Algorithmic analysis of a basic evolutionary algorithm for continuous optimization

In practical optimization, applying evolutionary algorithms has nearly become a matter of course. Their theoretical analysis, however, is far behind practice. So far, theorems on the runtime are limited to discrete search spaces; results for continuous search spaces are limited to convergence theory or even rely on validation by experiments, which is unsatisfactory from a theoretical point of view. The simplest, or most basic, evolutionary algorithms use a population consisting of only one individual and use random mutations as the only search operator. Here the so-called (1+1) evolution strategy for minimization in R n is investigated when it uses isotropically distributed mutation vectors. In particular, so-called Gaussian mutations are analyzed when the so-called 1/5-rule is used for their adaptation. Obviously, a reasonable analysis must respect the function to be minimized, and furthermore, the runtime must be measured with respect to the approximation error. A first algorithmic analysis of how the runtime depends on n, the dimension of the search space, is presented. This analysis covers all unimodal functions that are monotone with respect to the distance from the optimum. It turns out that, in the scenario considered, Gaussian mutations in combination with the 1/5-rule indeed ensure asymptotically optimal runtime; namely, Θ(n) steps/function evaluations are necessary and sufficient to halve the approximation error.