Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

Population standard deviation in x values
1
1e-10
1e-20
1st variable
SBX
30th variable
15th variable
1e-30
1e-30
1e-40
1e-40
0 500 1000
F2-4:MinimizeNXi=1ix2i+10(1?cos(xi):
1500 2000
0 500 1000 1500 2000
Generation Number
Generation Number
Figure 12: Mutation strengths for variables x1,x15, andx30are shown with self-adaptive
(10/10,100)-ES for the function F2-1.
(22)
Figure 11: Population standard deviation in variablesx1,x15,
andx30are shown with realparameter
GAs with the SBX operator for the
function F2-1.
Mutation strengths for variables
1
1e-10
1e-20
1st variable
(10/10,100)-ES
30th variable
15th variable
Figure 15 shows that the (10/10,100)-ES with self-adaptation is able to converge near the optimum. The
same is true with real-parameter GAs with SBX, but the rate of convergence is smaller compared to that
of the self-adaptive ES.
We now construct a multi-modal test function by adding a cosine term as follows:
MinimizeNXi=10@iXj=1xj1A2:
Figure 16 shows the performance of real-parameter GAs and self-adaptive ESs with identical parameter
setting as on the elliptic function F2-3. Now, self-adaptive ES seems not able to converge to the global
attractor (allxi=0having function value equal to zero), for the same reason as that described in section
5.1.4—the initial mutation strength being too large to keep the population within the global basin.
When all initial mutation strengths reduced to one-tenth of their original values, non-isotropic ESs had no
trouble in finding the true optimum with increased precision. We observe no such difficulty in GAs with
SBX operator and a performance similar to that in Function F2-3 is observed here.
5.3 Correlated function
Next, we consider a function where pair-wise interactions of variables exist. The following Schwefel’s
function is chosen:
F3-1: (23)
The population is initialized atxi2[?1:0;1:0]. Figure 17 shows the performance of real-parameter GAs
with SBX (=1) and tournament size 3, non-isotropic (4,100)-ES, and correlated (4,100)-ES. Although
all methods have been able to find increased precision in obtained solutions, the rate of progress for the
real-parameter GAs with SBX is slower compared to that of the correlated self-adaptive ESs. However,
GAs with SBX makes a steady progress towards the optimum and performs better than the non-isotropic
(4,100)-ES. The reason for the slow progress of real-parameter GAs is as follows. In the SBX operator,
variable-by-variable crossover is used with a probability of 0.5. Correlations (or linkage) among the variables
are not explicitly considered in this version of SBX. Although some such information comes via
the population diversity in variables, it is not enough to progress faster towards the optimum in this problem
compared to the correlated ES, where pairwise interactions among variables are explicitly considered.
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