This paper presents a theoretical study of the behaviour of the Univariate Marginal Distribution Algorithm for continuous domains (UMDAc ) in dimension n. To this end, the algorithm with tournament selection is modelled mathematically, assuming an infinite number of tournaments. The mathematical model is then used to study the algorithm's behaviour in the minimization of linear functions L(x) = a0 + i=1 a i x i and quadratic function Q(x) = i , with x = (x1 , . . . , xn ) and a i IR, i = 0, 1, . . . , n. Linear functions are used to model the algorithm when far from the optimum, while quadratic function is used to analyze the algorithm when near the optimum. The analysis shows that the algorithm performs poorly in the linear function L1 (x) = i=1 x i . In the case of quadratic function Q(x) the algorithm 's behaviour was analyzed for certain particular dimensions. After taking into account some simplifications we can conclude that when the algorithm starts near the optimum, UMDAc is able to reach it. Moreover the speed of convergence to the optimum decreases as the dimension increases.

Abstract. Recently, there has been a growing interest in developing evolutionary algorithms based on probabilistic modeling. In this scheme, the offspring population is generated according to the estimated probability density model of the parent instead of using recombination and mutation operators. In this paper, we have proposed probabilistic model-building genetic algorithms (PMBGAs) in permutation representation domain using edge histogram based sampling algorithms (EHBSAs). Two types of sampling algorithms, without template (EHBSA/WO) and with template (EHBSA/WT), are presented. The results were tested in the TSP and showed EHBSA/WT worked fairly well with a small population size in the test problems used. It also worked better than well-known traditional two-parent recombination operators. 1

this paper the convergence behavior of the Population Based Incremental Learning algorithm (PBIL) is analyzed using discrete dynamical systems. A discrete dynamical system is associated with the PBIL algorithm. We demonstrate that the behavior of the PBIL algorithm follows the iterates of the discrete dynamical system for a long time when the parameter # is near zero. We show that all the points of the search space are fixed points of the dynamical system, and that the local optimum points for the function to optimize coincide with the stable fixed points. Hence it can be deduced that the PBIL algorithm converges to the global optimum in unimodal functions. 1. Introduction

It has previously been shown analytically and experimentally that continuous Estimation of Distribution Algorithms (EDAs) based on the normal pdf can easily suffer from premature convergence. This paper takes a principled first step towards solving this problem. First, prerequisites for the successful use of search distributions in EDAs are presented. Then, an adaptive variance scaling theme is introduced that aims at reducing the risk of premature convergence. Integrating the scheme into the iterated density–estimation evolutionary algorithm (IDEA) yields the correlationtriggered adaptive variance scaling IDEA (CT-AVS-IDEA). The CT-AVS-IDEA is compared to the original IDEA and the Evolution Strategy with Covariance Matrix Adaptation (CMA-ES) on a wide range of unimodal test-problems by means of a scalability analysis. It is found that the average number of fitness evaluations grows subquadratically with the dimensionality, competitively with the CMA-ES. In addition, CT-AVS-IDEA is indeed found to enlarge the class of problems that continuous EDAs can solve reliably.

Abstract. This paper discusses the performance of a hybrid system which consists of EDP and GP. EDP, Estimation of Distribution Programming, is the program evolution method based on the probabilistic model, where the probability distribution of a program is estimated by using a Bayesian network, and a population evolves repeating estimation of distribution and program generation without crossover and mutation. Applying the hybrid system of EDP and GP to various problems, we discovered some important tendencies in the behavior of this hybrid system. The hybrid system was not only superior to pure GP in a search performance but also had interesting features in program evolution. More tests revealed how and when EDP and GP compensate for each other. We show some experimental results of program evolution by the hybrid system and discuss the characteristics of both EDP and GP.

Michigan-style learning classifier systems (LCSs), such as the accuracy-based XCS system, evolve distributed problem solutions represented by a population of rules. Recently, it was shown that decomposable problems may require effective processing of subsets of problem attributes, which cannot be generally assured with standard crossover operators. A number of competent crossover operators capable of effective identification and processing of arbitrary subsets of variables or string positions were proposed for genetic and evolutionary algorithms. This paper effectively introduces two competent crossover operators to XCS by incorporating techniques from competent genetic algorithms (GAs): the extended compact GA (ECGA) and the Bayesian optimization algorithm (BOA). Instead of applying standard crossover operators, here a probabilistic model of the global population is built and sampled to generate offspring classifiers locally. Various offspring generation methods are introduced and evaluated. Results indicate that the performance of the proposed learning classifier systems XCS/ECGA and XCS/BOA is similar to that of XCS with informed crossover operators that is given all information about problem structure on input and exploits this knowledge using problemspecific crossover operators.

To solve a wide range of different problems, the research in black-box optimization faces several important challenges. One of the most important challenges is the design of methods capable of automatically discovering the regularities in the problem and utilizing these to ensure efficient and reliable search. This paper discusses the Bayesian optimization algorithm (BOA) that uses Bayesian networks to model promising solutions and guide exploration of the search space. Using Bayesian networks in combination with population-based genetic and evolutionary search allows the algorithm to discover and utilize regularities in the form of a problem decomposition. The paper analyzes the applicability of the methods for learning Bayesian networks in context of genetic and evolutionary search. In particular, the population sizing ensuring that BOA learns a proper decomposition of the problem is analyzed. The paper concludes that the combination of the two approaches in BOA yields a robust, efficient, and accurate search.

In this paper we propose a crossover operator for evolutionary algorithms with real values that is based on the statistical theory of population distributions. The operator is based on the theoretical distribution of the values of the genes of the best individuals in the population. The proposed operator takes into account the localization and dispersion features of the best individuals of the population with the objective that these features would be inherited by the offspring. Our aim is the optimization of the balance between exploration and exploitation in the search process. In order to test the efficiency and robustness of this crossover, we have used a set of functions to be optimized with regard to different criteria, such as, multimodality, separability, regularity and epistasis. With this set of functions we can extract conclusions in function of the problem at hand. We analyze the results using ANOVA and multiple comparison statistical tests. As an example of how our crossover can be used to solve artificial intelligence problems, we have applied the proposed model to the problem of obtaining the weight of each network in a ensemble of neural networks. The results obtained are above the performance of standard methods. 1.

Recent studies have examined the effectiveness of using probabilistic models to guide the sample generation process for searching high dimensional spaces. Although

Abstract. In evolutionary continuous optimization by building and using probabilistic models, the multivariate Gaussian distribution and their variants or extensions such as the mixture of Gaussians have been used popularly. However, this Gaussian assumption is often violated in many real problems. In this paper, we propose a new continuous estimation of distribution algorithms (EDAs) with the variational Bayesian independent component analyzers mixture model (vbICA-MM) for allowing any distribution to be modeled. We examine how this sophisticated density estimation technique has influence on the performance of the optimization by employing the same selection and population alternation schemes used in the previous EDAs. Our experimental results support that the presented EDAs achieve better performance than previous EDAs with ICA and Gaussian mixture- or kernel-based approaches. 1