\be
\item For each of the following NLPs, estimate the solution by performing
three iterations of the SQPT using the provided initial values
\be
\item
\begin{align*}
 \mbox{minimize} f(x_1, x_2) &=x_1^2+2x_1x_2x_3-x_3^2 \\
	\mbox{ subject to} \\
	 x_1^2+x_2^2+x_3^2 &= 10 \\
    2x_1+x_2-x_3 &=1
	\end{align*}
along with $\ds \vecx_1=\begin{bmatrix} -1 \\ 1 \\2 \end{bmatrix}$
and  $\ds \muvec_1=\begin{bmatrix} -2 \\ 3 \end{bmatrix}$. (Verify that
your sequence of values,
$\ds \left\{ (\vecx_1,\muvec_1), (\vecx_2,\muvec_2),\ldots \right\}$, agrees
with that obtained by applying Newton's Method to the associated Lagrangian function.)

\item
\begin{align*}
 \mbox{minimize} f(x_1, x_2) &=x_1^2+2x_1-x_1x_2-3x_2^2 \\
	\mbox{ subject to} \\
	 x_1 -\sin(x_2) &\leq 0 \\
    2x_1^2+x_2 &\leq 1 \\
    x_2 &\geq 0 \\
	\end{align*}
along with $\ds \vecx_1=\begin{bmatrix} 1 \\ -1 \end{bmatrix}$,
$\ds \lambdavec_1=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$,
and $\ds \mu_1=1$.

\ee
Consider the NLP
\begin{align*}
 \mbox{minimize} f(x_1, x_2) &=-x_1^2+x_2^2 \\
	\mbox{ subject to} \\
    2x_1^2+x_2^2=6. \\
	\end{align*}
Solve this NLP and demonstrate, by choosing different initial values,
how it is possible to generate a sequence of SQPT iterates
that converges to a KKT point that is not the solution.

\item Use the SQPT to estimate the solution of the \CMC NLP, (\ref{E:ConProNLP_overview_NLP}).

\item Recall \emph{Pam's Pentathlon Training Program} NLP, as discussed
in Exercise 2, from Section \ref{S:NLP_Overview}.

\begin{align*}
 \mbox{maximize } f(x_1,x_2,x_3)&= .11193(254-(180-.5x_1-x_2-x_3))^{1.88} \\
 & \qquad + 56.0211(5.2+.1x_1)^{1.05} \notag \\
\mbox{ subject to} & \notag\\
&6 \leq x_1+x_2+x_3 \leq 10 \notag \\
&2 \leq x_1 \leq 4 \notag \\
&3 \leq x_2 \leq 4 \notag \\
&.6x_1  -.4x_3 \leq 0 \notag \\
& x_1,x_2, x_3 \geq 0 \notag.
\end{align*}

\noi The decision variables, $x_1$, $x_2$, and $x_3$, denote the total number hours per week Pam devotes to weight lifting, distance running, and speed workouts, respectively. The objective function represents the portion of Pam's total pentathlon score stemming from her performances in the 800 meter run and shot put. It is based upon International Amateur Athletic Federation scoring formulas, together with fact Pam currently completes the 800 meter run in 3 minutes and throws the shot put 6.7 meters.
Including sign restrictions, the NLP has ten constraints, which reflect requirements as to how Pam allots her training time.

\newpar
Use the MSQPT, with initial values
$\ds \vecx_1=\begin{bmatrix} 2 \\ 2 \end{bmatrix}$ and
$\ds \lambdavec_1=.5 \vece$, where $\vece$ is the vector in $\R^{10}$
all of whose entries are one, 
to estimate the solution of this NLP.
\ee





