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{\large\bf C\&O 367 \\ 
Midterm  Feb. 27, 2001.
}
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{\large   Instructor H. Wolkowicz \\
(  The exam is from 1:00PM - 2:15PM.  )\\
{\bf  NO CALCULATORS.}\\
{\bf  Please come to class on Thursday Mar. 1. TRACE questionnaire will
be handed out!}
}
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\section{(20 points)}
\begin{enumerate}
\item
Define the following:
\begin{enumerate}
\item
convex set in $\Re^n$
\item
convex function defined on $\Re^n$
\item
closed convex cone in $\Re^n$
\item
dual (nonnegative polar) cone of a set $C$ in $\Re^n$
\end{enumerate}
\item
Show that the following two sets are convex cones. Then
derive an explicit form for the dual (polar) cone $C^+=C^*$.
\begin{enumerate}
\item
\label{c1}
$C_1 = \left\{ (x,y) \mid 0 \le y\le x \right\}$.
\item
\label{c2}
$C_2 = \left\{ (x,y) \mid y \ge  |x| \right\}$.
\end{enumerate}
\item
Show that each of the above cones does not contain the point
$(\bar{x},\bar{y})=(0,-1)$.
For each of the above cones find
a separating hyperplane with the point
$(\bar{x},\bar{y})=(0,-1)$.
\item
\label{funcf}
Consider the quadratic objective function
\[ 
f(x)=3x_1^2-4x_1x_2+2x_2^2+7x_1-x_2.
\]
\begin{enumerate}
\item
Find appropriate $A,b$ and write $f(x)=x^T A x + b^T x$. 
\item
Show that $f$ is a strictly convex function.
\item
Use the geometric characterization of optimality (Pshenichnyi
condition). Show that the origin in $\Re^2$
 minimizes $f$ (defined
above) subject to $x \in C_1$ (defined in (\ref{c1}) above).
\end{enumerate}
\end{enumerate}




\section{(10 points)}
\begin{enumerate}
\item
Let $f(x)$ be twice continuously differentiable on $\Re^n$.
\begin{enumerate}
\item
Define local and global minimum for $f(x)$.
\item
Define Newton's method for minimization of $f(x)$.
\end{enumerate}
\item
Show that the function $f(x)$ defined on $\Re^1$ by
\[  f(x)=x^{\frac 43}
\]
has a unique global minimizer at $x^*$ but that, for any nonzero initial
point $x^{(0)}$, the Newton's Method sequence $\left\{x^{(k)}\right\}$ with
initial point $x^{(0)}$ for minimizing $f(x)$ diverges.
\end{enumerate}



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