Analysis of Non Convex Polynomial Programs by the Method of Moments

In this work we propose a general procedure for estimating the global minima of mathematical programs given in the general form as follows: min P0 (t) s.t. Pi (t) ≤ 0 for i = 1, . . . , k (Π) , where the functions Pi : R n → R are n-dimensional polynomials which are supposed to be non convex. The theory behind the Method of Moments guarantees that all global minima of the non convex program (Π) can be estimated by solving an equivalent convex program. One difficult question about the Method of Moments is how to characterize one particular convex set V of moment vectors. In this work we solve the Multidimensional Truncated Moment Problem in algebraic compact sets. Later we use this result to find the global minima of non convex polynomial programs (Π) by using only one equivalent semidefinite program. In particular, we solve the general non convex quadratic program. Theory and several examples are explained in full detail.