How put Minimize in place of min in a typical optimization problem?

Using amsmath you can set this up as a starred version of a math operator so that the limits get correctly placed.

\documentclass{article}

\usepackage{amsmath}

\DeclareMathOperator*{\minimize}{minimize}

\begin{document}
$\displaystyle{\minimize_{x\in S} f(x)}$
\end{document}

This solution requires no preamble definitions:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
$\underset{x \in S}{\text{minimize}} f(x)$
\end{document}