Semidefinite Representations of Noncompact Convex Sets

We consider the problem of the semidefinite representation of a class of noncompact basic semialgebraic sets. We introduce the conditions of pointedness and closedness at infinity of a semialgebraic set and show that under these conditions our modified hierarchies of nested theta bodies and Lasserre's relaxations converge to the closure of the convex hull of S. Moreover, if the Putinar-Prestel's Bounded Degree Representation (PP-BDR) property is satisfied, our theta body and Lasserre's relaxation are exact when the order is large enough; if the PP-BDR property does not hold, our hierarchies converge uniformly to the closure of the convex hull of S restricted to every fixed ball centered at the origin. We illustrate through a set of examples that the conditions of pointedness and closedness are essential to ensure the convergence. Finally, we provide some strategies to deal with cases where the conditions of pointedness and closedness are violated.