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): Rⁿ 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.