i=0
(k+ 1)iftrap
k(sk×i+1sk×i+2· · · sk×i+k)
Power law:
m−1
i=0
(i+ 1)3ftrap
k(sk×i+1sk×i+2· · · sk×i+k)
Uniform:
m−1
i=0
ftrap
k(sk×i+1sk×i+2· · · sk×i+k)
By adopting different scaling setups, we can compare the original ECGA with our approach under different degrees of linkage sensibilities. By varying k and m, we can observe the behavior of the proposed method with respect to different problem and subproblem sizes in a controlled manner. Furthermore, various selection pressures are also taken into consideration to make a more thorough observation.
The purpose of the following experiments is to understand the impact of the pro-posed method on the computational resource (population size and function evaluations) required to solve a problem. Thus, we do not use solution quality as a measure of comparison but treat it as a minimum requirement. More precisely, we use a bisection method (Sastry, 2001) to bound the minimum population size capable of achieving reliable convergence to the optimum. Of course, solution quality can be an important indicator for evaluating a newly invented approach. However, the primary goal of this study is to design a more economic approach for solving problems, and the experiments are designed to evaluate the ability of the proposed approach in this aspect.
6.1 Effect of Selection Pressure
This section describes the experiments designed for observing the effect of selection pressure on both the original ECGA and the ECGA combined with the proposed ap-proach. The purpose of these experiments is twofold.
• First, we want to determine the range of selection pressure with which the pro-posed approach works as we designed. Appropriate selection pressure is quite
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8 12 16 20 24
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
Tournament Sizes
Population Sizes
ECGA, = 40 ECGA, = 80 ECGA+MP, = 40 ECGA+MP, = 80
(a) Population Sizes
8 12 16 20 24
0 1 2 3 4 5 6 7 8 9 10 11 12
x 104
Tournament Sizes
Function Evaluations
ECGA, = 40 ECGA, = 80 ECGA+MP, = 40 ECGA+MP, = 80
(b) Function Evaluations
Figure 1: Empirical results of the proposed method and original ECGA on 40- and 80-bit (k= 4, m = 10 and 20) exponential scaled problems. Five tournament sizes ranging from 8 to 24 were used to observe the behavior of the algorithms under different selection pressures.
important to the proper functioning of our approach because the pruning mech-anism is designed according to the statistical inconsistencies between the two subpopulations.
• Second, because the proposed approach will be compared with the original ECGA in the subsequent experiments, in order to make a fair and meaningful compar-ison, the selection pressure must be set to an appropriate value for the original ECGA to work under good conditions.
6.1.1 Experimental Settings
Because tournament selection is adopted, the selection pressure is altered by changing the tournament size. We consider tournament sizes ranging from 8 to 24, and the problem instances used to make the observations are of length 40 bits and 80 bits with 4-bit trap functions as subproblems (k= 4, m = 10 and 20, respectively).
For simplicity, the splitting of population is performed in the way that the two resulting subpopulations are disjoint and of equal size. The stopping criterion is set such that a run is terminated when all solutions in the population converge to the same fitness value. For each tournament size, the minimum required population size is determined by a bisection method (Sastry, 2001) such that on average, m− 1 building blocks converge to the correct values in 50 runs for each of the two problem instances.
6.1.2 Results and Observations
The results for exponential, power law, and uniformly scaled problems are presented in Figures 1, 2, and 3, respectively. It can be observed from Figures 1(b), 2(b), and 3(b) that for all three scalings, the original ECGA works best (in terms of the number of function evaluations) under tournament size 12 or 16. Based on that, we will use these two tournament sizes in the following sets of experiments to ensure that the improvement of our approach over the original ECGA is not a result of improper selection pressure. In fact, we also performed experiments using a tournament size of 4, of which the results
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Figure 2: Empirical results of the proposed method and original ECGA on 40- and 80-bit (k= 4, m = 10 and 20) power law scaled problems. Five tournament sizes ranging from 8 to 24 were used to observe the behavior of the algorithms under different selection pressures.
Figure 3: Empirical results of the proposed method and original ECGA on 40- and 80-bit (k= 4, m = 10 and 20) uniformly scaled problems. Five tournament sizes ranging from 8 to 24 were used to observe the behavior of the algorithms under different selection pressures.
are listed in Table 5. This demonstrates that adopting a lower selection pressure does not yield better performance for ECGA or for our approach.
The results of these experiments give some insights into the pruning mechanism. It can be observed that the appropriateness of a particular selection pressure is related to the linkage sensibility of the problem at hand. This property could cause inconvenience in choosing selection pressure for the algorithm because when dealing with black box optimization, we usually do not have any information about the problem at hand.
Fortunately, Figures 1(b), 2(b), and 3(b) also suggest that under tournament sizes ranging
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Table 5: Empirical results of the proposed method and original ECGA using a tour-nament size of 4. Experiments were conducted on 40- and 80-bit problems formed by concatenating 4-bit trap functions with three different scalings. The symbols , n, and fevdenote problem size, population size, and function evaluations, respectively.
n fev std. of fev
Exponential ECGA 40 1,719 44,487.72 2,682.02
80 3,748 187,549.92 5,912.06
ECGA+MP 40 1,405 37,373.00 2,027.11
80 4,221 210,881.16 8,568.54
Power law ECGA 40 1,604 32,946.16 2,105.37
80 5,507 163,557.90 6,017.21
ECGA+MP 40 1,248 27,755.52 1,929.44
80 4,361 141,034.74 5,884.63
Uniform ECGA 40 1,346 17,228.80 1,489.44
80 3,479 58,308.04 3,411.61
ECGA+MP 40 2,181 30,446.76 2,411.81
80 5,598 100,540.08 5,535.96
from 8 to 16, our approach works better than the original ECGA in the exponential and power law scaled cases. Under this range of tournament sizes (8 to 16), the behavior of the proposed approach in uniformly scaled cases is relatively stable compared to that under higher selection pressure. This observation demonstrates that for a broad range of selection pressure, the improvement obtained by using the pruning mechanism can be expected in cases of limited linkage sensibility, while in cases for which linkage information is completely sensible, the overhead is relatively stable.