Principled Design and Runtime Analysis of Abstract Convex Evolutionary Search

Geometric crossover is a formal class of crossovers which includes many well-known recombination operators across representations. In previous work, it was shown that all evolutionary algorithms with geometric crossover (but with no mutation) do the same form of convex search regardless of the underlying representation, the specific selection mechanism, offspring distribution, search space, and problem at hand. Furthermore, it was suggested that the generalised convex search could perform well on generalised forms of concave and approximately concave fitness landscapes, regardless of the underlying space and representation. In this article, we deepen this line of enquiry and study the runtime of generalised convex search on concave fitness landscapes. This is a first step towards linking a geometric theory of representations and runtime analysis in the attempt to (i) set the basis for a more general, unified approach for the runtime analysis of evolutionary algorithms across representations, and (ii) identify the essential matching features of evolutionary search behaviour and landscape topography that cause polynomial performance. We present a general runtime result that can be systematically instantiated to specific search spaces and representations, and present its specifications to three search spaces. As a corollary, we obtain that the convex search algorithm optimises LeadingOnes in O (n log n) fitness evaluations, which is faster than all unbiased unary black-box algorithms.