Self-Adaptive Genetic Algorithms with Simulated Binary Crossover

SBX performance
1e+10
1
1e-10
Best function value
Standard deviation in 15-th var
1e-20
0 500 1000 1500 2000 2500
Generation number
3000 3500 4000
Figure 13: Population-best objective function
value and the standard deviation ofx15variable
are shown for real-parameter GAs with SBX operator
on function F2-2.
Population best function value
1
1e-10
SBX
Self-Adaptive ES
1e-20
0 200 400 600
Generation number
800 1000
Figure 15: Population-best objective function
value for real-parameter GAs and self-adaptive
ESs are shown for the elliptic function F2-3.
Self-adaptive (15/15,100)-ES Performance
1e+10
1
1e-10
Best function value
Mutation strength in 15-th var
1e-20
0 500 1000 1500 2000 2500
Generation number
3000 3500 4000
Figure 14: Population-best objective function
value and the mutation strength forx15variable
are shown for self-adaptive (15/15,100)-ES on
function F2-2.
Population best function value
1
1e-10
Self-Adaptive ES
SBX
1e-20
0 200 400 600
Generation number
800 1000
Figure 16: Population-best objective function
value with real-parameter GAs and self-adaptive
ESs are shown for the multi-modal function F2-4.
Clearly, a better crossover operator handling the linkage issue but with the concept of probability distribution
to create children solutions is in order to solve such problems faster. One such implementation is
suggested in Section 6.
Besides the linkage issue discussed above, there is another mismatch between the GAs with SBX and
the correlated self-adaptive ESs used above. In GAs, a selection pressure of 3 (best solution in a population
gets a maximum of three copies after the tournament selection operation), whereas in the correlated selfadaptive
(4,100)-ESs, only 4 best solutions are picked from 100 children solutions. In order to alleviate
this mismatch in selection pressures, we use a different algorithm (we call ES-SBX) where a(;)-ES
is used, but children solutions are created from parent solutions only by the action of SBX operator
alone. Each parent (x(1;t)) mates with other parents in exactly=crossovers, everytime creating one
child solution using equation 5. Like the SBX operator used in GAs, every variable is crossed with a
probability 0.5. If a variable is not to be crossed, its value (x(i1;t)) in the first parent is directly passed
on to the child. No explicit mutation operator is used. The rest of the algorithm is exactly the same as
that in a(;)-ES. As shown in Figure 17, this new algorithm with (4,100)-ES-SBX is able to achieve
16