\input{opt-def}

\begin{document}

%\lecture{n�zev}{��slo}{datum p�edn�ky}{datum zm�ny}{jm�no}

\lecture{1}{�vod, probl�m line�rn�ho programov�n�}{24. 2. 2012}{\today}{jm�no zapisuj�c�ho}

\section{�vod}

\section{�loha line�rn�ho programov�n�}


\[
\begin{LPmax}{\bfc^T\bfx}
A\bfx & \geq & \bfb
\end{LPmax}
\]

Dopravn� probl�m

\[
\begin{LPmaxpro}{\sum_{i,j}c_{ij}^Tx_{ij}}{\bfx\geq0}
\sum_jx_{ij} &\leq& a_i& \forall i
\\
\sum_ix_{ij} &\geq& b_j& \forall j
\end{LPmaxpro}
\]

Nejkrat�� cesta

\[
\begin{LPmax}{y_t-y_s}
y_v-y_u &\leq &c_{uv} &\textrm{pro v�echny hrany $uv$} 
\end{LPmax}
\]

P��klad se zarovn�v�n�m sloupc� pro t�i prom�nn�

\[
\begin{LPpro}[3]{maximalizuj}{\bfc^T\bfx}{\bfx\in\set{0,1}^3}
x_1 &&& + & 3x_3 & \geq & 1
\\
2x_1 & - & x_2& + & 3x_3 & \geq & 1
\\
 & & 100x_2& + & 3x_3 & \geq & 1 & \textrm{pro ...}
\end{LPpro}
\]

\begin{definice}
Toto je definice pojmu {\bf definice}.
\end{definice}

\begin{veta}
Toto je v�ta.
\end{veta}
\begin{proof}
Toto je d�kaz.
\end{proof}

Mno�iny zapisujte pomoc� p��kaz�
\verb+\set{1,2}+, tedy $\set{1,2}$, nebo
 \verb+\setcond{x}{x\in\RR}+,
tedy $\setcond{x}{x\in\RR}$.  M��ete tak� pou��t varianty s velk�mi
z�vorkami \verb+\bigset+ a \verb+\bigsetcond+.
Pokud nechcete pou��vat tyto p��kazy,
nen� dob�e pou��vat 
\verb+\{x|x\in\RR\}+, proto�e vygeneruje formuli s nep�kn�mi mezerami:
$\{x|x\in\RR\}$. Lep�� je \verb+\{x\mid x\in\RR\}+.

\subsection{Blah blah blah}


\subsection{Blah blah blah}
Here is a subsection.

\subsubsection{Blah blah blah}
Here is a subsubsection. You can use these as well.

\subsection{Using Boldface}
Make sure to use lots of boldface.

\paragraph{Question:}
How would you use boldface?

\paragraph{Example:}
This is an example showing how to use boldface to 
help organize your lectures.


\paragraph{Some Formatting.}
Here is some formatting that you can use in your notes:
\begin{itemize}
\item {\em Item One} -- This is the first item.
\item {\em Item Two} -- This is the second item.
\item \dots and here are other items.
\end{itemize}

If you need to number things, you can use this style:
\begin{enumerate}
\item {\em Item One} -- Again, this is the first item.
\item {\em Item Two} -- Again, this is the second item.
\item \dots and here are other items.
\end{enumerate}

%\bibliography{mybib}
%\bibliographystyle{alpha}

\end{document}
