Convergence of Lasserre's hierarchy: the general case

TL;DR

This paper provides a general proof of convergence for Lasserre's hierarchy in approximating solutions to the generalized moment problem, applicable across various fields like polynomial optimization, control, and PDEs.

Contribution

It offers a universal convergence proof for Lasserre's hierarchy under standard assumptions, enhancing understanding of its applicability and duality properties.

Findings

Proves convergence of moment relaxations to the GMP's optimal solution.

Establishes strong duality in both infinite and finite-dimensional formulations.

Applicable to a wide range of problems including polynomial optimization and PDEs.

Abstract

Lasserre's moment-SOS hierarchy consists of approximating instances of the generalized moment problem (GMP) with moment relaxations and sums-of-squares (SOS) strenghtenings that boil down to convex semidefinite programming (SDP) 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 (POP), the optimal control problem (OCP), the volume computation problem, stability sets approximation problems, and solving nonlinear partial differential equations (PDE). 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.