  This paper concerns a method for finding the minimum of a polynomial on a
semialgebraic set, i.e., a set in $\re^m$ defined by finitely many polynomial
equations and inequalities, using the Karush-Kuhn-Tucker (KKT) system and sum
of squares (SOS) relaxations. This generalizes results in the recent paper
\cite{njw_grad}, which considers minimizing polynomials on algebraic sets,
i.e., sets in $\re^m$ defined by finitely many polynomial equations. Most of
the theorems and conclusions in \cite{njw_grad} generalize to semialgebraic
sets, even in the case where the semialgebraic set is not compact. We discuss
the method in some special cases, namely, when the semialgebraic set is
contained in the nonnegative orthant $\re^n_+$ or in box constraints $[a,b]_n$.
These constraints make the computations more efficient.
