Absorption in Model-based Search Algorithms for Combinatorial Optimization

Model-based search is an abstract framework that unifies the main features of a large class of heuristic procedures for combinatorial optimization, it includes ant algorithms, cross entropy and estimation of distribution algorithms. Properties shown for the model-based search therefore apply to all these algorithms. A crucial parameter for the long term behavior of model-based search is the learning rate that controls the update of the model when new information from samples is available. Often this rate is kept constant over time. We show that in this case after finitely many iterations, all model-based search algorithms will be absorbed into a state where all samples consist of a single solution only. Moreover, it cannot be guaranteed that this solution is optimal, at least not when the optimal solution is unique.