Constraint Handling Heuristics for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 115

The state transitions between the two states Γ 1 and Γ 2 are analyzed in the
following.

• Γ 1 Γ 1 . The probability for a success is p , which is relatively low, see

lemma 7.1. A success results in a step size increase σ t +1 = γσ t . A successful
mutation may lie arbitrarily close to the constraint boundary ( d/σ

→

0) and
therefore decrease the success probability p rapidly. But a success may also
lead to an increase of the distance to the constraint boundary, at most by
σ tan β . The constrained case is not left, if the step size increase is higher
than the distance increase to guarantee d/σ < 1:

→

d + σ tan β
σγ

d
σ +tan β<γ

< 1 ≡ (7.15)

As d/σ < 1itmusthold γ> 1 + (tan β ) to fulfill the above condition for the
proof of step size reduction. So, the probability for staying in state Γ 1 is

P ( Γ 1 → Γ 1 | Γ 1 )= p, σ t +1 = γσ t . (7.16)

• Γ 2 → Γ 1 : The probability for a success if the last step was a failure is again p ,

the step size is increased and the constrained case is not left for γ> 1+tan β .

P ( Γ 2 → Γ 1 | Γ 2 )= p, σ t +1 = γσ t . (7.17)

• Γ 1 → p . It results in a step decrease

σ t +1 = γ − 1 σ t . If the step decrease leads to d/σ t +1 > 1, the constraint bound-
ary is left. In this case step size decrease and increase occur with the same
probability and the expected change of σ becomes E ( γ )= γ

Γ 2 : The probability for a failure is 1 −

γ − 1 =1.But
the constraint boundary will be reached again within the following steps.
Hence, we summarize

·

σ t +1 = γ − 1 σ t . P ( Γ 1 → Γ 2 | Γ 1 )=1 − p, (7.18)

• Γ 2 → Γ 2 : Similar to transition Γ 1 → Γ 2 the probability to stay in the state

Γ 2 is the probability 1 − p for a failure.

σ t +1 = γ − 1 σ t . P ( Γ 2 → Γ 2 | Γ 2 )=1 − p, (7.19)

This yields the following state transition probability matrix T for states Γ 1 and
Γ 2 :

T = p

(1 − p )

(7.20)

p (1 − p )

It is worth to mention that in the case of a success with an overwhelming prob-
ability of p =(1 − β/ (2 π )) → 1for β → 0, the distance d to the constraint