Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 52

4.1.7 Self-Adaptive Mutation for Binary Representations

From the point of view of GAs the mutation operator has always been seen as a
background operator [59]. Traditionally, the main variation operator for GAs is
the crossover operator. Beyer and Meyer-Nieberg point out that standard bit flip
mutation for binary representations introduces a bias [91]. Self-adaptation for
binary representations was introduced by Back [7], [6]. He encoded the mutation
rate as a bit-string. The self-adaptation works as follows: the mutation rate is
decoded to a value in the interval [0 , 1] and used to mutate its own bit-string
representation. The mutation rate is applied to the bit-strings of the objective
variables. An analysis of the asymptotic behavior neglecting recombination and
selectionshowedthatthemutationrateresultsinaMarkovchainwiththe
absorbing state zero. Back and Schutz [9] showed that the mutation rate encoded
as a bit-string is counterproductive for self-adaptation. They introduced a real-
coded mutation rate to overcome this problem. Smith and Fogarty [144] extended
the work of Back and examined the behavior of a ( μ +1)-GA.

4.1.8 Mutation Operators for Strategy Parameters

There are two main variants for the mutation of strategy parameters [16]: Meta-
EP mutation has up to now mainly been applied to the angle adaptation of the
correlated ES,

Λ := Λ + γ ·N (0 , 1) . (4.27)

The exogenous parameter γ is a learning parameter and defines the mutation
strength. A second operator is the log-normal mutation operator by Schwefel
[132]. It has already been introduced in this chapter.

Λ := Λ e γ·N (0 , 1) · (4.28)

InthecaseofstepsizesofES, γ is called τ . Variants like the extended log-normal
mutation exist, see section 4.1.2. Rechenberg [115] introduced a discrete mutation
operator using a symmetric two-point distribution, the two-point-operator

⎧
⎨
⎩

Λ (1 + γ )if u (0 , 1] ≤ 1 / 2

Λ := (4.29)

Λ/ (1 + γ ) f u (0 , 1] > 1 / 2 .

Again, parameter γ is a learning parameter, u is a uniformly distributed random
number.

4.2 The Biased Mutation Operator

In this section we introduce our BMO, starting with the main operator. The two
variants sphere and cube BMO are introduced subsequently.