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Convergence of Lasserre’s hierarchy: the general case

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Abstract

Lasserre’s moment-SOS hierarchy consists in approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming problems. Due to the generality of the initial GMP, applications of this technology are countless, and one can cite among them the polynomial optimization problem, the optimal control problem, the volume computation problem, stability sets approximation problems, and solving nonlinear partial differential equations. The solution to the original GMP is then approximated with finite truncatures of its moment sequence. For each application, proving convergence of these truncatures towards the optimal moment sequence gives valuable insight on the problem, including convergence of the relaxed values to the original GMP’s optimal value. This note proposes a general proof of such convergence, regardless the problem one is faced with, under simple standard assumptions. As a byproduct of this proof, one also obtains strong duality properties both in the infinite dimensional GMP and its finite dimensional relaxations.

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Notes

  1. Indeed an equality constraint \(\int \varphi _\alpha \; d\mu = a_\alpha \) can always be decomposed into two inequality constraints \(\int \varphi _\alpha \; d\mu \le a_\alpha \) and \(\int (-\varphi _\alpha ) \; d\mu \le -a_\alpha \).

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Acknowledgements

The author would like to thank Didier Henrion and Jean Bernard Lasserre for fruitful discussions.

Funding

This work was funded by the French Company Réseau de Transport d’Électricité.

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Authors and Affiliations

  1. LAAS-CNRS, 7 Avenue du Colonel Roche, BP 54200, 31031, Toulouse Cedex 4, France

    Matteo Tacchi

  2. Réseau de Transport d’Electricité (RTE), 7c Place du Dôme, Immeuble Window, 92073, Paris la Défense Cedex, France

    Matteo Tacchi

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Tacchi, M. Convergence of Lasserre’s hierarchy: the general case. Optim Lett 16, 1015–1033 (2022). https://doi.org/10.1007/s11590-021-01757-6

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  • DOI: https://doi.org/10.1007/s11590-021-01757-6

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