Modeling dynamics of a real-coded CHC algorithm in terms of dynamical probability distributions

Abstract

Some theoretical models have been proposed in the literature to predict dynamics of real-coded evolutionary algorithms. These models are often applied to study very simplified algorithms, simple real-coded functions or sometimes these make difficult to obtain quantitative measures related to algorithm performance. This paper, trying to reduce these simplifications to obtain a more useful model, proposes a model that describes the behavior of a slightly simplified version of the popular real-coded CHC in multi-peaked landscape functions. Our approach is based on predicting the shape of the search pattern by modeling the dynamics of clusters, which are formed by individuals of the population. This is performed in terms of dynamical probability distributions as a basis to estimate its averaged behavior. Within reasonable time, numerical experiments show that is possible to achieve accurate quantitative predictions in functions of up to 5D about performance measures such as average fitness, the best fitness reached or number of fitness function evaluations.

Appendix: A formulation for hypersphere volumes

The volume of a hypersphere $V_{H}$ of radius $r=d/2$ is given by $V_{H}(r)= C_n r^n$ where $n$ is the dimension, $C_n=\pi^{n/2}/\Upgamma(1+n/2)$ is the unit sphere, and $\Upgamma(.)$ is the Gamma function.

The volume for a hypersphere cap $V_C$ of radius $r$ and height $h$ is given by

$$ V_C(r,h)=C_n r^n \left(\frac{1}{2} - \frac{r-h}{r}\frac{\Upgamma\left(1+\frac{n}{2}\right)}{\pi^{\frac{1}{2}} \Upgamma\left(\frac{n+1}{2}\right)} {}_{2}F_{1}(a,b;c;z) \right) $$

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where $a=\frac{1}{2}, b=\frac{1-n}{2}, c=\frac{3}{2}, z=( \frac{r-h}{r})^2. \, {}_2F_1$ is a hypergeometric function defined by the next series:

$$ _2F_1(a,b;c;z)= \sum_{i=0}^{\infty} \frac{ (a)_n (b)_n z^n}{(c)_n n!} $$

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when it meets $z<1,$ that in this case it is $h>0.$ Therefore, $(x)_n$ is given as follows:

$$ (x)_n = \prod_{i=0}^{n-1} (x+i) $$

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The volume of intersection of two hyperspheres $V_I$ can be found by adding the volume of two hyperspherical caps. Let $r1$ and $r2$ be the radius of both hyperspheres that intersect, and $\delta$ the distance between their centers. Then,

$$V_I(r1,r2,\delta) = V_C(r1,h1) + V_C(r2,h2) $$

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where the heights of the caps $h1$ and $h2$ are

$$ h1=\frac{(r1+r2-\delta)(r2-r1+\delta)}{2 \delta} $$

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$$ h2=\frac{(r1+r2-\delta)(r1-r2+\delta)}{2 \delta} $$

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Table 6 Model predictions and average simulation data at each generation for the function F ₁

Table 7 Model predictions and average simulation data at each generation for the function F ₂

Table 8 Model predictions and average simulation data at each generation for the function F ₂

Table 9 Model predictions and average simulation data at each generation for the function F ₃

Table 10 Model predictions and average simulation data at each generation for the function F ₄

Table 11 Model predictions and average simulation data at each generation for the function F ₅

Table 12 Model predictions and average simulation data at each generation for the function $F_5$ All values are normalized