Algebraic certificates for the truncated moment problem

Didier Henrion (1, 2) , Simone Naldi (3) , Mohab Safey El Din (4)

1 Department of Control Engineering, Faculty of Electrical Engineering [Prague]
2 LAAS-POP - Equipe Polynomial OPtimization
3 XLIM - XLIM
4 PolSys - Polynomial Systems

The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure on a given set K of a finite-dimensional Euclidean space. This problem has plenty of applications e.g. in optimization, control theory and statistics. When K is a compact semialgebraic set, the duality between the cone of moments of non-negative measures on K and the cone of non-negative polynomials on K yields an alternative: either y is a moment vector, or y is not a moment vector, in which case there exists a polynomial strictly positive on K making a linear functional depending on y vanish. Such a polynomial is an algebraic certificate of moment unrepresentability. We study the complexity of computing such a certificate using computer algebra algorithms.

Mots clés: moments; sums of squares; semialgebraic sets; real algebraic geometry; algorithms; complexity

Domaines: Géométrie algébrique [math.AG]; Optimisation et contrôle [math.OC]

Didier Henrion, Simone Naldi, Mohab Safey El Din. Algebraic certificates for the truncated moment problem. 2023. ⟨hal-03987123⟩