Exercises in this section are well-suited for using the \mf{golden\_section}, \mf{SDM}, and \mf{Newtons\_Method} procedures.
\be
\item Apply Newton's Method with $x_0=1$ to the function\newline $f(x)=x^4-8x^3+14x^2+8x$. Explain what happens. Why does this result not contradict Theorem \ref{Th:unconstrained_numeric_Zoutendijk}?
%The function f is not Lipschitz continuous on the lower level set f(1)<=3

\item The function $\ds f(x_1,x_2)=-10x_1^2x_2+5x_1^2+4x_2^2+x_1^4+2x_2^4$ with
$\ds \vecx_0=\begin{bmatrix} -1 \\ -1 \end{bmatrix}$ and $\epsilon=.1$. If the function $\phi$ does not have
a local or global minimum, use its smallest positive critical value, $t_0$. What happens?

\item Rosenbrock's ``banana function,'' $\ds f(x_1,x_2)=100(x_2-x_1^2)^2+(1-x_1)^2$ is useful for comparing the efficiency of various numeric optimization algorithms. A contour diagram for $f$ is shown in Figure \ref{Fi:Exercises_unconstrained_opt_numeric_Rosenbrock}. Superimposed on this diagram are contours for the positive definite quadratic
approximation of $f$ based at $\ds \vecx_0=\begin{bmatrix} -1 \\ 0 \end{bmatrix}$.

\begin{figure}[h]
\centering
\psfrag{x1axislabel}[Bc][Bc]{$x_1$}
\psfrag{x2axislabel}[Bc][Bc]{$x_2$}
\psfrag{x0}[Bc][Bc]{$\ds \vecx_0= \begin{bmatrix} -1 \\ 0 \end{bmatrix}$}
\psfrag{z1}[Bc][Bc]{$z=104$}
\psfrag{z2}[Bc][Bc]{$z=30$}
\psfrag{z3}[Bc][Bc]{$z=10$}
\psfrag{z4}[Bc][Bc]{$z=2$}
\includegraphics[width=.7\linewidth]{images/f08x01x05.eps}
\caption{Contour diagram for Rosenbrock's banana function, together with contours of its quadratic approximation}
\label{Fi:Exercises_unconstrained_opt_numeric_Rosenbrock}
\end{figure}
	\be
	\item At the point $\ds \vecx_0=\begin{bmatrix} -1 \\ 0 \end{bmatrix}$ on the diagram, sketch the optimal
	descent direction and the Newton direction of $f$.
	\item Use your vectors from (a) to estimate, graphically, the first iterates $\vecx_1$ using both the methods of
	Steepest Descent
	and Newton.
	\item Determine each of the previous iterates $\vecx_1$ algebraically.
	\item Verify that when $\vecx_0$ is used as the initial choice, the hypotheses of Theorem
	\ref{Th:unconstrained_numeric_Zoutendijk} are met, for both methods. (Note: For Newton's Method, you will
	need to compute the condition number of $H_f(\vecx)$.)
	\item Use Maple to determine how many iterations of each method are required before
	$\ds \left\|  \nabla f\left(\vecx_k\right)\right\| < .01$. Which numeric method appears more efficient?
	\item Finally, calculate the global minimum of $f$ algebraically using appropriate methods
	from Chapter \ref{C:First_Second_Order_Conditions}. (Note that $f$ is not convex on $\R^2$!)
	\ee
%\asshole1
%\item Easom's function is given by
%  $\ds f(x_1,x_2)=-\cos(x_1)\cos(x_2)e^{-(x_1-\pi)^2-(x_2-\pi)^2}$. % \assholeEasom Compare the results
%of using both the Steepest Descent and Newton Methods with
% \asshole2 $\ds \vecx_0=\begin{bmatrix} 2 \\ 2 \end{bmatrix}$.
%Do the respective sequences of iterates appear to converge? If so, are the limits the same? Can you determine the nature of the limits as local maxima, minima, or saddle points?
%\asshole3
% \asshole3a \item Suppose $f:S \to \R$ is twice continuously differentiable, and assume
% $\ds \lambda_{\max,f,\vecx}$, the dominant eigenvalue of $\ds  H_f(\vecx)$,
%\asshole3b is bounded by a constant that is independent of the choice of $\vecx$ in the set $\mathcal{O}$.
%\asshole4 Show that then $\nabla f$ is Lipschitz continuous on $\mathcal{O}$. Here the set $\mathcal{O}$ is as
%defined in Theorem \ref{Th:unconstrained_numeric_Zoutendijk}. Hint: Use the Mean Value Theorem. 	
%\asshole4a
%\ee
% \asshole5
\ee
\endinput

