How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms

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 15-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:Rn→R:Rn→R with x↦x⊤Ixx↦x⊤Ix (where I∈Rn×nI∈Rn×n 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 f:Rn→Rf:Rn→R with x↦x⊤Qxx↦x⊤Qx, where Q∈Rn×nQ∈Rn×n, inducing ellipsoidal fitness landscapes that are “far away from being spherically symmetric” are exemplarily investigated, namely f(x)=ξ·x12+⋯+xn/22+xn/2+12+⋯+xn2with ξ=poly(n)ξ=poly(n) such that 1/ξ→01/ξ→0 as n→∞n→∞. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime to halve the approximation error is Θ(ξ·n)Θ(ξ·n) compared to Θ(n)Θ(n) for SPHERE.