Abstractly modelling complex systems

Rattray, Charles

doi:10.1007/978-3-7091-6451-8_1

Cited by 6 publications

(5 citation statements)

References 9 publications

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“…The former section has been motivated from previous work by Rattray [9,10], using the basic idea of categorical shape theory due to Cordier and Porter [3]. As it is known, category theory provides basic notation and a universal language for explaining, investigating and discussing concepts and constructions from different fields, in a uniform way [2].…”

Section: Discussionmentioning

confidence: 99%

A Categorical View to Structural Complexity

Leal

Claudio

Toscani

2004

Proc Appl Math and Mech

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In the present work we are interested in to provide a universal language for supporting formalisms to specify the approximation hierarchy system for an abstract NP-hard optimization problem. This work grew from the idea of providing a categorical view of structural complexity to optimization problems. The direction is aimed towards actually exploring the connections among the structural complexity aspects and categorical concepts, which may be viewed in a high-level, in a structuralistic sense. After introducing the optimization problems categories OPTS and OPT, as well as related questions, a formal system modelling the approximation hierarchy of a given optimization problem is provided, based on categorical shape theory.

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Section: Discussionmentioning

confidence: 99%

A Categorical View to Structural Complexity

Leal

Claudio

Toscani

2004

Proc Appl Math and Mech

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“…All other relevant properties of the population after crossover can be found from the maximum entropy ansatz. A more general method is to follow the evolution of a number of cumulants explicitly, as in references [17,18], but this is unnecessary here because of the special form of crossover used, which is not appropriate in problems with stronger spatial interactions.…”

Section: The Statistical Mechanics Formalismmentioning

confidence: 99%

“…To make the problem tractable, it is assumed that before selection the population is at maximum entropy with constraints on the mean overlap and correlation within the population, as discussed in Appendix A. The calculation presented here is similar to that presented elsewhere [18], except for a minor refinement which seems to be important when considering problems with noise under selection. The correlation of an infinite population after selection from a finite population is given by,…”

Section: The Correlation After Selectionmentioning

confidence: 99%

“…A formalism has been developed by Prügel-Bennett, Shapiro and Rattray which describes the dynamics of a simple GA using methods from statistical mechanics [15,16,17,18]. This formalism has been successfully applied to a number of simple Ising systems and has been used to determine optimal settings for some of the GA search parameters [23].…”

Section: Introductionmentioning

confidence: 99%

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The dynamics of a genetic algorithm for a simple learning problem

Rattray

Shapiro

1996

J. Phys. A: Math. Gen.

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Abstract.A formalism for describing the dynamics of Genetic Algorithms (GAs) using methods from statistical mechanics is applied to the problem of generalization in a perceptron with binary weights. The dynamics are solved for the case where a new batch of training patterns is presented to each population member each generation, which considerably simplifies the calculation. The theory is shown to agree closely to simulations of a real GA averaged over many runs, accurately predicting the mean best solution found. For weak selection and large problem size the difference equations describing the dynamics can be expressed analytically and we find that the effects of noise due to the finite size of each training batch can be removed by increasing the population size appropriately. If this population resizing is used, one can deduce the most computationally efficient size of training batch each generation. For independent patterns this choice also gives the minimum total number of training patterns used. Although using independent patterns is a very inefficient use of training patterns in general, this work may also prove useful for determining the optimum batch size in the case where patterns are recycled.

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“…Among all these selection mechanisms, Boltzmann selection has an important place because it has some nice theoretical properties in some models of evolutionary algorithms [9]. For example, Boltzmann selection is extensively used in statistical mechanics theory of evolutionary algorithms [12,13,14,15]. Moreover, Boltzmann selection scheme is not used often in solving practical problems because, similar to simulated annealing, it needs an annealing schedule for perturbing the value of inverse temperature parameter used in Boltzmann selection, which is difficult to choose [9].…”

Section: Introductionmentioning

confidence: 99%

Cauchy annealing schedule: an annealing schedule for Boltzmann selection scheme in evolutionary algorithms

Dukkipati¹,

Murty²,

Bhatnagar³

Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753)

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Boltzmann selection is an important selection mechanism in evolutionary algorithms as it has theoretical properties which help in theoretical analysis. However, Boltzmann selection is not used in practice because a good annealing schedule for the 'inverse temperature' parameter is lacking. In this paper we propose a Cauchy annealing schedule for Boltzmann selection scheme based on a hypothesis that selection-strength should increase as evolutionary process goes on and distance between two selection strengths should decrease for the process to converge. To formalize these aspects, we develop formalism for selection mechanisms using fitness distributions and give an appropriate measure for selection-strength. In this paper, we prove an important result, by which we derive an annealing schedule called Cauchy annealing schedule. We demonstrate the novelty of proposed annealing schedule using simulations in the framework of genetic algorithms.

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Abstractly modelling complex systems

Cited by 6 publications

References 9 publications

A Categorical View to Structural Complexity

A Categorical View to Structural Complexity

The dynamics of a genetic algorithm for a simple learning problem

Cauchy annealing schedule: an annealing schedule for Boltzmann selection scheme in evolutionary algorithms

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