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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 Pólya’s Positivstellensatz. As an extension of a recent result by Fang and Fawzi (Math Program, 2020.  10.1007/s10107-020-01537-7 ) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65–92, 2008.  10.1007/s10107-006-0085-1 ) over the sphere has a convergence rate of    $$O(1/r^2)$$  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 et al. (J Theor Comput Sci 361(2–3):210–225, 2006).

Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere