  In this paper we study necessary optimality conditions for the optimization
problem $$\textrm{infimum}f_0(x) \quad \textrm{ subject to } \quad x \in S,$$
where $f_0 \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is a polynomial function
and $S \subset \mathbb{R}^n$ is a set defined by polynomial inequalities.
Assume that the problem is bounded below and has the Mangasarian--Fromovitz
property at infinity. We first show that if the problem does {\em not} have an
optimal solution, then a version at infinity of the Fritz-John optimality
conditions holds. From this we derive a version at infinity of the
Karush--Kuhn--Tucker optimality conditions. As applications, we obtain a
Frank--Wolfe type theorem which states that the optimal solution set of the
problem is nonempty provided the objective function $f_0$ is convenient.
Finally, in the unconstrained case, we show that the optimal value of the
problem is the smallest critical value of some polynomial. All the results are
presented in terms of the Newton polyhedra of the polynomials defining the
problem.
