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infeasible search
space
feasible search
space
infeasible search
space
feasible search
space
P
t-1
P
t
optimum at
vertex of boundaries
optimum at
vertex of boundaries
optimum of
unconstrained
objective function
optimum of
unconstrained
objective function
Fig. 7.5.
Concept of the TSES. The figure shows the effect of intermediary recombina-
tion. Individuals can reach a constrained optimum from both sides of the boundaries.
Left: The individuals with sex
o
(pink dots) enter the infeasible region. Right: After
intermediary recombination all individuals get closer to the optimum on a vertex of
the feasible region.
dots). The measurement
G
for the fulfillment of constraints has already been
defined in equation 7.2. By means of intermediary recombination, all individuals
get closer to the optimum of the problem, but still are found on opposite sides
of the boundaries between the feasible and the infeasible search space. For the
initialization feasible starting points are not required.
7.5.2
Modifications of the Basic Two Sexes Evolution Strategy
Several modifications of the basic concept of the TSES are necessary until the al-
gorithm provides successful results. The usual self-adaptation process effects an
explosion of the mean step sizes: The invasion far into the feasible search space
is rewarded with high fitness values for the
c
-individuals. Approaching the un-
constrained optimum of the objective function is rewarded for the
o
-individuals.
Modifications of the population ratios of the TSES aim at reducing the diversity
in the population to avoid the over-adaptation of the step sizes. Several experi-
ments with different sex ratios and selection operators on problem 2.40 lead to
the following heuristic modifications:
•
sex ratio and birth surplus with settings around
λ
o
≈
15 and
λ
c
≈
85, Kramer
& Schwefel [75] recommend (8 + 8
,
13 + 87) ,
•
two-step selection operator for the sex
c
, according to the metric penalty
function by Hoffmeister and Sprave [57]: Firstly, selection by fulfillment of
constraints, secondly, if enough feasible solutions exist, selection by objective
function,
•
introduction of a finite life span 1
<κ<
∞
, see section 2.2.2, for individuals
with the sex
c
.
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