The Moment-SOS Hierarchy

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints " f (x) $\ge$ 0 for all x $\in$ K " and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the " Generalized Problem of Moments " (GPM) whose list of important applications in various domains is endless. We describe this methodology and outline some of its applications in various domains.

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