2-3.tex

较好的一种新的关于遗传算法的源程序

\documentclass{article}
\oddsidemargin=0cm
\evensidemargin=0cm
\textwidth=14.8 true cm
\textheight=23.2 true cm
\def \x{{\mbox{\boldmath $x$}}}
\def \y{{\mbox{\boldmath $y$}}}
\def \a{{\mbox{\boldmath $a$}}}
\def \d{{\mbox{\boldmath $d$}}}
\def \p{{\mbox{\boldmath $p$}}}
\def \r{{\mbox{\boldmath $r$}}}
\def \t{{\mbox{\boldmath $t$}}}
\def \w{{\mbox{\boldmath $w$}}}
\def \fx{{\mbox{\boldmath $\tilde x$}}}
\def \fy{{\mbox{\boldmath $\tilde y$}}}
\def \hxi{{\mbox{\boldmath $\xi$}}}
\def \heta{{\mbox{\boldmath $\eta$}}}
\def \htau{{\mbox{\boldmath $\tau$}}}
\def \Pos{{\rm Pos}}
\def \Ch{{\rm Ch}}
\begin{document}

\noindent Liu B., {\em Uncertain Programming}, John Wiley \& Sons, New York, 1999

\vskip 1.5cm
\noindent {\bf Example 2.3} This is a biobjective optimization on a nonconvex set,
\begin{equation}
\left\{\begin{array}{l}
\max \,\,f_1(\x)=x_1^2+x_2^2\\[0.1cm]
\max \,\,f_2(\x)={x_3}/{(1+x_1+x_2)}\\[0.1cm]
\mbox{subject to:}\\[0.12 cm]
\qquad x_1^2+x_2^2+x_3^2\ge  1\\[0.1 cm]
\qquad x_1^2+x_2^2+x_3^2\le 4\\[0.1 cm]
\qquad x_1,x_2,x_3> 0.
\end{array}\right.
\end{equation}
We assign 0.4 and 0.6 as the weighting factors to the objective functions $f_1(\x)$ and $f_2(\x)$, respectively.
\end{document}