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%%%% Definicion de Macros %%%%%

%simbolos usuales:

\renewcommand{\R}{\mathbbm{R}}    %conjunto numeros reales
\newcommand{\N}{\mathbbm{N}}    %conjunto numeros naturales
\newcommand{\Z}{\mathbbm{Z}}    %conjunto numeros enteros
\newcommand{\Q}{\mathbbm{Q}}    %conjunto numeros racionales
\newcommand{\I}{\mathbbm{I}}    %conjunto numeros irracionales

\newcommand{\D}{\mathcal{D}}    %problema dual
\renewcommand{\P}{\mathcal{P}}  %problema primal

\newcommand{\ssi}{\Leftrightarrow} %si y solo si
\newcommand{\To}{\Rightarrow}      %implica
\newcommand{\tq}{\mid }            % tal que

%operadores matematicos:

\newcommand{\sen}{\operatorname{sen}}       %seno
\newcommand{\argmin}{\operatorname{argmin}}       %seno
\newcommand{\senh}{\operatorname{senh}}       %seno
\newcommand{\asen}{\operatorname{arcsen}}
\newcommand{\diag}{\operatorname{diag}}         %diagonal
\newcommand{\sop}{\operatorname{Sop}}       %soporte
\newcommand{\Dom}{\operatorname{Dom}}       %Dominio
\newcommand{\Rec}{\operatorname{Rec}}       %Recorrido
\newcommand{\ind}{\operatorname{\mathbbm{1}}} %vector de unos
\newcommand{\id}{\operatorname{\mathbf{I}}} %matriz identidad
\newcommand{\Int}{\operatorname{Int}}   %Interior
\newcommand{\img}{\operatorname{Im}}    %imagen
\newcommand{\Tr}{\operatorname{Trace}}  %traza
\newcommand{\conv}{\operatorname{conv}}%envoltura convexa
\newcommand{\aff}{\operatorname{aff}} %afin de un conjunto
\newcommand{\esssup}{\operatornamewithlimits{ess\,sup}}%supremo escencial

\newcommand{\prob}{\mathbbm{P}} % probabilidad
\newcommand{\E}{\mathbbm{E}}    %esperanza variable aleatoria
\newcommand{\var}{\operatorname{Var}}   %varianza

\providecommand{\abs}[1]{\lvert#1 \rvert}  %valor absoluto ej: $\abs{ x }$
\providecommand{\norm}[1]{\lVert#1 \rVert} %norma ej: $\norm{ x }$

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\newcommand{\nota}{{\bf Notaci\'on}:\ }
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\newcommand{\ej}{{\bf Ejemplo}:\ }

\newcommand{\NN}{\mathcal{N}} %vecindad de un punto
\newcommand{\RR}{\mathcal{R}} %
\newcommand{\QQ}{\mathcal{Q}} %func cuasianaliticas
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%definicion de ambientes:

\newtheorem{teor}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\Escuela}{Universidad de Chile\\Facultad de Ciencias F\'{\i}sicas y Matem\'aticas\\Departamento de 
Ingenier\'{\i}a
Matem\'atica\\}
\newcommand{\titulo}{Minimization of the expected compliance as an alternative approach to multiload truss optimization.}
\newcommand{\subtitulo}{ Estudio te\'orico y num\'erico.}
\newcommand{\Titulo}{\MakeUppercase{\titulo}}
\newcommand{\subTitulo}{\MakeUppercase{\subtitulo}}
\title{\Titulo}
%\subtitle{\subtitulo}
\author{ Felipe Alvarez Daziano \\ Miguel Carrasco Briones}
\institution{\Escuela}
\Logo(0.8,7.1){\includegraphics[width=0.7cm]{logoU.eps}}
\slideCaption{ Optimal design of trusses.}
\begin {document}
\maketitle

