We discuss algorithms for optimizing polynomials on
semialgebraic sets using representation theorems from real algebraic
geometry for positive polynomials. In the case of compact semialgebraic
sets, the method of Lasserre generates a sequence of SDP relaxations
which converge to the solution, however this method does not always
work in the non-compact case. We will discuss work of Demmel, Nie,
and Sturmfels in the global case and joint work with Demmel and Nie
in the case of non-compact semialgebraic sets.