Theoretical basis of parameter tuning for finding optima near the boundaries of search spaces in real-coded genetic algorithms

Abstract

Studies on parameter tuning in evolutionary algorithms are essential for achieving efficient adaptive searches. This paper discusses parameter tuning in real-valued crossover operators theoretically. The theoretical analysis is devoted to improving robustness of real-coded genetic algorithms (RCGAs) for finding optima near the boundaries of bounded search spaces, which can be found in most real-world applications. The proposed technique for crossover-parameter tuning is expressed mathematically, and thus enables us to control the dispersion of child distribution quantitatively. The universal applicability and effect have been confirmed theoretically and verified empirically with five crossover operators. Statistical properties of several practical RCGAs are also investigated numerically. Performance comparison with various parameter values has been conducted on test functions with the optima placed not only at the center but also in a corner of the search space. Although the parameter-tuning technique is fairly simple, the experimental results have shown the great effectiveness.

Appendices

Appendix 1: Definition of CEC 2008 benchmark functions

The CEC 2008 benchmark functions are described in Table 4. The function value of the optimum of every function except ${{\fancyscript{F}}}_{7}$ is zero. Since the optimum of ${{\fancyscript{F}}}_{7}$ is unknown, the best function value, −1548, found in the CEC 2008 was treated as the offset in Sect. 5.2.

Table 4 Description of the CEC 2008 benchmark functions

Appendix 2: Theoretical curves of sampling bias with SPX

The crossover operation using SPX with $\theta=2$ within a one-dimensional search space [0, 1] is considered. The parental vectors and their centroid are thus regarded as scalar values: $x_{\theta}^{{(1)}}, x_{\theta}^{(2)},$ and $g_{\theta},$ where $0 \le x_{\theta}^{(1)} < x_{\theta }^{(2)} \le 1.$ The proof can be regarded as a generalized version of that for BLX-α with $\alpha=0.5$ presented in Someya and Yamamura (2002).

From a pair of the parents, a child $x_{\lambda}$ is generated within the range

$$ g_{\theta}-\frac{w}{2}<x_{\lambda}<g_{\theta}+\frac{w}{2}, $$

(59)

where w represents the width of the child distribution in a single crossover operation. Substituting $g_{\theta}= \frac{{x_{\theta}^{{(1)}}+ x_{\theta}^{{(2)}}}}{2}$ and $w=\varepsilon({x_{\theta}^{{(2)}}- x_{\theta}^{{(1)}}})$ into the above inequality leads to

$$ -1<\frac{2x_{\lambda}-({x_{\theta}^{(1)}+x_{\theta }^{(2)}})}{\varepsilon({x_{\theta}^{(2)}- x_{\theta}^{{(1)}}})}< 1. $$

(60)

This can be divided into the following two inequalities:

$$ \frac{({\varepsilon+1}){x_{\theta}^{(1)}-2x_{\lambda}}}{{\varepsilon-1}}< {x_{\theta }^{{(2)}}}, $$

(61)

$$ \frac{({\varepsilon-1}){x_{\theta}^{(1)}+2x_{\lambda}}}{{\varepsilon+1}}< {x_{\theta }^{{(2)}}}. $$

(62)

These show that the p.d.f. of children $p(x_{\lambda})$ can be expressed with a normalization constant K as follows:

$$ p(x_{\lambda})=K\{{p_{A}(x_{\lambda})+p_{B}(x_{\lambda})+ p_{C} (x_{\lambda})}\}, $$

(63)

$$ \left\{{\begin{array}{l} {\begin{aligned}p_{A}(x_{\lambda})&=\int\limits_{{x_{\lambda} }}^{\gimel} {\int\limits_{{\frac{{\left(\varepsilon+1\right)x_{\theta}^{{(1)}}- 2x_{\lambda }}}{{\varepsilon-1}}}}^{1}{w^{{-1}}\,{\hbox{d}}x_{\theta }^{{(2)}}\,{\hbox{d}}x_{\theta}^{{(1)}}}}\\ &\qquad\qquad\quad\quad\quad\quad:x_{\lambda}\le x_{\theta}^{{(1)}}, p_{B} (x_{\lambda})=\int\limits_{0}^{{x_{\lambda}}} {\int\limits_{{x_{\lambda}}}^{1} {w^{{-1}}\,{\hbox{d}}x_{\theta }^{{(2)}}\,{\hbox{d}}x_{\theta}^{{(1)}}}}\\&\qquad\qquad\quad\quad\quad\quad:x_{\theta }^{{(1)}}\le x_{\lambda}\le x_{\theta }^{{(2)}}, p_{C}(x_{\lambda})=\int\limits_{0}^{{x_{\lambda}}} {\int\limits_{{\frac{{\left(\varepsilon-1\right)x_{\theta}^{{(1)}}+2x_{\lambda }}}{{\varepsilon+1}}}}^{{x_{\lambda}}}{w^{{- 1}}\,{\hbox{d}}x_{\theta}^{{(2)}}\,{\hbox{d}}x_{\theta}^{{(1)}}}}\\&\qquad\qquad\quad\quad\quad\quad: x_{\theta }^{{(2)}}\le x_{\lambda},\end{aligned}}\\ \end{array}}\right. $$

