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We introduce an evolutionary algorithm called recombinator-$k$-means foroptimizing the highly non-convex kmeans problem. Its defining feature is thatits crossover step involves all the members of the current generation,stochastically recombining them with a repurposed variant of the $k$-means++seeding algorithm. The recombination also uses a reweighting mechanism thatrealizes a progressively sharper stochastic selection policy and ensures thatthe population eventually coalesces into a single solution. We compare thisscheme with state-of-the-art alternative, a more standard genetic algorithmwith deterministic pairwise-nearest-neighbor crossover and an elitist selectionpolicy, of which we also provide an augmented and efficient implementation.Extensive tests on large and challenging datasets (both synthetic andreal-word) show that for fixed population sizes recombinator-$k$-means isgenerally superior in terms of the optimization objective, at the cost of amore expensive crossover step. When adjusting the population sizes of the twoalgorithms to match their running times, we find that for short times the(augmented) pairwise-nearest-neighbor method is always superior, while atlonger times recombinator-$k$-means will match it and, on the most difficultexamples, take over. We conclude that the reweighted whole-populationrecombination is more costly, but generally better at escaping local minima.Moreover, it is algorithmically simpler and more general (it could be appliedeven to $k$-medians or $k$-medoids, for example). Our implementations arepublicly available at\href{https://github.com/carlobaldassi/RecombinatorKMeans.jl}{https://github.com/carlobaldassi/RecombinatorKMeans.jl}.

Recombinator-k-Means: An Evolutionary Algorithm That Exploits k-Means++ for Recombination