Flat Truncation in Polynomial Optimization: a Geometric and Algebraic Perspective

In Polynomial Optimization, finite convergence of the Lasserre's Moment and Sums of Squares hierarchies is usually observed in applications, but it is not completely investigated theoretically. In practice, finite convergence is certified using Flat Truncation, a rank condition on the moment matrix of the sequence of moments that realize the minimum. We investigate the Flat Truncation property, studying Lasserre's spectrahedral outer approximations of the convex cone of measures supported on a semialgebraic set. We present different pathological examples and introduce a new generic algebraic condition that is necessary and sufficient for Flat Truncation. Finally, we deduce convergence rates for Lasserre's spectrahedral outer approximations to the cone of measures from a new version of the Effective Putinar's Positivstellensatz. Based on joint works with Bernard Mourrain and Adam Parusinski.