(64)

where $\gimel$ is the right-hand side of $x_{\theta}^{(1)}< \frac{2(x_{\lambda}-1)}{\varepsilon+1}+1,$ derived by substituting $x_{\theta}^{(2)}=1$ into (61). By calculating the inner integrals of the three above cases, the following can be obtained:

$$ \left\{ {\begin{array}{l} \begin{aligned}\varepsilon \cdot p_{A} (x_{\lambda } )& = \int\limits_{{x_{\lambda } }}^{\gimel } \left\{ {\ln | {1 - x_{\theta }^{{(1)}} } | - \ln \left| {\frac{{2\left(x_{\theta }^{(1)}- x_{\lambda}\right)}}{{\varepsilon- 1}}} \right|} \right\}\,{\hbox{d}}x_{\theta }^{{(1)}}\\& \qquad\qquad\quad\quad\quad\quad:x_{\lambda } \le x_{\theta }^{{(1)}}, \varepsilon \cdot p_{B} (x_{\lambda } ) = \int_{0}^{x_\lambda } \{{\ln | {1 - x_{\theta }^{{(1)}} } | - \ln | {x_{\lambda} - x_{\theta }^{{(1)}} }|}\}\,{\hbox{d}}x_{\theta }^{{(1)}}\\& \qquad\qquad\quad\quad\quad\quad:x_{\theta }^{{(1)}} \le x_{\lambda } \le x_{\theta }^{{(2)}} , \varepsilon \cdot p_{C} (x_{\lambda } ) = \int\limits_{0}^{{x_{\lambda } }} \left\{\ln | {x_{\lambda } - x_{\theta }^{{(1)}} } | - \ln \left| {\frac{{- 2({x_{\theta }^{{(1)}}- x_{\lambda } } )}}{{\varepsilon + 1}}} \right|\right\}\,{\hbox{d}}x_{\theta}^{{(1)}}\\& \qquad\qquad\quad\quad\quad\quad:x_{\theta }^{{(2)}} \le x_{\lambda},\end{aligned}\end{array}}\right. $$

(65)

where $\left|{\star}\right|$ is the absolute value. Integrating gives

$$ \begin{aligned} \frac{\varepsilon}{K} \cdot p(x_{\lambda}) & = \int\limits_{0}^{\gimel} {\{ {\ln ({1 - x_{\theta }^{{(1)}} } ) - \ln 2} \}\,{\hbox{d}}x_{\theta }^{{(1)}}}\\ &\quad-\int\limits_{{x_{\lambda}}}^{\gimel}{\{{\ln ( {x_{\theta}^{{(1)}}-x_{\lambda}})-\ln(\varepsilon-1)} \}\,{\hbox{d}}x_{\theta}^{{(1)}}}\\ &\quad- \int\limits_{0}^{{x_{\lambda}}}{\{{\ln({- x_{\theta }^{{(1)}} + x_{\lambda}})-\ln(\varepsilon+1)} \}\,{\hbox{d}}x_{\theta}^{{(1)}}}.\\ \end{aligned} $$

(66)

By calculating each integral, this can be simplified to

$$ p(x_{\lambda})=\frac{K}{\varepsilon}\{\ln({\varepsilon+1}) -\ln{2}-(1-x_{\lambda})\ln{(1-x_{\lambda})}-x_{\lambda} \ln x_{\lambda}\}, $$

(67)

where

$$ K = \frac{2\varepsilon}{2\{\ln({\varepsilon+1})-\ln{2}\}+1}, $$

(68)

determined for

$$ {\int\limits_{0}^{1}}{p(x_{\lambda}){\hbox{dx}_{\lambda}}}=\frac{K}{\varepsilon}\left\{\ln({\varepsilon+1})-\ln{2} +\frac{1}{2}\right\}=1. $$

(69)