  We consider the problem of the semidefinite representation of a class of
non-compact 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 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 convergent 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.
