\be
\item Suppose $f:\R^2 \to \R$ is given by
$\ds f(x_1,x_2)=x_1^4-8x_1^3+24x_1^2-32x_1+4x_2^2-4x_2$. For what initial
values, $\vecx_0$, will Newton's method fail to produce a sequence of iterates?

\item \emph{Rosenbrock's valley 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. Figure \ref{Fi:f08x02x03}. illustrates the graph of this function, which varies significantly in magnitude in a neighborhood
    of the origin.

\begin{figure}[h]
\centering
\psfrag{x1_axis_label}[Bc][Bc]{$x_1$}
\psfrag{x2_axis_label}[Bc][Bc]{$x_2$}
\includegraphics[width=.7\linewidth]{images/f08x02x03.eps}
\caption{Rosenbrock's valley function.}
\label{Fi:f08x02x03}
\end{figure}
	\be
	\item Verify that $\ds \vecx_{\star}= \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is both a local and global minimum.
    \item Suppose $\ds \vecx_0=\begin{bmatrix} -1 \\ 0 \end{bmatrix}$. Calculate the Newton direction of
    $f$ at $\ds \vecx_0$. Use your result to calculate the first Newton Method iterate, $\vecx_1$.
    \item How many iterations of Newton's Method are required to estimate the minimum of $f$
    with a tolerance of $.01$?
    \item Now determine the optimal descent direction of $f$ at $\ds \vecx_0$. Use your result to calculate the first Steepest Descent Method iterate, $\vecx_1$.
    \item Calculate two more Steepest Descent Method iterates, $\vecx_2$ and $\vecx_3$. Using the eigenvalues of
    $\ds H_f(\vecx_{\bigstar})$, verify that formula
    (\ref{E:SteepestDescent_ConvergenceRate_Inequality1}) from Section \ref{S:SteepestDescent}
    is satisfied for these iteration values.
    \item How many iterations of the Steepest Descent Method are required to estimate
    the minimum of $f$ with a tolerance of $.01$?
    \ee

\item Suppose that $f:\R^2 \to \R$ is defined by
$\ds f(x_1,x_2)=\ln(1+x_1^2)+x_2^2$ and that $\ds \vecx_0=\begin{bmatrix} 2 \\ 2 \end{bmatrix}$.
Recall that the Steepest Descent Method, starting with $\vecx_0$, produces a sequence
of iterates converging to the global minimum of $f$ at the origin.
Show that Newton's Method fails to produce such a convergent sequence. Then experiment with other initial values, $\vecx_0$, and try to account for the varying outcomes by using the Hessian of $f$.

\item Suppose that $f:\R^2 \to \R$ is defined by
$\ds f(x_1,x_2)=x_1^4+x_2^2$. 
\be
\item Verify that $f$ is convex on $\R^2$ and that, starting from any $\vecx_0$, in $\R^2$,
the sequence of Newton Method iterates converges to the global minimum $\ds \vecx_{\bigstar}= \begin{bmatrix} 0 \\ 0 \end{bmatrix}$. 
\item Show explicitly that convergence is not quadratic, and explain why the
hypothesis of Theorem \ref{Th:NewtonsMethodConvergenceRateTheorem} is not satisfied. (Hint: To construct a sequence that fails to exhibit quadratic convergence, consider
setting $x_2=0$. Then Newton's Method yields a sequence of points on the $x_1$-axis that converge to zero, but not at a quadratic rate.
\ee
\ee
\endinput

