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

TL;DR

This paper analyzes the convergence rates of RLT and Lasserre hierarchies for the generalized moment problem over the simplex and sphere, establishing polynomial rates of convergence under certain conditions.

Contribution

It proves convergence rates of O(1/r) for RLT hierarchies over the simplex and O(1/r^2) for Lasserre hierarchies over the sphere, extending previous results and connecting hierarchies.

Findings

RLT hierarchy over the simplex converges at rate O(1/r).

Lasserre hierarchy over the sphere converges at rate O(1/r^2).

Linear RLT relaxation generalizes previous hierarchies for polynomial optimization.

Abstract

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].