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The Moment-SOS hierarchy: Applications and related topics

Part of: Integral transforms, operational calculus General theory of linear operators Hamilton-Jacobi theories, including dynamic programming Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  04 September 2024

Jean B. Lasserre Open the ORCID record for Jean B. Lasserre [Opens in a new window]
Jean B. Lasserre*
Affiliation:
LAAS-CNRS and Toulouse School of Economics (TSE), University of Toulouse, 7 Avenue du Colonel Roche, BP54200, 31031 Toulouse cédex 4, France E-mail: [email protected]
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Abstract

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The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.

MSC classification

Primary: 44A60: Moment problems 90C22: Semidefinite programming
Secondary: 42C05: Orthogonal functions and polynomials, general theory 47A57: Operator methods in interpolation, moment and extension problems 49L25: Viscosity solutions
Type
Research Article
Information
Acta Numerica , Volume 33 , July 2024 , pp. 841 - 908
Copyright
© The Author(s), 2024. Published by Cambridge University Press

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