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

Abstract

AbstractWe 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. https://doi.org/10.1007/s10107-020-01537-7) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65–92, 2008. https://doi.org/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).