Global Optimization with Polynomials and the Problem of Moments

We consider the problem of finding the unconstrained global minimum of a realvalued 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.