Global optimization with polynomials and the problem of moments

We consider the problem of finding the unconstrained global minimum of a real-valued polynomial p(x) : R-n-->R, as well as the global minimum of p (x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality( LMI) problems. A notion of Karush-Kuhn-Tucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided.