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 optim...
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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 linearprogramming, RLT-type hierarchy, where r is the level of the hierarchy, using a quantitative version of Polya's Positivstellensatz. As an extension of a recent result by Fang and Fawzi (Math Program, 2020. https://***/10.1007/s10107-020-01537-7) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65-92, 2008. https://***/10.1007/s10107-006-(X)85-1) 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 et al. (J Theor Comput Sci 361(2-3):210-225, 2006).
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