  We consider the generalized moment problem (GMP) over the simplex and the
sphere. This is a rich setting and it contains NP-hard problems as special
cases, like constructing optimal cubature schemes and rational optimization.
Using the Reformulation-Linearization Technique (RLT) and Lasserre-type
hierarchies, relaxations of the problem are introduced and analyzed. For our
analysis we assume throughout the existence of a dual optimal solution as well
as strong duality. For the GMP over the simplex we prove a convergence rate of
$O(1/r)$ for a linear programming, RLT-type hierarchy, where $r$ is the level
of the hierarchy, using a quantitative version of P\'olya's Positivstellensatz.
As an extension of a recent result by Fang and Fawzi [Math. Program., 2020,
https://doi.org/10.1007/s10107-020-01537-7] we prove the Lasserre hierarchy of
the GMP [Math. Program., Vol. 112, 65-92, 2008] over the sphere has a
convergence rate of $O(1/r^2)$. Moreover, we show the introduced linear
RLT-relaxation is a generalization of a hierarchy for minimizing forms of
degree $d$ over the simplex, introduced by de Klerk, Laurent and Parrilo [J.
Theoretical Computer Science, Vol. 361, 210-225, 2006].