\begin{slide}{}
\begin{itemize}
\item[$\bullet$]  \hyperlink{parte1}{Optimization of trusses}
\begin{itemize}
\item[$\bullet$]  Standard minimum compliance truss design.
\item[$\bullet$]  Instabilities and the standard multiload model.
\item[$\bullet$]  Random Perturbations: Minimizing the expected compliance.
\item[$\bullet$]  Numerical examples.
\end{itemize}
\end{itemize}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\part{Optimization of trusses \hypertarget{parte1}{}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{Introduction}
\begin{itemize}
\item[$\bullet$]  Definition.
\item[$\bullet$]  Problem: find the best structure capable to carry a external nodal force.
\item[$\bullet$]  Constraints: mechanical equilibrium , total volume and others \ldots
\item[$\bullet$]  Minimize the compliance.
\end{itemize}
\begin{center}
\includegraphics[height=1.5cm]{imagenes/configN2.eps}
\end {center}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{Minimum compliance truss design}
\begin{align*}
\tag{$\D$} \min_{y,\lambda,u}&\ \frac{1}{2}f^Tu\\
 s.t. &\ K(y,\lambda)u=f\\
 &\ \lambda\in \Delta_m \\
&\  y \in V_1\times\ldots \times  V_N.
\end{align*}
Design variables
\begin{itemize}
\item[\blue $\bullet$]  $\lambda$:vector of bar volumes (topology).%``aspecto topol\'ogico''.
\item[\blue $\bullet$]  $y$: nodal positions (geometry).%``aspecto geom\'etrico''.
\end{itemize}
%La cantidad $\frac{1}{2}f^Tx$ se conoce como ``complacencia''.\\
$\red \frac{1}{2}f^Tu$ is called \emph{Compliance}.
\end{slide}

\begin{slide}{Matematical formulation}
\begin{minipage}{6.5cm}
\begin{itemize}
\item[$\bullet$] $\lambda\in \R^m;$ $m$ number of bars.
\item[$\bullet$] $y\in \R^N;$ $N$ number of nodes.
\item[$\bullet$]  $n$ grades of fredom.
\item[$\bullet$]  $f\in \R^n$ nodal load vector.
\item[$\bullet$]  $u\in \R^n$ nodal displacements.
\end{itemize}
\end{minipage}\ \  
\begin{minipage}{2cm}
\fbox{\vbox{
{\small% 
\begin{align*}%
\min_{y,\lambda,u}&\ \frac{1}{2}f^Tu\\%
 s.t. &\ K(y,\lambda)u=f\\%
 &\ \lambda\in \Delta_m \\%
 &\  y \in V_1\times\ldots \times  V_N.%
\end{align*}%
}
}}
\end{minipage}
\begin{itemize}
\item[$\bullet$]   $\Delta_m =\{ \lambda\in \R^m \tq  \lambda\ge 0, \
\sum\limits_{i=1}^m \lambda_i=1 \}$
\item[$\bullet$]  $K(y,\lambda)\in \R^{n\times n},$  \emph{stiffnes matrix}.
\end{itemize}
%$\frac{1}{2}f^Tu$ is  independent of the selection $u$ s.t.
%$K(u,\lambda)u=f.$%DECIRLO%%%%%%%%%%%%%%%%%%%%%
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{Stiffnes Matrix}
\[ K(y,\lambda)= \sum_{i=1}^m \lambda_i K_i(y)
\]
 where
\[
 K_i(y)= \frac{E_i}{l_i(y)^2}\gamma_i(y)\gamma_i(y)^T \in \R^{n\times n}
\]
remmark  $K_i$ is dyadic ($K_i=b_ib_i^T$).
\begin{itemize}
\item[$\bullet$]  $E_i$: Young modulus.
\item[$\bullet$]  $l_i(y)$: length of  bar $i.$
\item[$\bullet$]  $\gamma_i(y)$: \emph{cosines/sines} vector.
\end{itemize}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\overlays{3}{
\begin{slide}{ground structure approach}
\FromSlide{1}
\begin{itemize}
\item[\red $\bullet$]  geometric variable $\To$ $(\D)$ is very dificult
%{
%\small\begin{itemize}
%%\item[$\bullet$]  Altamente non lineal.
%\item[$\bullet$]  Kinky mathematical structure.
%\item[$\bullet$]  Hard to solvednumerically.
%\end{itemize}}
\FromSlide{2}
\item[\blue $\bullet$]  solution: \emph{ground structure}
{\small\begin{itemize}
\item[\blue $\bullet$] Nodal positions are fixed in the reference configuration ($y$ is fixed).
\item[\blue $\bullet$]  We use a mesh full of nodes and bars.
\end{itemize}}
\end{itemize}
\FromSlide{3}
\begin{center}
\includegraphics[height=2cm, width=5cm]{imagenes/michNodenso.eps}
\end {center}
\end{slide}
}

