A mathematical analysis of EDAs with distance-based exponential models

Estimation of Distribution Algorithms have been successfully used to solve permutation-based Combinatorial Optimization Problems. In this case, the algorithms use probabilistic models specifically designed for codifying probability distributions over permutation spaces. One class of these probability models are distance-based exponential models, and one example of this class is the Mallows model. In spite of its practical success, the theoretical analysis of Estimation of Distribution Algorithms for permutation-based Combinatorial Optimization Problems has not been developed as extensively as it has been for binary problems. With this motivation, this paper presents a first mathematical analysis of the convergence behavior of Estimation of Distribution Algorithms based on Mallows models. The model removes the randomness of the algorithm in order to associate a dynamical system to it. Several scenarios of increasing complexity with different fitness functions and initial probability distributions are analyzed. The obtained results show: a) the strong dependence of the final results on the initial population, and b) the possibility to converge to non-degenerate distributions even in very simple scenarios, which has not been reported before in the literature.This research has been partially supported by Spanish Ministry of Science and Innovation through the projects PID2019-104966GB-I00/AEI/10.13039/501100011033, PID2019-104933GB-I00/AEI/10.13039/501100011033, PID2019-106453GA-I00/AEI/10.13039/501100011033 and BCAM Severo Ochoa accreditation SEV-2017-0718; and by the Basque Government through the program BERC 2022-2025 and the projects IT1504-22 and IT1494-22; and by UPV/EHU through the project GIU20/054. Imanol holds a grant from the Department of Education of the Basque Government (PRE_2021_2_0224). Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature

A mathematical analysis of edas with distance-based exponential models

Estimation of Distribution Algorithms have been successfully used for solving many combinatorial optimization problems. One type of problems in which Estimation of Distribution Algorithms have presented strong competitive results are permutation-based combinatorial optimization problems. In this case, the algorithms use probabilistic models specifically designed for codifying probability distributions over permutation spaces. One class of these probability models is distance-based exponential models, and one example of this class is the Mallows model. In spite of the practical success, the theoretical analysis of Estimation of Distribution Algorithms for permutation-based combinatorial optimization problems has not been extensively developed. With this motivation, this paper presents a first mathematical analysis of the convergence behavior of Estimation of Distribution Algorithms based on the Mallows model by using an infinite population to associate a dynamical system to the algorithm. Several scenarios, with different fitness functions and initial probability distributions of increasing complexity, are analyzed obtaining unexpected results in some cases

A mathematical analysis of EDAs with distance-based exponential models

Estimation of Distribution Algorithms have been successfully used to solve permutation-based Combinatorial Optimization Problems. In this case, the algorithms use probabilistic models specifically designed for codifying probability distributions over permutation spaces. One class of these probability models are distance-based exponential models, and one example of this class is the Mallows model. In spite of its practical success, the theoretical analysis of Estimation of Distribution Algorithms for permutation-based Combinatorial Optimization Problems has not been developed as extensively as it has been for binary problems. With this motivation, this paper presents a first mathematical analysis of the convergence behavior of Estimation of Distribution Algorithms based on Mallows models. The model removes the randomness of the algorithm in order to associate a dynamical system to it. Several scenarios of increasing complexity with different fitness functions and initial probability distributions are analyzed. The obtained results show: a) the strong dependence of the final results on the initial population, and b) the possibility to converge to non-degenerate distributions even in very simple scenarios, which has not been reported before in the literature.Spanish Ministry of Science and Innovation through the projects PID2019-104966GB-I00/AEI/10.13039/501100011033, PID2019-104933GB-I00/AEI/10.13039/501100011033, PID2019-106453GA-I00/AEI/10.13039/501100011033 and BCAM Severo Ochoa accreditation SEV-2017-0718; and by the Basque Government through the program BERC 2022-2025 and the projects IT1504-22 and IT1494-22; and by UPV/EHU through the project GIU20/054. Imanol holds a grant from the Department of Education of the Basque Government (PRE_2021_2_0224)

Contributions to the mathematical modeling of estimation of distribution algorithms and pseudo-boolean functions

134 p.Maximice o minimice una función objetivo definida sobre un espacio discreto. Dado que la mayoría de dichos problemas no pueden ser resueltos mediante una búsqueda exhaustiva, su resolución se aproxima frecuentemente mediante algoritmos heurísticos. Sin embargo, no existe ningún algoritmo que se comporte mejor que el resto de algoritmos para resolver todas las instancias de cualquier problema. Por ello, el objetivo ideal es, dado una instancia de un problema, saber cuál es el algoritmo cuya resoluciones más eficiente. Las dos líneas principales de investigación para lograr dicho objetivo son estudiar las definiciones de los problemas y las posibles instancias que cada problema puede generar y el estudio delos diseños y características de los algoritmos. En esta tesis, se han tratado ambas lineas. Por un lado,hemos estudiado las funciones pseudo-Booleanas y varios problemas binarios específicos. Por otro lado,se ha presentado un modelado matemático para estudiar Algoritmos de Estimación de Distribuciones diseñados para resolver problemas basados en permutaciones. La principal motivación ha sido seguir progresando en este campo para comprender mejor las relaciones entre los Problemas de Optimización Combinatoria y los algoritmos de optimización