## Schemata, Distributions and Graphical Models in Evolutionary Optimization (1999)

Venue: | Journal of Heuristics |

Citations: | 89 - 8 self |

### BibTeX

@MISC{Mühlenbein99schemata,distributions,

author = {Heinz Mühlenbein and Thilo Mahnig and Alberto Ochoa Rodriguez},

title = {Schemata, Distributions and Graphical Models in Evolutionary Optimization },

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper the optimization of additively decomposed discrete functions is investigated. For these functions genetic algorithms have exhibited a poor performance. First the schema theory of genetic algorithms is reformulated in probability theory terms. A schema denes the structure of a marginal distribution. Then the conceptual algorithm BEDA is introduced. BEDA uses a Boltzmann distribution to generate search points. From BEDA a new algorithm, FDA, is derived. FDA uses a factorization of the distribution. The factorization captures the structure of the given function. The factorization problem is closely connected to the theory of conditional independence graphs. For the test functions considered, the performance of FDA- in number of generations till convergence- is similar to that of a genetic algorithm for the OneMax function. This result is theoretically explained.

### Citations

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Citation Context ...arch method. A set of points is generated, promising points are selected, and new points are generated using the genetic operators recombination /crossover and mutation. The simple genetic algorithm (=-=Goldberg 1989-=-) selects promising points according to proportionate selection p s (x; t) = p(x; t) f(x)sf(t) : (1) Here x = (x 1 ; x 2 ; : : : ; x n ) denotes a vector of discrete variables (genotype), p(x; t) is t... |

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Citation Context ...n. 1 muehlenbein@gmd.de 2 Also with the Centre of Artificial Intelligence. ICIMAF. Cuba. ochoa@cidet.icmf.inf.cu 3 Real World Computing Partnership 1 A possible structure for aggregation is a schema (=-=Holland 1992-=-). We just give an example for a schema. Extending the usual notation, H(x i ; x k ) = (; : : : ; ; x i ; ; : : : ; ; x k ; ; : : : ) (3) defines a schema where the values of the variables x i and x k... |

1105 |
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Citation Context ...i d i := S i j=1 s j (16) b i := s i n d i\Gamma1 (17) c i := s i " d i\Gamma1 (18) 6 We set d 0 = ;. In the theory of decomposable graphs, d i are called histories, b i residuals and c i separat=-=ors (Lauritzen 1996-=-). Theorem 3 (Factorization Theorem). Let p(x) be a Boltzmann distribution on X with p(x) = Exp u f(x) F u with u ? 1 arbitrarily. (19) If b i 6= ; 8i = 1; : : : ; l; d l = ~ X; (20) 8is2 9j ! i such ... |

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Citation Context ...ustrated plaquette cannot attain the 4-term minimal energy of-4, whereas an unfrustrated can. It has been shown earlier that there are polynomial algorithms to compute a matching with minimum weight (=-=Lawler, 1976-=-). We have used this method to compute the exact solution of a fairly difficult free boundary problem on a 11*11 grid. We recall that an exact factorization requires sets of order O( p n). Therefore w... |

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Citation Context ...time. The simplest approximation uses first order schemata. These schemata define univariate marginal distributions. This approximation is used by the Univariate Marginal Distribution Algorithm UMDA (=-=Muhlenbein 1998-=-). New points are generated according to p(x; t + 1) = n Y i=1 p s (x i ; t): (5) p s (x i ; t) denotes the probability (after selection) of a first order schema defined at loci i. If the distribution... |

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1 | Learning in Graphical Models Cambridge:MIT - Jordan - 1999 |