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{\Large {\bf Homework problems, set 1}}
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\begin{enumerate}
\item
Find the minimum
\begin{equation}\label{}
\min x^T G  x + c^T x ~~\mbox{ subject to} ~~ A x= b
\end{equation}
where
\begin{equation}
G=\pmatrix{
2& 0 & 1 \cr
0 & 4 & 2 \cr
1 & 2 & 4 \cr
},~~ c=\left [\matrix{
1\cr
0\cr
5
}\right ], ~~
A=\pmatrix{
1 & 1 & 1 \cr
},
~~b= 10
\end{equation}
\item
Find the minimum
\begin{equation}\label{}
\min x^T G  x + c^T x ~~\mbox{ subject to }~~ A x \geq b
\end{equation}
where
\begin{equation}
G=\pmatrix{
2& 0 & 1 \cr
0 & 4 & 2 \cr
1 & 2 & 4 \cr
},~~ c=\left [\matrix{
1\cr
0\cr
5
}\right ], ~~
A=
\pmatrix{
1 & 0 & 1 \cr
0 & 1 & 1 \cr
}~~b= \left (\matrix{
1\cr
-1\cr
}\right )
\end{equation}
\item
Find the values of $ b$ for which  that the restriction is active.

Draw the picture.
\begin{equation}\label{}
\min x^T G  x + c^T x ~~\mbox{ subject to} ~~ A x \geq b
\end{equation}
where

\begin{equation}
G=\pmatrix{
 4 & 2 \cr
 2 & 2 \cr
},~~ c=\left [\matrix{
1\cr
2\cr
}\right ], ~~
A=\pmatrix{
1 & -1 \cr
}
\end{equation}


\end{enumerate}











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