\be
\item For each of the following NLP's, verify that the problem is convex and
 the determine its solution.
\be
\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=x_1^2-6x_1+x_2^2-4x_2  \\
		\mbox{ subject to} \notag\\
		x_1+x_2 &\leq 4  \notag\\
		x_1 &\leq 1 \notag \\
		x_2 &\geq 0\notag
		\end{align*}

\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=(x_1-6)^2+(x_2-4)^2  \\
		\mbox{ subject to} \notag\\
		-x_1+x_2^2 &\leq 0  \notag\\
		x_1 &\leq 4 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}

\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=x_1^2-\ln(x_2+1)  \\
		\mbox{ subject to} \notag\\
		2x_1+x2 &\leq 3  \notag\\
		x_1,x_2 &\geq 0\notag
		\end{align*}


\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=4x_1^2-12x_1+x_2^2-6x_2  \\
		\mbox{ subject to} \notag\\
		-x_1^2+x_2^2 &\leq 2  \notag\\
        2x_1+x_2&=1  \notag\\
		\end{align*}
\ee

\item Suppose that $A$ is an $n$ by $n$ matrix, and define $f:\R^n \to \R$ by
$\ds f(\vecx)=\vecx^tA\vecx$, where $\vecx \in \R^n$. Consider the NLP
\begin{align}\label{E:Exercises_Karush_Kuhn_Tucker_QuadraticForm}
		 \mbox{minimize } f(\vecx)& \\
		\mbox{ subject to} \notag\\
		\|\vecx\| &\leq 1 \notag
		\end{align}

Show that the optimal solution occurs at the origin, unless $A$ possesses a positive
eigenvalue. Show that in this latter case, the solution
occurs at the eigenvector $\vecx=\vece$, suitably normalized, which corresponds
to the largest eigenvalue, $\lambda$. Verify that the
corresponding objective value is given by $f(\vecx)=-\lambda$.

\item A company produces a certain product at two different factories and delivers the finished product to its two retail outlets. \footnote{Based upon Sharp, Snyder, and Greene's, ``A Decomposition Algorithm for Solving the Multifacility
 Production-Transportation Problem with Nonlinear Production Costs,'' \emph{Econometrica}, \textbf{38}, 3, 1970. \cite{Sharp_Snyder_Greene}.}
 Assume that the product is of the type that can be produced in non-integer quantities. At factory \# 1,
 the cost of producing $s$ units is given by $P_1(s)=s^2$ dollars; at factory \# 2 the corresponding cost formula is $P_2(s)^2=.8s^2$. The cost to ship one product unit from factory \# 1 to outlet \# 1
 is 6 dollars and from factory \# 2 to outlet \# 1 is 9 dollars. The corresponding costs for outlet
 \# 2 are 8 dollars and 10 dollars, respectively. Assume that the demand at each outlet is at
 least 20 units of product. If the company wishes to minimize the sum of its production and
 shipping costs, how many units of the product should be shipped from each factory to each outlet?

\ee

