Biased Mutation for Evolution Strategies - Self-Adaptive Heuristics for Evolutionary Computation - page 50

4.1.5 Cauchy Mutation

Instead of the Gaussian mutation the so called fast evolutionary programming
(FEP) [163] makes use of the Cauchy distribution. The one-dimensional Cauchy
density function is defined by

p ( x )= 1
π

t
t 2 + x 2 ,

−∞ <x< ∞ (4.15)

with a scale parameter t> 0 [163]. The corresponding distribution is

π arctan x

.

1

2 + 1

P = (4.16)

t

The mutation of an individual x follows the principle of the uncorrelated Gaus-
sian mutation:

x := x + z (4.17)

with the mutation

z := σ ( P 1 ( t ) ,..., P N ( t )) (4.18)

in the case of one step size and the mutation

z := ( σ 1 P 1 ( t ) ,...,σ N P N ( t )) (4.19)

in the case of N step sizes.

4.1.6 Covariance Matrix Adaptation (CMA)

The covariance matrix adaptation evolution strategy (CMA-ES) by Hansen
and Ostermeier [54] is a successful evolutionary optimization method for real-
parameter optimization of non-linear functions. The basis of this approach is a
derandomized step-size adaptation. The mutation distribution is altered deter-
ministically so that the probability to reproduce steps in the search space, which
have led to the current population, is increased. The CMA-ES makes use of a
population of search points, which are sampled with a normal distribution, see
equation 4.20. The idea of the CMA is similar to quasi-Newton [52]: approxi-
mation of the inverse Hessian matrix, i.e. fitting the search distribution to the
objective function curvature. The complete algorithm and the full explanation of
all symbols can be found in the tutorial of Hansen [52]. Here we shortly repeat the
core algorithm that can also be found in the tutorial of Hansen [52], pages 22-24.

We use the following symbols:

thesampleof λ search points x ( t +1)
k

IR N • ∈ for k =1 ,...,λ of generation

t +1,