On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

Abstract

Consider a finite system of non-strict polynomial inequalities with solution set $S\subseteq \mathbb R^n$. Its Lasserre relaxation of degree d is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most d. It defines a spectrahedron that projects down to a convex semialgebraic set containing S. In the best case, the projection equals the convex hull of S. We show that this is very often the case for sufficiently high d if S is compact and “bulges outwards” on the boundary of its convex hull. Now let additionally a polynomial objective function f be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree d is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if S is compact and d exceeds some bound that depends on the description of S and certain characteristics of f like the mutual distance of its global minimizers on S.