On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets

Abstract

Consider a finite system of non-strict real polynomial inequalities and suppose its solution set S⊆Rn is convex, has nonempty interior and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of S. Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on S (which is very natural because of the convexity of S) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on S (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation.In their seminal work of 2009, Helton and Nie showed under the same conditions that S is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a non-constructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of S.