\overlays{2}{
\begin{slide}{}
\begin{center}
\FromSlide{1}
\includegraphics[height=2.5cm]{imagenes/michel.eps}\\
\FromSlide{2}
\includegraphics[height=2.5cm]{imagenes/michelLP.eps}\\
Tipically a large number of bar vanish at the optimum.
\end {center}
\end{slide}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}{}
\begin{align*}
\tag{$\D$} \min_{\lambda\in \R^m,u\in\R^n}&\ \red \frac{1}{2}f^Tu\\
 s.t. &\blue \ K(\lambda)u=f\\
 &\ \lambda\in \Delta_m
\end{align*}
$\red \frac{1}{2}f^Tu$ is  independent of the choice  $u$ satisfying
$\blue K(\lambda)u=f.$%DECIRLO%%%%%%%%%%%%%%%%%%%%%

\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{Minimax Formulation}
\[\tag{$\P$} \min_{x\in \R^n} \max_{1\leq i\leq m}\{\frac{1}{2}x^TK_i x - f^T x\}.\]
 Dual of $(\P)$ is the design problem $(\D)$.\\
{\bf \green advantages}
\begin{itemize}
\item[$\green \bullet$]  reduction of the dimention.
%\item[$\green \bullet$] Es posible recuperar los volumenes con los multiplicadores de lagrange del problema.
\item[$\green \bullet$]  The proof of equivalence between $(\P)$ and $(\D)$
don't use the dyadic structure of matrix $K_i.$
% La demostraci\'on de la
%equivalencia entre $(\P)$ y $(\D)$ no hace uso de la estructura
%particular de las matrices $K_i.$
\end{itemize}
{\bf \red disadvantages}
\begin{itemize}
\item[$\red \bullet$]  Lost of differentibility of the objective function.
\end{itemize}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% MODELO ALEATORIO %%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\part{Instabilities and the standard multiload model}
\overlays{3}{
\begin{slide}{Example}
\FromSlide{1} The design problem $(\D)$ may produce insatisfactory
results in respect to mechanical stability
\begin{center}
\FromSlide{2}
\includegraphics[height=3cm]{imagenes/j2d.eps}
\FromSlide{3}
\includegraphics[height=3cm]{imagenes/j2dLP.eps}
\end {center}
\end{slide}
}
\begin{slide}{Multiload model}
{\bf Standard multiload model}
\begin{align*}
\min_{\lambda\in \R^m}& \frac{1}{2}\sum_{j=1}^k \alpha_jf^{jT} u^j \\
        s.t.&\  K(\lambda)u^j=f^j,\quad j=1,\ldots, k\\
        & \ \lambda\in \Delta_m.
\end{align*}
We minimize  a weighted average of the compliances.
\end{slide}

\part{Random Perturbations: Minimizing the expected compliance.}

\begin{slide}{Random load model}
Let $\xi\in \R^n$ be a perturbation on the load vector $f.$
\[ \Psi(\xi,\lambda)=\begin{cases}
\frac{1}{2}(f +\xi)^Tx &\textrm{if $\lambda\in \Delta_m$ and $u\in \R^n$ such that}\\
& K(\lambda)u=f+\xi,\\
 +\infty & \textrm{ otherwise }
\end{cases}\]
\[
\Psi \colon (\R^n,\B(\R^n)) \to (\R\cup \{+\infty\},\bar{\B}(\R))
\]
Results to be proper, lower semicontinuous and convex.
\end{slide}
\begin{slide}{Random load model}
For heach $\lambda\in \Delta_m$
\[\begin{array}{cccc}
\Psi(\cdot , \lambda) \colon &(\R^n,\B(\R^n))& \to &(\R\cup \{+\infty\},\bar{\B}(\R))\\
 &\xi & \mapsto & \Psi(\xi,\lambda)
\end{array}
\] is measurable.

