Clustered genetic search in continuous landscape exploration

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Abstract

We present the strategy of clustered genetic search as a case-oriented tool for global optimization problems. The strategy is performed in two quite different manners—as an iterative discrete hill-crunching approach and as a continuous method based on a finite mixture model. We set up the main points of theory on convergence, stop condition and error estimation. We illustrate the properties of both methods in two examples, including the problem of optimal geometry for argon cluster.

Section snippets

Statement of the problem

There is a wide range of global optimization methods which are aimed at finding a single local minimizer (or maximizer) x+ of the real valued objective Φ in an admissible set D⊂Rn. However, there is often a need of locating all available minimizers (maximizers) as well as detecting their basins of attraction (see definition below). One can mention inverse parametric problems being classical ambiguous problems with many minimizers, such as tasks appearing in the ultrasonographic defectoscopy for

Selected results of genetic algorithms asymptotic theory

This section is to present some theoretical results that will be used in Section 3 as a background for analysis of asymptotic properties of two clustered genetic search methods introduced there. In Section 2.1 we introduce some basic elements of Simple Genetic Algorithm (SGA) theory. We interpret SGA as a measure-transforming system and conclude with two theorems which are tools for asymptotic analysis of both clustered genetic search methods. Section 2.1 describes also comprehensively the idea

Two examples of clustered genetic search

For each strategy below we define the positive fitness function f=Φ̃∘code in the space of genetic codes Ω (genetic universum), that expresses the source minimization problem for Φ on D. The encoding function code:Ω→D represents the N-dimensional vector on D defined in the standard way (see eg. Goldberg, 1989).

Applications

In order to illustrate some properties of the presented methods, we show results of tests for two functions: two-dimensional sinus and a simple chemical interaction potential. The tests for the sinus function ((10) and Fig. 2a)f1(x,y)=sin(xy)+1(x,y)∈[−3;3]illustrate the ability of both methods to recognize many basins in the case of a multimodal function (see Fig. 2, Fig. 3). The minima of the function constitute some one-dimensional manifolds, which are an additional difficulty.

However this

Concluding remarks

The combination of genetic algorithms and clustering methods in which the genetic sample is classified exhibits high positive synergy in solving global optimization problems. There are the following advantages of this approach:

  • central parts of the basins can be detected and approximated;

  • evolution landscape can be partially explored and recognized;

  • there is a possibility to analyze the stability of minimizers;

  • the number of local time-expansive searches can be delimit to one in each basin when the

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