Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 56

O (2 N ) operations, but is based on the simpler Gaussian mutation. In compar-
ison to correlated mutation the BMO saves

N · ( N− 1)
2

N additional strategy
parameters. Chapter 4.5 reveals the power of the different approaches.

−

4.3 The Descent Direction Mutation Operator (DMO)

The BMO enables to bias mutations into beneficial directions. Under the as-
sumption that the optimum lies on the extension of the direction between the
center of consecutive generations, the idea arises to adjust the bias along this
direction. More precisely, the descent direction mutation operator (DMO) ad-
justs the bias to the descent direction of the centers of two successive generations.
Figure 4.5 illustrates the idea of the DMO. Let χ t be the center of the population
of generation t

μ

χ t = x i (4.39)

i =1

The normalized descent direction ξ of two successive population centers χ t and
χ t +1 is

χ t +1 − χ t

ξ = (4.40)

| χ t +1 − χ t |

Similar to the BMO, the DMO now becomes

z := ( σ 1 N 1 (0 , 1) + ξ 1 σ 1 ,...,σ N N N (0 , 1) + ξ N σ N ) (4.41)

=( σ 1 N 1 ( ξ 1 , 1) ,...,σ N N N ( ξ N , 1)) (4.42)

center of P t

P t

center of P t-1

P t-1

Fig. 4.5. Principle of the DMO in two dimensions: The vector reaching from the center
of population P t− 1 to the center of the population P t defines the descent direction of
the DMO, i.e. the shift of the center of the mutation ellipsoid