We asume that $\xi $ is a random variable corresponding to an uncertain nodal load perturbation
\[\begin{array}{cccc}
\xi \colon & (\Omega,\A)& \to &(\R^n,\B(\R^n))\\
&\omega & \mapsto & \xi(\omega)
\end{array}
\]
\end{slide}

\begin{slide}{Random load model}
Minimum expected compliance design problem
\[\tag{$\D;\prob$} \min_{\lambda \in \Delta^m} \E_\xi [\Psi(\xi,\lambda)]\]
Remmark: $V_{\lambda} = \img K(\lambda)=\{ K(\lambda)u \tq u\in \R^n\}$\\
\[ \E_\xi[\Psi(\xi,\lambda)]=\int_{V_\lambda}\Psi(\xi,\lambda)d\prob(w) +\infty\prob\{ V_\lambda^c\} \]
if $V_\lambda \neq \R^n$ it may ocurr that $\E_\xi [\Psi(\xi,\lambda)]=+\infty$ whenever $\prob\{ f +\xi \in V_\lambda 
\}>0$ $ \To\  \lambda $ is not a feasible member.

\end{slide}


\begin{slide}{Example: discrete perturbation}
\begin{enumerate}
\item $\prob_1\{ \xi \in B\}=1$ if $0\in B$ and $0$ otherwise.\\
We obtein the clasical single load model.
\item $\prob_2$ with finite support $\sop(\prob_2) = \{ \xi^1,\ldots, \xi^k \}.$
\begin{align*}
(\D;\prob_2)\qquad \min_{\lambda\in \R^m}& \frac{1}{2}\sum_{j=1}^k {\red \alpha_j}f^{jT} u^j \\
        s.t.&\  K(\lambda)u^j=f^j,\quad i=1,\ldots, k\\
        & \ \lambda\in \Delta_m.
\end{align*}
$(\D;\prob_2)$ is the \emph{standard multiload model} where
 $\red \alpha_j =\prob(\xi=\xi^j)$ and $f^j=f+\xi^j,$ $j=1,\ldots , k.$
\end{enumerate}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\part{Continuos case}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}{Continuous case}
\begin{teor}\nonumber
 Let $\xi\colon \Omega \to \R^n$ be a continuous random
variable with mean vector $\E(\xi)=0$ and Variance--covariance
matrix $\var(\xi)=PP^T,$ with $P \in \R^{n\times k}$

 Then the corresponding minimum expected compliance design problem
$(\D;\prob)$ is given by
\begin{align*}
(\D;\prob) \quad  \min_{\lambda\in \R^m}&\
\underbrace{\frac{1}{2}f^T u}_{\text{\red mean load}}
+\underbrace{\frac{1}{2}\Tr(P^TU)}_{\text{\red variance}}  \\
        s.t.&\  K(\lambda)u= f\\
            &\  K(\lambda)U=P \\
            &\  \lambda\in \Delta_m.
\end{align*}
\end{teor}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{Continuous case}
Remmark $f^T u$ and $\frac{1}{2}\Tr(P^TU)$ are independant of the
choice $u$ and $U$ satisfaying $K(\lambda)u= f,$ and $
K(\lambda)U=P $ respectively.\\
Remmark  $p^j,\ j=1,\ldots,k,$ columns of  $P$ \\
Defining
\begin{align*}
\hat{f}&=(f^T, p^{1T},\ldots, p^{kT})^T\in \R^{n(k+1)},\\
\hat{K_i}&=\diag(K_i,K_i,\ldots, K_i)\in \R^{n(k+1)\times n(k+1)},\\
\hat{K}(\lambda)&=\sum_{i=1}^m\lambda_i\hat{K}_i,
\end{align*}
Then $(\D;\prob)$ may be written as a multiload model.
%%\obs Se pierde la estructura di\'adica.%%
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{slide}{}
%\section{Ejemplos de distribuciones continuas}
%\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}{Example}
\subsection*{Random one dimentional preturbation}
\begin{itemize}
\item[]  $\xi=\varepsilon d ,$ where 
\item[] $d\in\R^n,\ \E(\varepsilon)=0,$ $\var(\varepsilon)=\sigma^2.$
\item[] Then $\E(\xi)=0$ and $\var(\xi)=\sigma dd^T$
\end{itemize}
\begin{align*}
(\D;\prob_3) \qquad  \min_{\lambda\in \R^m}&\ \frac{1}{2} f^T u +\frac{1}{2}\sigma d^Tv  \\
        s.t.&\  K(\lambda)u=f\\
            &\  K(\lambda)v=\sigma d \\
            &\  \lambda\in \Delta_m.
\end{align*}
\end{slide}%
%
\begin{slide}{example}
\subsection*{Random multidimentional independant perturbation}
 $ \xi=\sum_{j=1}^k\varepsilon_j d^j,$ where\\
 $d^j\in\R^n,$ $\E(\varepsilon_j)=0 $ and $\var(\varepsilon_j)=\sigma^2_j.$
\begin{align*}
(\D;\prob_4) \qquad  \min_{\lambda\in \R^m}&\ \frac{1}{2} f^T u +\frac{1}{2}\sum_{j=1}^r\sigma_j d^{jT}u^j \\
        s.t.&\  K(\lambda)u=f\\
            &\  K(\lambda)u^j=\sigma_j d^j \quad j=1,\ldots, k\\
            &\  \lambda\in \Delta_m.
\end{align*}
\end{slide}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{slide}{Stochastic model including variance}
We consider
\[ \min_{\lambda\in \R^m } \alpha \E_\xi[\Psi(\xi,\lambda)] + \beta \var[\Psi(\xi, \lambda)] \]
Example: $\xi \thicksim \NN(0, P P^T).$\\
 Taking for simplicity $\alpha=0$ and $\beta=1$
\begin{align*}
(\D;\prob_{\text{var}})\ \min_{\lambda \in \R^m}&\frac{1}{2}\left( \Tr((P^TU)^2)+2f^T U U^Tf\right)  \\
        s.t.&\  K(\lambda)u=f\\
            &\  K(\lambda)U=P\\
            & \ \lambda\in \Delta_m.
\end{align*}
Apparent harder to solve.
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\part{Numerical results}	
\begin{slide}{Toy example}
%%%%%%%%JUGUETE
\slideCaption{Toy example, Ben-Tal and Nemirowski 1997}
\begin{center}
\includegraphics[height=2.3cm]{imagenes/juguete2d.eps}
\includegraphics[height=2.3cm]{imagenes/juguete2dLPa.eps}
%\caption{Toy example (Ben-Tal and Nemirowski)}
\end {center}

