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{\large\bf C\&O 350 \\ 
Assignment 2 
}
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{\large  Due: Wed. Feb 1 (\S 2, \S 3),
          Thurs. Feb 2 (\S 1 )
}
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\begin{enumerate}

\item
Chvatal, Problem 3.9 a,b,c, page 44.

\item
Read Chapter 4 in Chvatal. Then reproduce the Klee-Minty cycling example
on pages 47,48,49 by using the matlab package (lpmovie) on the computer.
Capture your output using the diary command and hand in a copy with your
assignment.

\item
Let $F \subset \Re^5$ be the set of $x$ satisfying

\[
\begin{array}{ccccccccccc}
2x_1&-&x_2&+&5x_3&+&x_4&& = & 4 \\
-x_1&+&x_2&-&3x_3&&+&x_5& = & -2 \\
&&&&&&& x_j \geq 0, & & \forall j \\
\end{array}
\]
For each of the following three cases, state whether or not the point
$\bar{x}$ is a basic feasible solution. In each case, if $\bar{x}$ is a
basic feasible solution, exhibit a corresponding dictionary, and if not,
find a nonzero vector $d \in \Re^5$ with both $\bar{x} + d$ and $\bar{x}
- d$ in $F$.
\begin{enumerate}
\item
\[
\bar{x} = (0~1~1~0~0)^T.
\]
\item
\[
\bar{x} = (\frac 12~0~\frac 12~\frac 12~0)^T.
\]
\item
\[
\bar{x} = (2~0~0~0~0)^T.
\]
\end{enumerate}

\item
Consider the LP
\[
\begin{array}{cc}
\min & cx \\
\mbox{ subject to} & Ax=b\\
& x \geq 0
\end{array}
\]
Suppose that this LP has a unique optimal solution. Show that this
optimal solution is an extreme point of the feasible set. (Use the
definition of extreme point given in the class.)
\end{enumerate}
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