Canonical Moments for Optimal Uncertainty Quantification on a Variety

The purpose of this work is to optimize an affine functional over positive measures. More precisely, we deal with a probability of failure (P.O.F). The optimization is realized over a set of distributions satisfying moment constraints, called moment set. The optimum is to be found on an extreme point of this moment set. Winkler's classification of those extreme points states they are finite discrete measures. The set of the support points of all discrete measures in the moment set is a manifold over which the P.O.F is optimized. We characterize the manifold's structure by proving it is an algebraic variety. It is the zero locus of polynomials defined thanks to the canonical moments. This reduces a highly constrained optimization over the moment set onto a constraint free manifold.