 8: Empirical results of the proposed method compared to the original ECGA for power law scaled problems composed of subproblems of sizes 3, 4, and 5 (k= 3, 4, and 5). In this experiment, tournament size t= 16 was used and the number of subfunctions forming the test problems was fixed at 10 (i.e., m= 10).
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Figure 9: Empirical results of the proposed method compared to the original ECGA for uniformly scaled problems composed of subproblems of sizes 3, 4, and 5 (k= 3, 4, and 5). In this experiment, tournament size t= 16 was used and the number of subfunctions forming the test problems was fixed at 10 (i.e., m= 10).
adopted. It presents the experimental results to illustrate the behavior under different scalings. The purpose for performing these experiments is twofold:
• First, we would like to observe how the splitting ratio is related to the scaling or linkage sensibility of a problem.
• Second, we wish to empirically study the change in performance obtained from decreasing or increasing the proportion of population for checking the model.
It is important in practice to spend function evaluations wisely. Since using too large a proportion of the population for pruning may result in a waste of resources, it should
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Figure 10: Empirical results of the proposed method for a 60-bit exponential scaled problem with different splitting ratios between the two subpopulations. The splitting ratio (|T |/|S + T |) ranging from 0.0 (ECGA without pruning) to 0.8 was used to observe the change in performance of the proposed approach.
be estimated to what degree the expense on checking the built model yields savings, and how the scaling of the problem is related to this matter.
6.4.1 Experimental Settings
The problem instances used in this set of experiments were of 60 bits formed by con-catenating 4-bit trap functions (k= 4, m = 15). The splitting ratio (|T |/|S + T |) ranged from 0.0 to 0.8. The ratio 0.0 represents the result of running the original ECGA (without pruning), which serves as a baseline. Two selection pressures were adopted by setting tournament size t to 12 and 16.
As in the previous experiments, the stopping criterion is set such that a run is terminated when all solutions converge to the same fitness value. For each splitting ratio, the minimum required population size was determined by a bisection method such that on average, m− 1 building blocks converge to the correct values in 50 runs.
6.4.2 Results and Observations
The empirical results for exponential scaled problems are presented in Figure 10. For both tournament sizes, the required population size decreases as the splitting ratio increases. However, the number of generations increases with the splitting ratio. The combined effect is that the minimum required function evaluation is obtained when the splitting ratio is 0.6, and the required function evaluation grows when the splitting ratio either increases or decreases.
Figure 11 shows the results for power law scaled problems. In contrast to the previous case, the required population size does not strictly decrease with the increment of the splitting ratio. The population size first decreases as the splitting ratio grows and then hits a turning point at 0.5 (t= 16) or 0.6 (t = 12). Similar to the exponential scaled case, the number of generations increases with the splitting ratio. The combined effect is that the number of function evaluations first decreases and then increases. For both tournament sizes, the minimum is obtained when the splitting ratio= 0.3.
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Figure 11: Empirical results of the proposed method for a 60-bit power law scaled problem with different splitting ratios between the two subpopulations. The splitting ratio (|T |/|S + T |) ranging from 0.0 (ECGA without pruning) to 0.8 was used to observe the change in performance of the proposed approach.
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Figure 12: Empirical results of the proposed method for a 60-bit uniformly scaled problem with different splitting ratios between the two subpopulations. The splitting ratio (|T |/|S + T |) ranging from 0.0 (ECGA without pruning) to 0.8 was used to observe the change in performance of the proposed approach.
Figure 12 shows the results for uniformly scaled problems. As expected, Figures 12(a) and 12(b) both share a common pattern in which the population size and the number of function evaluations increase with the splitting ratio. This is because in the uniformly scaled case, the linkage is always completely sensible, and there is no need to verify or prune the built probabilistic model.
These experimental results demonstrate that under different scaling setups, the behavior of the proposed approach corresponding to the splitting ratio varies differently.
The empirical results suggest that if the given problem is evidently with distinguishable prominence among the constituting subproblems, using higher splitting ratios will yield
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better performance. Lower ratios are more suitable if the problem at hand is composed of subproblems with roughly equal salience.
Another insight provided by this set of experiments is that reducing the size of the proportion of population spent on the pruning mechanism can considerably improve the performance. As shown in Figures 10(b) and 11(b), compared to the original ECGA (splitting ratio= 0.0 in the figures), significant performance gain can be obtained by using a mere 10% of the population to validate the built model. On the other hand, Figure 12(b) also demonstrates that using this small percentage of population on the pruning mechanism will not bring serious overhead for the overall performance.