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{\large\bf C\&O 367 \\ 
Assignment 2 
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{\large  Due on Tuesday, Oct. 12,~~~~   Instructor H. Wolkowicz
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({\bf Notes:}\\ TA: Aleksandar Jurisik in MC5159, x6249, ajurisic@violet, office
hour Tues. 11:30-12:30.\\
Instructor: henry@orion, office hour Thurs 10:30-11:30 or by appointment.\\
Questions can also be posed on the newsgroup uw.co.co367.)\\
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\begin{enumerate}

\item
Attempt to solve
\[
   0 = f(x) = (2x+4ax^3)e^{x^2+ax^4}
\]
with
\begin{enumerate}
\item
$a=10, x_0=1$;
\item
$a=10, x_0=2,3,10$.
\end{enumerate}
Use Newton's method and a reasonable tolerance. Do not use safeguarding
or a linesearch. (You can cancel the exponential term in the Newton
iteration to avoid overflow in matlab.)
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
Find an (simple) example of a function that is continuous on $[-1,1]$ but
not Lipschitz continuous. (No oscillating junk is permitted.) 

\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 $x: [-1,+1] \rightarrow \Re^n$, $x \in C^2([-1,+1])$, and let
$f\in C^2(\Re^n)$.  Compute the gradient and Hessian of $f(x(t))$ at $t =
0$.

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

\item
Consider the following two-variable problem
\[ \max f(x) = 2x_1x_2 + 2x_2 - x_1^2 -2x_2^2.
\]
Perform several iterations of steepest ascent using an exact line
search. Start with the initial point (0,0). What is the optimal solution?
\end{enumerate}
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