Limitations of Mutative sigma-Self-Adaptation 
         on Linear Fitness Functions

               Nikolaus Hansen

 This paper investigates sigma-self-adaptation for real valued
 evolutionary algorithms on linear fitness functions. We identify
 log(sigma) as key quantity to understand strategy behavior.
 Identifying the bias of mutation, recombination, and selection on
 log(sigma) is sufficient to explain strategy behavior in many cases,
 even in a non-linear or noisy environment.  On a linear fitness
 function, if intermediate multi-recombination is applied on the
 object parameters, the i-th best and the i-th worst individual have
 the same sigma-distribution. Consequently, the correlation between
 fitness and step-size is zero.  Assuming additionally that
 sigma-changes due to mutation and recombination are unbiased, and
 (mu,lambda)-truncation selection, where mu=lambda/2, then
 sigma-self-adaptation is unable to enlarge the step-size. Experiments
 show the relevance of the given assumptions. 

In Evolutionary Computation, 14(3), 2006.

http://www.mitpressjournals.org/toc/evco/14/3