\begin{center}
\includegraphics[height=2.5cm]{imagenes/juguete2dNLa.eps}
\includegraphics[height=2.5cm]{imagenes/juguete2dNLa2.eps}
\end {center}

\end{slide}

\begin{slide}{}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%TABLA
\begin{table}[!hbp]
\begin{center}
\begin{tabular}{|p{2.5cm}|p{1.5cm}|p{1.6cm}|p{1.6cm}|}
\hline
Problem & compl. mean &max  compl. & min compl.\\
\hline
%& & &  \\
Standard single load & $\nexists$ & $+\infty$ & $+\infty$\\
%Tradicional  &  & & \\
\hline \hline
 & & & \\
     (a)    &0.025 & 0.053 & 0.013\\
      & & & \\
\hline
 & & & \\
(b) & 0.023 & 0.051 & 0.012\\
 & & &\\
 \hline
\end{tabular}
\end {center}
\end{table}

\end{slide}
%%
%
\begin{slide}{Dome}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CUPULA
\slideCaption{Dome 3D}
\begin{center}
\includegraphics[height=6.5cm]{imagenes/Domemono.eps}
\end {center}
\end{slide}
\begin{slide}{Dome }%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CUPULA
\begin{center}
\includegraphics[height=6.8cm]{imagenes/Dome3DLP.eps}
\end {center}
\end{slide}
\begin{slide}{DOme}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CUPULA
\begin{center}
\includegraphics[height=6.8cm]{imagenes/Dome3DNL.eps}
\end {center}
\end{slide}

\begin{slide}{Electricity mast}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%TORRE
\slideCaption{Electricity mast , Achtziger et al. 1992}
\begin{center}
\includegraphics[height=3cm, width=2cm]{imagenes/Torre_pre1.eps}
\includegraphics[height=3cm, width=2cm]{imagenes/Torre_preLP.eps}
\end {center}
\begin{center}
\includegraphics[height=3cm, width=2cm]{imagenes/Torre1.eps}
\includegraphics[height=3cm, width=2cm]{imagenes/Torre2.eps}
\includegraphics[height=3cm, width=2cm]{imagenes/Torre5.eps}
\end{center}
\end{slide}

\begin{slide}{Michel 3D}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MICHEL 3D
\slideCaption{Michel 3D}
\begin{center}
\includegraphics[height=4.8cm]{imagenes/Mich3D.eps}
\includegraphics[height=4.9cm]{imagenes/Mich3DLP2.eps}
\end {center}
\end{slide}
\begin{slide}{Michel 3D}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MICHEL3D
\begin{center}
\includegraphics[height=6.8cm]{imagenes/Mich3DNL.eps}
\end {center}
\end{slide}

