doi:10.1016/j.tcs.2006.04.004 
Copyright © 2006 Elsevier B.V. All rights reserved.
How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms
Jens Jägersküpper
, 1, a, 
aDepartment of Computer Science 2, Dortmund University, 44221 Dortmund, Germany
Available online 22 May 2006.
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Abstract
The (1+1) evolution strategy (ES), a simple, mutation-based evolutionary algorithm for continuous optimization problems, is analyzed. In particular, we consider the most common type of mutations, namely Gaussian mutations, and the 
-rule for mutation adaptation, and we are interested in how the runtime/number of function evaluations to obtain a predefined reduction of the approximation error depends on the dimension of the search space.
The most discussed function in the area of ES is the so-called SPHERE-function given by SPHERE
with x
x
Ix (where 
is the identity matrix), which also has already been the subject of a runtime analysis. This analysis is extended to arbitrary positive definite quadratic forms that induce ellipsoidal fitness landscapes which are “close to being spherically symmetric”, showing that the order of the runtime does not change compared to SPHERE. Furthermore, certain positive definite quadratic forms 
with xxQx, where 
, inducing ellipsoidal fitness landscapes that are “far away from being spherically symmetric” are exemplarily investigated, namely
with
ξ=poly(n) such that
1/ξ→0 as
n→∞. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime to halve the approximation error is
Θ(ξ·n) compared to
Θ(n) for S
PHERE.
Keywords: Continuous optimization; Evolutionary direct search method; Probabilistic analysis of runtime
Mathematical subject codes: 68W40; 90C56; 90C59
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