Open Access
Comprehensive characterization of the behaviors of estimation of distribution algorithms
(Elsevier, 2015-04-20) Echegoyen Arruti, Carlos; Santana, Roberto; Mendiburu, Alexander; Lozano, José Antonio; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Estimation of distribution algorithms (EDAs) are a successful example of how to use machine learning techniques for designing robust and efficient heuristic search algorithms. Understanding the relationship between EDAs and the space of optimization problems is a fundamental issue for the successful application of this type of algorithms. A step forward in this matter is to create a taxonomy of optimization problems according to the different behaviors that an EDA can exhibit. This paper substantially extends previous work in the proposal of a taxonomy of problems for univariate EDAs, mainly by generalizing those results to EDAs that are able to deal with multivariate dependences among the variables of the problem. Through the definition of an equivalence relation between functions, it is possible to partition the space of problems into equivalence classes in which the algorithm has the same behavior. We provide a sufficient and necessary condition to determine the equivalence between functions. This condition is based on a set of matrices which provides a novel encoding of the relationship between the function and the probabilistic model used by the algorithm. The description of the equivalent functions belonging to a class is studied in depth for EDAs whose probabilistic model is given by a chordal Markov network. Assuming this class of factorization, we unveil the intrinsic connection between the behaviors of EDAs and neighborhood systems defined over the search space. In addition, we carry out numerical simulations that effectively reveal the different behaviors of EDAs for the injective functions defined over the search space {0,1}^3 . Finally, we provide a novel approach to extend the analysis of equivalence classes to non-injective functions.
Open Access
Mateda-2.0: estimation of distribution algorithms in MATLAB
(Foundation for Open Access Statistics, 2010-07-26) Larrañaga, Pedro; Santana, Roberto; Bielza, Concha; Lozano, José Antonio; Echegoyen Arruti, Carlos; Mendiburu, Alexander; Armañanzas, Rubén; Shakya, Siddartha; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
This paper describes Mateda-2.0, a MATLAB package for estimation of distribution algorithms (EDAs). This package can be used to solve single and multi-objective discrete and continuous optimization problems using EDAs based on undirected and directed probabilistic graphical models. The implementation contains several methods commonly employed by EDAs. It is also conceived as an open package to allow users to incorporate different combinations of selection, learning, sampling, and local search procedures. Additionally, it includes methods to extract, process and visualize the structures learned by the probabilistic models. This way, it can unveil previously unknown information about the optimization problem domain. Mateda-2.0 also incorporates a module for creating and validating function models based on the probabilistic models learned by EDAs.
Open Access
On the taxonomy of optimization problems under estimation of distribution algorithms
(MIT Press Journals, 2013-09-13) Echegoyen Arruti, Carlos; Mendiburu, Alexander; Santana, Roberto; Lozano, José Antonio; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
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 {0,1}^3

Echegoyen Arruti, Carlos

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