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Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere
Kirschner,Felix de Klerk,Etienne
Kirschner,Felix
de Klerk,Etienne
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 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/r2). 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).
Description
Funding Information: The authors are supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement N. 813211 (POEMA) Publisher Copyright: © 2022, The Author(s).
Date
2022-11
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Keywords
Generalized moment problem with polynomials, Linear programming hierarchies, Semidefinite programming hierarchies
Citation
Kirschner, F & de Klerk, E 2022, 'Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere', Optimization Letters, vol. 16, no. 8, pp. 2191–2208. https://doi.org/10.1007/s11590-022-01851-3