Understanding the relationship between a search algorithm and the space of problems is a fundamental issue in the optimization field. In this paper, we lay the foundations to elaborate taxonomies of problems under estimation of distribution algorithms (EDAs). By using an infinite population model and assuming that the selection operator is based on the rank of the solutions, we group optimization problems according to the behavior of the EDA. Throughout the definition of an equivalence relation between functions it is possible to partition the space of problems in equivalence classes in which the algorithm has the same behavior. We show that only the probabilistic model is able to generate different partitions of the set of possible problems and hence, it predetermines the number of different behaviors that the algorithm can exhibit. As a natural consequence of our definitions, all the objective functions are in the same equivalence class when the algorithm does not impose restrictions to the probabilistic model. The taxonomy of problems, which is also valid for finite populations, is studied in depth for a simple EDA that considers independence among the variables of the problem. We provide the sufficient and necessary condition to decide the equivalence between functions and then we develop the operators to describe and count the members of a class. In addition, we show the intrinsic relation between univariate EDAs and the neighborhood system induced by the Hamming distance by proving that all the functions in the same class have the same number of local optima and that they are in the same ranking positions. Finally, we carry out numerical simulations in order to analyze the different behaviors that the algorithm can exhibit for the functions defined over the search space .

Estimation of distribution algorithms (EDAs) that use marginal product model factorizations have been widely applied to a broad range of mainly binary optimization problems. In this paper, we introduce the affinity propagation EDA (AffEDA) which learns a marginal product model by clustering a matrix of mutual information learned from the data using a very efficient message-passing algorithm known as affinity propagation. The introduced algorithm is tested on a set of binary and nonbinary decomposable functions and using a hard combinatorial class of problem known as the HP protein model. The results show that the algorithm is a very efficient alternative to other EDAs that use marginal product model factorizations such as the extended compact genetic algorithm (ECGA) and improves the quality of the results achieved by ECGA when the cardinality of the variables is increased.

The question of finding feasible ways for estimating probability distributions is one of the main challenges for Estimation of Distribution Algorithms (EDAs). To estimate the distribution of the selected solutions, EDAs use factorizations constructed according to graphical models. The class of factorizations that can be obtained from these probability models is highly constrained. Expanding the class of factorizations that could be employed for probability approximation is a necessary step for the conception of more robust EDAs. In this paper we introduce a method for learning a more general class of probability factorizations. The method combines a reformulation of a probability approximation procedure known in statistical physics as the Kikuchi approximation of energy, with a novel approach for finding graph decompositions. We present the Markov Network Estimation of Distribution Algorithm (MN-EDA), an EDA that uses Kikuchi approximations to estimate the distribution, and Gibbs Sampling (GS) to generate new points. A systematic empirical evaluation of MN-EDA is done in comparison with different Bayesian network based EDAs. From our experiments we conclude that the algorithm can outperform other EDAs that use traditional methods of probability approximation in the optimization of functions with strong interactions among their variables.