The Theory of Evolution Strategies pp 113-142 | Cite as

# The (\(1\mathop + \limits_, \lambda \)) Quality Gain

## Abstract

The analysis of the quality gain for the (\(1\mathop + \limits_, \lambda \)) strategies can be carried out for a large number of fitness landscapes and mutation operators. This chapter will provide the theoretical background as well as first application examples. Firstly, an integral expression will be derived for \({\overline Q _{1\mathop {{\text{ }} + }\limits_, \lambda }}\) that can be properly approximated. The central problem in the derivation of analytical \({\overline Q _{1\mathop {{\text{ }} + }\limits_, \lambda }}\) formulae lies in the approximation of the distribution function of the mutation-induced fitness distribution *Q*(z). The Hermite polynomials are used for this purpose. Fourier coefficients occur in these derivations. They are expressed by statistical parameters, such as mean value, standard deviation, and the cumulants of kth order. These parameters are calculated for some special fitness landscapes, and the respective analytical \({\overline Q _{1\mathop {{\text{ }} + }\limits_, \lambda }}\) formulae are obtained, serving as application examples. These formulae are further compared with the results of ES simulations. The connection to the differential-geometric model is established. As a result, it is shown why two kinds of performance measures are necessary for the ES analysis.

## Keywords

Mutation Operator Sphere Model Progress Rate Hermite Polynomial Fitness Landscape## Preview

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