Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 113

Proof. The analysis of the probabilities for changing through the states is based
on a success rates analysis of the circle model for the three mentioned cases, see
figure 7.1. In our model the success rate p s is the relation between the length of
the circular arc l =2 πrα/ (2 π ) and the circumference 2 πr :

p s = α/ (2 π ) . (7.6)

The EA starts its run in the feasible part of the search space, σ<d .Asthe
fitness function is linear and the constraint boundary does not cut off the circle,
the angle over the circular arc with successful mutations is α = π . Hence,

( p s ) σ<d =1 / 2 . (7.7)

The step sizes are increased with probability 1 / 2 and decreased with probabi-
lity 1 / 2. The constrained case is much more important. For σ>d we have to
distinguish two cases:
1. σ<s , i.e. the circle cuts its contour line within the feasible region. Distance
s is the segment of the contour line between x and the constraint boundary
with s =( d/ sin β ). The success area on the edge of the circle is the circular
arc over α and the circular arc in the opposite direction. The success prob-
ability ( p s ) σ>d,σ<s can also be expressed with the help of ϑ

( p s ) σ>d,σ<s =1 / 2 − ϑ/π (7.8)

The triangle with a right angle yields cos ( ϑ )= d/σ and the success proba-
bility becomes

( p s ) σ>d,σ<s =1 / 2 − arccos( d/σ ) /π (7.9)

As 0 < ( d/σ ) < 1, it holds 0 < arccos( d/σ ) <π/ 2 and the desired property

( p s ) σ>d,σ<s < 1 / 2 . (7.10)

The success probability ( p s ) σ>d,σ<s is comparatively small. Analyzing the
asymptotic behavior we get

( p s ) σ>d,σ<s → 0for d/σ → 0 , (7.11)

i.e. for small distances d the success proba-

bility becomes 0. But one have to keep in mind that a small d/σ implies a
small angle β as σ<s .Itisobviousthat( p s ) σ>d,σ<s ≥

→ 0 or huge step sizes σ →∞

( p s ) σ>d,σ>s . Hence
the analysis of the case σ>s is more interesting as it considers β .