\section{Training for a Pentathlon}\label{S:pentathlontraining}
An athlete wishes to design a training plan for competing in a pentathlon. During a given week,
she can devote time to ten different activities. We will let $x_1$,
$x_2$, $x_3$, $x_4$, and $x_5$ denote the
{\underbar{number of hours per week}} devoted
to the general conditioning activities of weight lifting, sprinting, distance
running, calisthenics, and stair running, respectively.\footnote{Based upon a model presented by S. Ladany in
Optimization of Pentathlon Training Plans,
\emph{Management Science}, \textbf{21}, 1975, 1144-1155. Scoring formulas gathered from the International
Amateur Athletic Association \texttt{www.iaaf.org}.}

\newpar
Similarly,
$x_6$,
$x_7$, $x_8$, $x_9$, and $x_{10}$ denote the
{\underbar{number of hours per week}} devoted
to the technical training activities of shot putting, high jumping, long jumping,
starting from hurdle blocks, and hurdling.

\newpar
The pentathlon itself is made up of five events. The {\underbar{performance}} in each of these events
will be represented as follows:
\begin{align*}
y_1 &= \mbox{100-meter hurdles time in seconds} \\
y_2 &= \mbox{shot put distance in meters} \\
y_3 &= \mbox{high jump in meters} \\
y_4 &= \mbox{long jump distance in meters} \\
y_5 &= \mbox{200-meter dash time in seconds}.
\end{align*}

\newpar
Statistical studies have focused on the extent to which the amount of time spent on each of the training
activities predicts performance in each of these five events. Using multivariate
linear regression, one study has established the following five estimation formulas:

\begin{align*}
\begin{split} y_1 &=  15.475-.035x_1-.149x_2
 -.085x_3 \\ & -.020x_4-.048x_5  -.008x_6-.020x_8-.122x_9-.163x_{10}
 \end{split}\\
\begin{split}  y_2 &= 8.021+1.002x_1+.011x_2
+.013x_3 \\ & +.302x_4+.010x_5 +.899x_6+.012x_8+.193x_9-.011x_{10}
\end{split}  \\
y_3 &= 1.411+.004x_1+.002x_2 +.031x_4+.009x_5+.032x_7+.007x_8   \\
\begin{split} y_4 &= 4.816+.010x_1+.166x_2 \\ &+.002x_3+.050x_4+.033x_5+.015x_7+.149x_8+.132x_{10}\end{split}  \\
\begin{split} y_5 &= 27.00-.033x_1-.302x_2 -.336x_3 \\ & -.004x_4-.167x_5-.004x_6-.004x_8-.266x_9-.136x_{10}.
\end{split}
\end{align*}
For each regression formula, the coefficient of multiple determination is at least .98, which demonstrates a strong goodness-of-fit.

\newpar
Several constraints are taken into account as an athlete trains.
\be
\item The athlete cannot devote more than 15 hours per week to training.
\item In order to reduce the risk of injury, the total weekly time devoted to strength training consisting of weight lifting and upstairs running
should not surpass 5 hours.
\item The amount of weekly speed training, in order to achieve higher basic speed, should be at least the
amount of time devoted to endurance training.
\item The amount of conditioning, consisting of speed and long distance enduring training, should
be more than the amount of technical training.
\item In order to have proper command of a technique, yet to prevent injuries, the amount of time devoted to training for each technical event should be at least 30 minutes per week but no more than 2 hours per week.
\item The following inequalities must hold regarding weight lifting, distance running, and sprints, in order to prevent over-stress and over-training:
\begin{align*}
.5 &\leq x_1 \leq 3.0 \\
2.0 &\leq x_2 \leq 8.0 \\
1.25 &\leq x_3 \leq 5.0.
\end{align*}
\ee

\newpar
IAAF guidelines dictate that point values for womens' events are determined using the following formulas. Assume that
$t$ is time measured in seconds, $d_m$ (resp. $d_c$) is distance measured in meters (resp. centimeters), and $P$ represents the point value.
	\be
	\item 100-meter hurdles: $\ds P= 9.23(26.7-t)^{1.8375}$;
	\item shot put : $\ds P=56(d_m-1.5)^{1.05}$;
	\item high jump: $\ds P= 1.845(d_c-75)^{1.348}$;
	\item long jump: $\ds P= .189(d_c-210)^{1.41}$;
	\item 200-meter dash: $\ds P= 5(42.5-t)^{1.81}$.
	\ee

\newpar
\be
\item Formulate a nonlinear programming problem whose goal is to design a training regimen that maximizes the athlete's pentathlon score subject to the preceding constraints.
\item Show that the NLP you just constructed is a convex program. For the objective function, you may wish to
utilize the fact that it is a sum of composite function, where each composite involves a linear and exponential function.
\item Approximate the solution to this NLP using the Interior Penalty Method and an appropriate numeric algorithm such as the Levenberg-Marquardt
Method.
\item Check your answer using Maple's NLP command.
\ee

