On the exactness of Lasserre’s relaxation for polynomial optimization with equality constraints

We study exactness condition for Lasserre’s relaxation hierarchy for polynomial optimization problem over R with equality constraints defined by n polynomials g1, . . . , gn. Under the assumption that the quotient ring R[x]/(g1, . . . , gn) has dimension equal to ∏n i=1 deg(gi), we show that any polynomial optimization problem can be solved by Lasserre’s relaxation of order at most ∑n i=1 deg(gi)− n if the degree of the objective function is smaller than 2 (∑n i=1 deg(gi)− n ) . When the common zero locus are real and all of multiplicity one, for relaxation of order d < ∑n i=1 deg(gi)−n, we describe the exact region as the convex hull of the moment map image of a vector subspace. The convex hull lies in the β-tubular neighborhood of the moment map image with respect to the relative entropy distance for some universal constant β. For d = ∑n i=1 deg(gi) − n − 1, the convex hull coincides with the moment map image, and is diffeomorphic to its amoeba. Based on the theory of amoeba, we obtain an explicit description of the exact region for order ( ∑n i=1 deg(gi) − n − 1) relaxation, from which we further derive error estimations for relaxation of this order. Keywords— Lasserre’s hierarchy, polynomial optimization, degree bound, exactness condition, hyperplane amoeba MSC Classification— 65K05, 90C23, 90C60