Evolutionary Algorithms - Self-Adaptive Heuristics for Evolutionary Computation - page 19

with α 1 ,α 2 IR . Hence, a PSOA is an EA in which every individual of the

population survives and is recombined with the best individual of the population.

∈

Selection. In the equation 2.7 selection takes place: in the PSOA the best solu-
tion found so far is used for the construction of every new candidate solution.
This is similar to the survival of the best individual of the elitist plus-selection
scheme. But the elitist selection scheme is softened as not only the best solu-
tion b found so far is used, but also to some extend the best solution pb of a
particle's history. The comparison only reveals some structural similarities and
relationships between PSOAs and ES. But it is neither a prove for identical
convergence properties nor allows reasoning concerning the behavior on certain
function classes.

2.2 Evolution Strategies

ES are one of the four main variants of EAs besides GAs, EP and GP. They were
invented by Rechenberg and Schwefel in the middle of the sixties at the Technical
University of Berlin [114], [134]. ES can successfully be applied to engineering
problem domains. In this section the basic features of the so called ( μ/ρ + ,λ )-ES
are introduced.

The ( μ/ρ + ,λ )-ES

2.2.1

For a comprehensive introduction to ES see Beyer and Schwefel [18]. Here, the
most important features of the state of the art ( μ/ρ + ,λ )-ES 2 for continuous search
spaces are repeated. An ES uses a parent population with cardinality μ and an
offspring population with cardinality λ . Each individual consists of objective and
strategy variables a =( x 1 ,...,x N ,σ 1 ,...,σ N ,F ( x )), with problem dimension N .
At first, the individuals are initialized. The objective variables represent a poten-
tial solution to the problem whereas the strategy variables, which are step sizes
in the standard ( μ/ρ + ,λ )-ES, provide the variation operators with information
how to produce new results. During each generation λ individuals are produced in
the following way. In the first step ρ (1

μ ) parents are randomly selected
for reproduction. After recombination of strategy and objective variables, the ES
applies log-normal mutation to the step sizes σ =( σ 1 ,...,σ N ) and uncorrelated
Gaussian mutation to the objective variables, see also section 3.4.

≤ ρ ≤

x := x + z

(2.12)

z := ( σ 1 N 1 (0 , 1) ,...,σ N N N (0 , 1)) (2.13)

σ 1 e ( τ 1 N 1 (0 , 1) ,...,σ N e ( τ 1 N N (0 , 1)

σ := e ( τ 0 N 0 (0 , 1))

· (2.14)

2 The , -notation combines the notation for the selection schemes plus and comma
selection.