\begin{slide}{Cantilever}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CANTILEVER 3D
\slideCaption{Cantilever , Achtziger 1997}
\begin{center}
\includegraphics[width=5cm]{imagenes/cant3dcsa.eps}
\includegraphics[width=5cm]{imagenes/cant3dcsaLP.eps}
\end {center}
\end{slide}

\begin{slide}{}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%CANTILEVER 3D
\begin{center}
\includegraphics[ width=5cm]{imagenes/cant3dcsaNL.eps}
\includegraphics[ width=5cm]{imagenes/cant3DcsaNL3pert.eps}
\end {center}
\end{slide}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{slide}{Conclutions}
\begin{itemize}
\item[\yellow $\bullet$] Clasical model (one load) don't get good results .
\item[\green  $\bullet$] Random model helps to avoid this problem.
\item[\green $\bullet$] It's equivalent
to multiload one, but another interpretations in weight factors holds.
\item[\red $\bullet$] We have to know the influence of each perturbation.
\item[\red $\bullet$] Increase the dimention of the problem.
\end{itemize}
\end{slide}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{}
\includegraphics[width=4cm]{imagenes/michel.eps}
\includegraphics[width=4cm]{imagenes/michelLP.eps}
\end{slide}
\begin{slide}{stability result}
\begin{minipage}{10cm}
\begin{lemma}\label{L:stable} Let $\lambda\in \Delta_m$  and  $C$ be such
that  $C\geq \frac{1}{2}\max_{j} f_j^T u_j$ with
$K(\lambda)u_j=f_j$, $j=1,\ldots,k$. For each \\ $f \in
\conv\{f_1, \ldots, f_k,-f_1,\ldots,- f_k\},$ there exists $u\in \R^n $ satisfying $K(\lambda)u=f$
and moreover $C\geq\frac{1}{2}f^T u$.
\end{lemma}\end{minipage}
\begin{figure}[!ht]
\begin{center}
%\sidecaption
\includegraphics[width=3.5cm]{imagenes/convfi.eps}
\caption{Convex Hull of $f_1,\ldots,f_k, -f_1,\ldots, -f_k$}
\label{fig:convfi}
\end {center}
\end {figure}
\end{slide}

\begin{slide}{self weight}
\begin{center}
\includegraphics[width=7cm]{imagenes/DomeW.eps}
\end {center}
\end{slide}

\begin{slide}{self weight}
\begin{center}
\includegraphics[width=7cm]{imagenes/DomeW2pert.eps}
\end {center}
\end{slide}


\begin{slide}{}
Se cumple el siguiente resultado:
\begin{teor}
Sea $(x^*, \mu^*)$ soluci\'on del problema $(QP),$ entonces existen
$\sigma^+,\sigma^- \in \R^m_+$ tales que definiendo \[\lambda^* =
\frac{1}{\mu^*}(\sigma^+ +\sigma^-) \] se cumple que el par $(x^*,
\lambda^*)$ es soluci\'on de $(\D),$ (el problema de dise\~no).
\end{teor}
\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}{}
$(QP)$ es equivalente a $(LP)$
\begin{eqnarray*}
  (LP) &\min\limits_{x\in \R^n}& -f^T x \\
        & s.a.&  b_i^Tx\leq 1 \quad i=1,\ldots , m\\
        & &  -b_i^Tx\leq 1  \quad i=1,\ldots , m\\
\end{eqnarray*}
\end{slide}
\begin{slide}{}
Se cumple el siguiente resultado
\begin{teor}\label{teor:reduc_LP} Sea $\bar{x}$ soluci\'on del problema $(LP)$
con vectores multiplicadores $\rho^+, \rho^- \in \R^m_+,$ entonces
\begin{align*}
 x^* &:= \mu^* \bar{x}\\
\lambda_i^* &:= \frac{1}{\mu^*}(\rho^+_i +\rho^-_i)\ i=1,\ldots , m\\
\end{align*}
con $\mu^* := \sum_{i=1}^m(\rho^+_i  + \rho^-_i )$ resuelven
$(\D).$
\end{teor}
\end{slide}

\end{document}
