On the Behavior of the (1+1) Evolutionary Algorithm on Quadratic Pseudo-Boolean Functions∗

Evolutionary algorithms are randomized search heuristics, which are often used as function optimizers. In this paper the well-known (1+1) Evolutionary Algorithm ((1+1) EA) and its multistart variants are studied. Several results on the expected runtime of the (1+1) EA on linear or unimodal functions have already been presented by other authors. This paper is focused on quadratic pseudo-boolean functions, i. e., functions of degree 2. Whereas quadratic pseudoboolean functions without negative coefficients can be optimized efficiently by the (1+1) EA, there are quadratic functions for which the expected runtime is exponential. However, multistart variants of the (1+1) EA are very efficient for many of these functions. This is not the case with a special quadratic function for which the (1+1) EA requires exponential time with a probability exponentially close to 1. At last, some necessary conditions for exponential runtimes are examined, and an “easy” subclass within the class of quadratic functions is presented.