


\section{\bf 4:\ \ Minimization}

\subsection{\bf 4\hspace{1em} Minimization}
\subsection{\bf 4.1\hspace{1em} Statement of Problem}

\begin{slide}
\begin{itemize}
\item The basic problem is to minimize $f(\alpha_1 \ldots \alpha_n
)$---a function of $n$ variables $\alpha_1 \ldots \alpha_n$.

\item Equivalently, we must find the zero of an arbitrary function
$f^\prime (x_1 \ldots x_n)$ of $n$ variables.

\item This arises in previous sections when we wish either to minimize
$\chi^2$ or maximize the likelihood $L$ (equivalently to
minimize---$\ln L$).

\item The methods divide according to
\begin{description}
\item{(i)} Can calulate $f$ only
\item{(ii)} Can calculate $f$, $f^\prime$, and $f^{\prime\prime}$.
\end{description}

\item (i) is extremely time consuming (i.e., it takes a lot of function
evaluations to converge to desired minimum).

\item In all problems I have faced, it has been simple to calculate
derivatives and so I will only discuss (ii).
\end{itemize}
\end{slide}

