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{\large\bf C\&O 367 \\ 
Assignment 3 
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{\large  Due on Thursday, Feb. 13,~~~~   Instructor H. Wolkowicz
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{\bf NOTE:} Several exercises are marked: {\bf Do not hand in.}
\begin{enumerate}

\item
({\bf Do not hand in.}) 
Attempt to solve
\[
   0 = f(x) = (2x+4ax^3)e^{x^2+ax^4}
\]
with
\begin{enumerate}
\item
$a=10, x_0=10$;
\item
$a=10, x_0=1$.
\end{enumerate}
Use Newton's method and a reasonable tolerance.
Why is there such a difference in the iteration count?
Note that $f(x)$ is the derivative of $e^{x^2+ax^4}$.

\item
In the notes, we discuss the {\em scale invariance} of Newton's
algorithm with respect to an affine change of variables.  
Define this.  Show that
Newton's algorithm is invariant with respect to affine transformations.  Show
that the Lipschitz constant is not.  Define a relative measure of nonlinearity.
Show that your relative measure of nonlinearity is scale invariant.

\item
({\bf Do not hand in.})
Find an (simple) example of a function that is continuous on $[-1,1]$ but
not Lipschitz continuous. (No oscillating junk is permitted.) 

\item
({\bf Do not hand in.})
Find an example of a function that satisfies all of the conditions for the
convergence of Newton's algorithm {\em except} Lipschitz continuity of the
derivative, and for which Newton's algorithm does not converge quadratically.

\item
Compute the gradient and Hessian of the following functions:
\begin{enumerate}
\item $f(x) = x^t A x$, where $A$ is an $n\times n$ 
matrix ($A$ is not assumed to
be symmetric).
\item $f(x) = \half x^t A x + b^t x$ ($A$ symmetric).

\item $f(x) = g(x)^t g(x)$ where $g:\Re^n \rightarrow \Re^m$ and $g$ is twice
differentiable. 
\end{enumerate}

\item
Let $x: [-1,+1] \rightarrow \Re^n$, $x \in \C2([-1,+1])$, and let
$f\in\C2(\Re^n)$.  Compute the gradient and Hessian of $f(x(t))$ at $t =
0$.

\item
Consider the problem 
\[
	\min_{x,y}	 \{ f(x,y) = x^2 - xy + 2y^2 - 2x + {\rm e}^{x+y}\}.
\]
Is the first-order necessary condition also sufficient?

\item
 Find and classify the critical points of the functions:
\begin{mlist}
\item $f(x) = 2x_1^3 - 3x_1^2 - 6x_1 x_2(x_1 - x_2 - 1)$.

\item $f(x) = 2x_1^2 + x_2^2 - 2x_1x_2 + 2x_1^3 + x_1^4$.
\end{mlist}

\item
({\bf Do not hand in.})
 Let $f(x,y) = x^4 + y^2$.  Let $(x,y) = (1,1)$ and let $s$ be the Newton
direction.  Compare the stepsizes computed by
\begin{enumerate}
\item Pure Newton,
\item Backtracking Linesearch,
\item Exact Minimization (approximately).
\end{enumerate}


\end{enumerate}
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