Obtaining tractable characterizations of functions which are nonnegative on a set K 伡 Rn is a topic of primary importance. Indeed,such characterizations are highly desirable to help solve (or at least
approximate) many important problem in various areas, and in particular, the global optimization problem:P,because solving P is equivalent to solving f When f is a polynomial and K a basic semi-algebraic set, we have seen in the previous talk that Putinar's Positivstellensatz provides such
tractable characterizations. Those characterizations depend on the representation of K through its defining polynomials.
In this talk we consider another way to look at continuous functions that are nonnegative on a (non necessarily compact basic semi-algebraic) set K<- Rn . This time, knowledge on K is through a finite Borel measure with support sup = K , and whose all moments na y = ( y ), a <-N , are available. This new characterization permits to define convergent outer approximations of the convex cone C (K) d of polynomials of degree at most d, nonnegative on K, by a hierarchy of spectrahedra (convex sets defined by linear matrix inequalities) defined uniquely in terms of the coeffcients of f. Important examples of cones C (K) d are the cone of nonnegative polynomials on Rn and the cone of copositve matrices. Checking whether a fixed and known polynomial f is nonnegative on K reduces to solving a sequence of generalized eigenvalue problems for real symmetric matrices of increasing size.