Rigorous Runtime Analysis of the (1+1) ES: 1/5-Rule and Ellipsoidal Fitness Landscapes

We consider the (1+1) Evolution Strategy, a simple evolutionary algorithm for continuous optimization problems, using so-called Gaussian mutations and the 1/5-rule for the adaptation of the mutation strength. Here, the function f : R → R to be minimized is given by a quadratic form f(x) = xQx, where Q ∈ R is a positive definite diagonal matrix and x denotes the current search point. This is a natural extension of the well-known Sphere-function (Q = I). Thus, very simple unconstrained quadratic programs are investigated, and the question is addressed how Q effects the runtime. For this purpose, quadratic forms f(x) = ξ · ` x1 2 + · · ·+ xn/2 2 ́ + xn/2+1 2 + · · ·+ xn 2 with ξ = ω(1), i. e. 1/ξ → 0 as n → ∞, and ξ = poly(n) are investigated exemplarily. It is proved that the optimization very quickly stabilizes and that, subsequently, the runtime (defined as the number of f -evaluations) to halve the approximation error is Θ(ξ ·n). Though ξ ·n = poly(n), this result actually shows that the evolving search point indeed creeps along the “gentlest descent” of the ellipsoidal fitness landscape.