Population Sizing
5.1 Population size for DSMGA-II
At first, the initial supply of each optimum in each subproblem is increased by local search.
Thus, the initial supply must be modify by including the concept of local search. The number of supply patterns after local search is denoted as s′. For example, the initial supply of 11111 on a 5 bits trap function is215 before local search. However, the proportion of 11111 in the subproblem becomes 214 after GHC.
Second, the viewpoint of cross-competition needs to be taken into consideration. Cross-competition may ruins the optimum which is produced in the initial population. With the probability of cross-competition, Pc, the supply must be modified by emerging the cross-competition, and it can be expressed as:
s′ · (1 − Pc) + (s′)2· Pc (5.2)
To verify the population-sizing model, this thesis performs the adaptive sweep pro-cedure on DSMGA-II. The propro-cedure changes population size adaptively to search ap-propriate population size for different problems. The requirement of this procedure is 10 consecutive successful hits. To include concept of this procedure, we need to make some changes for the model. We assume that the success rate of 10-hits is 50%, and the 10 consecutive successful hits is equivalent to a 1-hit with success rate 93%. We modify the above-mentioned Equation 5.4 to
On concatenated trap, due to the acceptance criterion in the back mixing, the optimum can be eliminated if cross competition happens twice in the same two subproblem. With
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Figure 5.1: Required population size for concatenated trap. This experiment adopted an adaptive sweeping procedure to find the fewest function evaluation, and 10 consecutive successful hits is also considered.
this probability, Pc2, it is necessary for population to provide (s′)2 initial supply of the optimum. Therefore, the required population size is
Pc2· (s′)2+(
1− Pc2
)· s′ (5.4)
Also, the model is extended to folded trap and NK-landscape with non-overlapping.
We ignore the possibility of building incorrect model, so there is no cross-competition on folded trap function. We only consider the viewpoint of the initial supply for folded trap problems. For NK-landscape with non-overlapping problem, we modify our estimation by Popt = 0.1875, which is estimated in Section 3.3.
The empirical results are shown in Figures 5.1, 5.2 and 5.3. To show the effect of cross-competition, the blue line is the population sizing model we proposed in this thesis, and the green line is the supply model with local search. We can find that the theoretical required population size is accurate and empirically tight for concatenated trap and folded trap. Owing to the equal criterion, the difference between these two lines is much slight.
However, the errors of NK-landscape with non-overlapping become greater as the size of
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Figure 5.2: Required population size for folded trap. This experiment adopted an adaptive sweeping procedure to find the fewest function evaluation, and 10 consecutive success-ful hits is also considered. Note that the probability of cross-competition is zero on this problem, so only blue line is shown in this figure.
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Figure 5.3: Required population size for NK-landscape with non-overlapping. This ex-periment adopted an adaptive sweeping procedure to find the fewest function evaluation, and 10 consecutive successful hits is also considered.
problem sizes, the probability of ruining the optimum’s initial supply can not be estimated accurately.
On problems with overlapping structures in ring topology, the initial supply is insuffi-cient for optimizing cyclic trap and 0-1 trap. In the Section 4.1, the optimum 11111 can be eliminated in situation 1 if the optimal fragment is not connected with the adjacent frag-ment. Thus, it needs (s′)2initial supply with the probability of cross-competition. Based
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Cyclic Trap
Population Size
Problem Size
Supply model Theory with CC Experiment
Figure 5.4: Required population size for cyclic trap. This experiment adopted an adaptive sweeping procedure to find the fewest function evaluation, and 10 consecutive successful hits is also considered.
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0−1 Trap
Population Size
Problem Size
Supply model Theory with CC Experiment
Figure 5.5: Required population size for 0-1 trap. This experiment adopted an adaptive sweeping procedure to find the fewest function evaluation, and 10 consecutive successful hits is also considered.
on the analyses above, the probability, Pc, is estimated by
Pc=(
P11111· P000002
) (5.5)
Based on the probability, the required population size for solving cyclic trap is derived.
Figure 5.4 shows comparison of theoretical value and experimental value. We can find that the difference between them becomes greater as the size of subproblems larger. Since P11111and P00000are changed during the mixing, Pccannot be estimated very accurately.
For 0-1 trap, the patterns in 1-trap is the global optimal fragments only if the two adjacent patterns are both the global optimal, that is to say, 11111 in 1-trap is always con-nected with 11111 in 0-trap. Thus, the situation 1 in Figure 4.6 cannot ruin the initial optimal fragments. We only take the supply of 1-trap into consideration. Based on the es-timation of P11111,1-trap = 0.094479 in Section 4.2, the population sizing model is derived and the result is shown in Figure 5.5. Also, the possibility of building incorrect model in this section since we only consider the viewpoint of initial supply but not the viewpoint of model building.
In the empirical results, our population sizing model is not accurate enough on problem with the phenomenon of cross-competition. However, the theoretical value is closer with empirical value than supply model.