\be
\item For each of the following NLP's, determine all KKT points. Then use appropriate
means to determine the global optimal solution of the NLP. You may wish to check your solution
using the Maple \mf{GlobalSolve} command in the \mf{GlobalOptimization} toolbox.
\be
\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*} %Solution: (4,2)

\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=(x_1-2)^2+(x_2-1)^2  \\
		\mbox{ subject to} \notag\\
		x_1-x_2 &\leq 1  \notag\\
		x_1+x_2 &\leq 2 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*} %Solution: (1.5,.5); Winston p. 679, #6


\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=x_1-x_2  \\
		\mbox{ subject to} \notag\\
		x_1^2+x_2^2 &=1  \notag\\
		\end{align*} %Solution: (0,1)

\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=(2x_1-25)^2+6(x_2-10)^2  \\
		\mbox{ subject to} \notag\\
		(x_1-5)^2+x_2^2&\leq 50  \notag\\
		3x_1^2+x_2^2 &\leq 200 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*} %Solution: (6,7)

\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 6  \notag\\
		x_1&\leq 3 \notag\\
		x_2&\leq 4 \notag \\
		x_1,x_2 &\geq 0\notag
		\end{align*}	%Solution: (3,2); Winston, p. 679, 8

\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=x_1^3-2x_1-x_2  \\
		\mbox{ subject to} \notag\\
		x_1+x_2 &\leq 1  \notag\\
		x_1,x_2 &\geq 0\notag
		\end{align*} %Solution: (.422,.577); HL, p. 615 13.6_10b

\item \begin{align*}
		\mbox{minimize } f(x_1,x_2)&=x_1^3-3x_1+x_2^2-6x_2  \\
		\mbox{ subject to} \notag\\
		x_1+x_2 &\leq 1  \notag\\
        2x_1+x_2 &\leq 2  \notag\\
		x_1,x_2 &\geq 0\notag
		\end{align*} %Solution: (0,1); HL, p. 615 13.6_15

        \item
		\begin{align*}
		 \mbox{maximize } f(x_1,x_2)&= \frac{x_1}{x_2+1}  \\
		\mbox{ subject to} \notag\\
		x_1-x_2&= -2 \notag \\
        x_1,x_2 &\geq 0\notag
		\end{align*} %Solution: (0,2); HL, p. 615 13.6_9

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

        \item
		\begin{align*}
		 \mbox{minimize } f(x_1,x_2)&=x_1x_2 \\
		\mbox{ subject to} \notag\\
		2x_1+x_2&\leq 2 \notag\\
        x_1,x_2 &\geq 0\notag
		\end{align*}	%Solution is (.33,.83), which utilizes pseudoconvexity of Lagrangian
\ee

\item Suppose that $A$ is an $n$ by $n$ matrix, and define $f:\R^n \to \R^n$ 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}
Let $\lambda$ denote the smallest eigenvalue of $A$, and let $\vece$, denote the corresponding
eigenvector, normalized so that $\|\vece \|=1$. Show that $\vecx_0=\vece$ is the global optimal solution (\ref{E:Exercises_Karush_Kuhn_Tucker_QuadraticForm}) with corresponding objective value
equal to $\lambda$.

% Quadratic PD objective and linear constraints gives x=0 as a global min; (See Simon and Blume, p. 386)

% Sharp, Snyder, and Greene: Production-Transportation article; %
% See Maple file in KKT Examples Folder: Production_Transportation_Gradients_Contours_10_14

% Quadratic Portfolio
\item Suppose you wish to create a portfolio consisting of three stocks and a
certificate of deposit (CD). Table \ref{Ta:Exercises_KKT_Portfolio} provides data for
the four investments collected over an extended period of time.

\begin{table}[h]
	\centering
\caption{Mean and variance in the daily rate of return for four investments}\label{Ta:Exercises_KKT_Portfolio}
	\begin{tabular}{|l|c|c|}
	\hline
	Investment & Average Daily Rate of Return & Variance of Return \\ \hline
	Stock A & .0361	& .00208 \\ \hline
	Stock B & .0385	& .00287 \\ \hline
	Stock C & .0142	& .00095  \\ \hline
	CD (2 year, APY of 2.13\%)  & .000057	& 0 \\ \hline
    \end{tabular}
	\end{table}

\noindent
The ``Average Daily Rate of Return'' is based upon closing stock prices collected over
a period of several days. It provides the means for estimating how an investment
will do on any given day. The ``Variance of Return'' is the square of the standard deviation.
It measures the variability or ``volatility'' of an investment.
Notice how investments with higher rates of return also have higher variances. This trend is quite common.
The CD has a very low return but is risk-free.

\newpar
Suppose one wishes to construct a portfolio, in which $w_i$ denotes the fraction
of the money invested in investment $i$. Then these ``weights'' must sum
to 1, i.e.
	\[ g(w_1,w_2,w_3,w_4)=w_1+w_2+w_3+w_4=1.\]
The average return for the portfolio is given by
	\[ 0361w_1 + .0385w_2+ .0142w_3+ .000057w_4,\]
and, according to properties of variance, the risk, or variance, of the portfolio is measured by
	\[ .00208w_1^2 + .00287w_2^2 + .00095w_3^2.\]

\newpar
Unfortunately, we cannot simultaneously maximize average portfolio return and minimize
portfolio risk. We can however, maximize what is known as the \emph{net value} of the
portfolio:
	\begin{multline*}
f(w_1,w_2,w_3,w_4)=  \left( 0361w_1 + .0385w_2+ .0142w_3+ .000057w_4 \right) \\
- \alpha \left( .00208w_1^2 + .00287w_2^2 + .00095w_3^2 \right).
    \end{multline*}

In this formula, the quantity $\alpha$ represents a \emph{risk-aversion parameter}. The larger the value of $\alpha$, the more conservative the investor, as he or she incorporates the variance to a greater extent when constructing
a portfolio. Using various values of $\alpha$, e.g. $\alpha=10,15,20,25,30$, determine the solution of the NLP
\begin{equation*}
 \mbox{Maximize } f(w_1,w_2,w_3,w_4) \mbox{ subject to } g(w_1,w_2,w_3,w_4)=1.
\end{equation*}
Describe any trends you notice. For example, as $\alpha$ increases,
what happens to the fraction of the money invested in the CD? Also, if a weight value, $w_i$, is negative, try to provide an
interpretation of this result in terms of investing.

\item \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.}} A company produces a certain product at two different factories and delivers the finished product
 to its three retail outlets. Assume that the product is of the type that can be produced in non-integer quantities. At factory \# 1, material costs per unit production, $s$, are given by $g_1(s)=s^2$; at factory \# 2 the corresponding formula is $g_2(s)^2=1.5s^2$. Each unit of product requires \$ 2 of labor cost as well \$ 20 transportation cost for delivery to an outlet. In addition to production,
 labor, and transportation costs, each factory requires overhead costs of \$ 450 and \$ 400, respectively. Assume that the demands at the three outlets are given by 18, 24 and 14 units, respectively.

 \newpar The company seeks to determine the quantity of the product to be manufactured at each factory as well as the fraction of that quantity delivered to each of the three outlets. Set up and solve a convex NLP in eight variables whose objective is the total cost to the company due to production, labor, transportation of goods, and overhead. Decision variables for the NLP should tell the company how much of the product to manufacture at each factory and how much of that product that is to be delivered to each of the three outlets.


\ee

