Algebraic certificates for the truncated moment problem

TL;DR

This paper explores algebraic certificates for the truncated moment problem, providing methods to determine whether a given vector of numbers corresponds to a measure on a set K, with applications in optimization and control.

Contribution

It introduces algebraic certificates for the truncated moment problem and analyzes the complexity of computing these certificates using computer algebra algorithms.

Findings

Established a duality framework between moment cones and polynomial positivity.

Developed algorithms for certifying non-moment vectors via algebraic certificates.

Analyzed the computational complexity of finding such certificates.

Abstract

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.