IEEE Computational Intelligence Magazine - August 2023 - 64

FIGURE 2 For the same initial population, the evolutionary information onf8 in a single run: (a) The times of ineffective co-evolution and (b) the best
fitness.
immediately re-diversify the convergent
subpopulations but with a probability
[see (8)]. The subpopulations responsible
for optimizing the interrelated variables
can be re-diversified one by one. For a
variable, when the values of its interrelated
variables are modified by the subpopulation's
evolution, CCFR2-AEPD can
immediately turn a false global optimum
to be a local optimum. The best overall
objective value can be improved by the
subpopulation re-diversification. The
contribution made by the subpopulation
re-diversification will not be zero. These
subpopulations can be selected to
undergo evolution in the future.
Steps 7 to 13 in Algorithm 2 were
replaced with S fPiji ¼ arg maxZig.
The new algorithm with this replacement
was named CCFR2-AEPD-Test,
where the convergent or stagnant subpopulation
with the greatest contribution
is immediately re-diversified. If a
convergent subpopulation, which is
trapped in a false global optimum, is
selected to undergo evolution in the
following co-evolutionary cycle, it will
be re-diversified and evolves. Because
the subpopulation has been trapped in a
false global optimum, it is ineffective
that the subpopulation spends computational
resources for its re-diversification
and co-evolution. The ineffective coevolution
cannot improve the best
overall objective value, which results in
the fact that the subpopulation's
contribution will be wrongly set to zero
even though it has not found the real
global optimum.
Figure 2 shows the evolutionary
information of CCFR2-AEPD and
CCFR2-AEPD-Test on f8 in a single
run. It can be seen in Figure 2(a) that
CCFR2-AEPD-Test took ineffective
co-evolution much more times than
CCFR2-AEPD as the evolution progressed.
CCFR2-AEPD-Test took
totally 156 times, while CCFR2-AEPD
took totally only six times. The experimental
results show that the ineffective
co-evolution resulted in the wrong contribution
evaluations for 131 subpopulations
in CCFR2-AEPD-Test and for
six subpopulations in CCFR2-AEPD.
CCFR2-AEPD could make better use
of computational resources than
CCFR2-AEPD-Test. CCFR2-AEPD
converged faster than CCFR2-AEPDTest
in the later period of the single
run (see Figure 2(b)). Re-diversifying a
subpopulation with a probability but
not immediately is beneficial for solving
the optimization problem where
the nonseparable variables are wrongly
identified as separable variables.
f11 is a partially separable function
with 20 subcomponents of nonseparable
variables. ERDG can correctly
group the 20 subcomponents of nonseparable
variables (see Table I). Figure 3
shows the evolutionary information on
f11 in a single run. It can be seen in
64 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2023
Figure 3(a) that CCFR2 without population
re-diversification obtained the
global optimum in all dimensions at the
end of the single run, while CCFR2AEPD
obtained the global optimum in
907 out of1000 dimensions. The population
re-diversification would disturb
the normal evolutionary process of
CCFR2. This would decrease the convergence
speed of the algorithm. For
the optimization problem CCFR2 can
efficiently solve, it can reduce the negative
effect ofthe population re-diversification
on the convergence speed that
the convergent and stagnant subpopulations
are regenerated with a probability
but not immediately. Figure 3(b) shows
the convergence of CCFR2-AEPD on
optimizing the 20th subcomponent of
nonseparable variables in the single run.
During the run, subpopulation P20,
which is responsible for optimizing the
20th subcomponent of nonseparable
variables, was re-diversified four times.
The best overall objective value became
better after 105, 227, and 314 evolutionary
generations due to the first, second,
and third population rediversifications,
respectively. For different
subpopulation re-diversifications,
P20 spent different numbers of evolutionary
generations improving the best
overall objective value. This phenomenon
was also observed in other subpopulations'
evolution. The experimental
study indicated that it is necessary that

IEEE Computational Intelligence Magazine - August 2023

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