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%\rfoot{\small \copyright{\ } by Siemens AG, 2007 all rights reserved\\
%I. Fischer, M. Paffrath, A. Sohr, U. Wever }
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\begin{titlepage}
\title{ {\Large {\bf Nonlinear chance-constrained optimization and applications}}}
\author{M. Paffrath\footnote{Address: Siemens AG, Corporate Technology, Otto-Hahn-Ring 6, D81730 Munich, 
Germany, email: meinhard.paffrath@siemens.com}, B. Weber, U. Wever}
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\date{\today}
\end{titlepage}
\begin{document}
\maketitle
%\tableofcontents
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\section{Summary}
In practical applications, optimization of structures 
has to deal with fluctuating parameters,
scattering environment data (e.g. temperatures), uncertain material and
geometry parameters. Under these prerequisites, a deterministic optimization may not be
adequate but a stochastic optimization should be performed instead.
Let $\vec X = (X_1,..,X_n)^T$ be a vector of independent stochastic input variables $X_i$ 
with mean $\mu_i$, standard deviation $\sigma_i$ and 
distribution $D_i(\mu_i,\sigma_i)$, $\vec \mu = (\mu_1,...,\mu_n)^T$,
$\vec \sigma = (\sigma_1,...,\sigma_n)^T$.
% with mean
% $\vec \mu$ and standard deviation $\vec \sigma$.
A general chance-constrained optimization problem may be stated as follows
% \begin{tabbing} xxxxxxxxxxxxxxxxxx \= xxxxxxxxxxx \= xxxxxxxxxxxx \= \kill 
% \> Minimize $S(f,\vec\mu,\vec\sigma)$ \\
% \> w.r.t. \\
% \> $P(g_i,\vec\mu,\vec\sigma) \le \epsilon_i, \qquad i=1,..,m$
% \end{tabbing}
\begin{eqnarray}
& & \mbox{Minimize} \; S(f,\vec\mu,\vec\sigma) \nonumber \\
& & \mbox{w.r.t.} \label{chance_opt} \\
& & P(g_i,\vec\mu,\vec\sigma) \le \epsilon_i, \qquad i=1,..,m \nonumber
\end{eqnarray}
with design parameters $\vec\mu, \vec\sigma$,
linear or nonlinear functions
\begin{eqnarray*}
f   &:& \R^n   \rightarrow \R \\
    &&  \vec x \mapsto     f(\vec x) \\
g_i &:& \R^n \rightarrow \R \\
    &&  \vec x \mapsto     g_i(\vec x) \qquad i=1,..,m
\end{eqnarray*}
and failure probabilities
\begin{equation}\label{eq_versagen}
P(g_i,\vec\mu,\vec\sigma) = \int_{g_i(\vec x)\le 0} 
\rho(\vec x,\vec\mu,\vec\sigma) d\vec x \qquad i=1,..,m
\end{equation}
$\vec x$ denotes the vector of input parameters (realizations of $\vec X$),
and $\rho$ the joint probability density function of $\vec X$.
Here we restrict ourselves to probability distributions with compact support
$\Omega(\vec \mu,\vec \sigma)$ (a cuboid) which are the most important distributions
in practical applications. 
The operator $S(f,\vec\mu,\vec\sigma)$ in the objective function may be a stochastic moment of $f$
or again a failure probability. So e.g. minimization of mean and variance of $f$ is given by:
\begin{eqnarray}
\mbox{Minimize}\; \bar f := E(f,\vec\mu,\vec\sigma) = 
\int\limits_{\Omega(\vec \mu,\vec \sigma)} 
f(\vec x) \rho(\vec x,\vec\mu,\vec\sigma) d\vec x \label{eq_min_mean} \\
\mbox{Minimize}\; Var(f,\vec\mu,\vec\sigma) = 
\int\limits_{\Omega(\vec \mu,\vec \sigma)} 
(\bar f - f(\vec x))^2 \rho(\vec x,\vec\mu,\vec\sigma) d\vec x \label{eq_min_var}
\end{eqnarray}
%
Numerically critical is the solution of the multi-dimensional integral in 
(\ref{eq_versagen}) for it generally requires the use of computationally
expensive sampling methods. Therefore we propose, as first step of the solution procedure of 
(\ref{chance_opt}),
the solution of an easier manageable problem where the chance constraints are replaced by
absolute reliability constraints:
\begin{displaymath}
P(g_i,\vec\mu,\vec\sigma) = 0, \qquad i=1,..,m
\end{displaymath}
which is equivalent to  
\begin{displaymath}
\min_{\vec x\in \Omega(\vec\mu,\vec\sigma)}  g_i(\vec x) \ge 0 \quad i=1,..,m
\end{displaymath}
Now the optimization problem has the form
%\begin{tabbing} xxxxxxxxxxxxxxxxxx \= xxxxxxxxxxx \= xxxxxxxxxxxx \= \kill 
%\> Minimize $S(f,\vec\mu,\vec\sigma)$ \\
%\> w.r.t. \\
%\> $\min_{\vec x\in \Omega(\vec\mu,\vec\sigma)}  g_i(\vec x) \ge 0&& \quad i=1,..,m$
%\end{tabbing}
\begin{eqnarray}
& & \mbox{Minimize} \; S(f,\vec\mu,\vec\sigma) \nonumber \\
& & \mbox{w.r.t.} \label{absrel_opt} \\
& & \min_{\vec x\in \Omega(\vec\mu,\vec\sigma)}  g_i(\vec x) \ge 0 \quad i=1,..,m \nonumber
\end{eqnarray}
In our presentation, we will concentrate on efficient numerical methods for evaluation of  
the integrals in (\ref{eq_min_mean}), (\ref{eq_min_var}), the solution of (\ref{absrel_opt})
and some applications in the field of mathematical engineering.
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\end{document}