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constraint boundary
infeasible search space
direction
to optimum
3
s
s
2
lines of constant fitness
s
1
feasible search space
Fig. 7.1. Success rates at the boundary of the feasible search space. Three cases have
to be considered, i.e. 1. σ<d ,2. σ>d , σ<s and 3. σ>d , σ>s . The bold circular
arcs are the regions where successful mutations are produced.
and
σ t +1 = γσ t
if f ( X t + σ t Z t ) <f ( X t )
g ( X t + σ t Z t )=0
(7.5)
γ 1 σ t otherwise
with step size σ t and mutation parameter γ> 1. The function g measures the
constraint violation. Each random vector Z t ,t
0 is independent and identically
distributed in the following way: We assume that mutations σ t Z t are produced
on the edge of the circle around X t with radius σ t . When a successful mutation
is produced, the step length σ t is increased and decreased otherwise. We are
interested in the development of the step size σ t and the distance d t to the
constraint boundary. For the sake of better readability we write σ instead of σ t
and d instead of d t where possible. In the following lemma 7.1 we analyze the
success probabilities for the three cases.
Lemma 7.1. Let ( p s ) be the success probability for individual X t of the (1+1)-
EA, with step size σ and distance d to the constraint boundary. Then it holds
( p s ) σ<d =1 / 2 and ( p s ) σ>d < 1 / 2 .Ford/σ
0 it holds ( p s ) σ>d
β/ (2 π ) .
 
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