Exact Moment Representation in Polynomial Optimization

TL;DR

This paper analyzes the conditions under which moment matrix hierarchies precisely represent measures in polynomial optimization, providing theoretical insights and bounds for exactness and flat truncation.

Contribution

It establishes the equivalence between flat truncation and zero-dimensional support, and extends the duality between MomentMatrix and Sum of Squares hierarchies with real radical support considerations.

Findings

Flat truncation occurs if and only if the support of the quadratic module is zero-dimensional.

The dual of the MomentMatrix hierarchy coincides with the Sum of Squares hierarchy extended with the real radical.

Boundary Hessian Conditions ensure the generic exactness of the MomentMatrix hierarchy.

Abstract

We investigate the problem of representing moment sequences by measures in the context ofPolynomial Optimization Problems, that consist in finding the infimum of a real polynomial ona real semialgebraic set defined by polynomial inequalities. We analyze the exactness of MomentMatrix (MoM) hierarchies, dual to the Sum of Squares (SoS) hierarchies, which are sequences ofconvex cones introduced by Lasserre to approximate measures and positive polynomials. Weinvestigate in particular flat truncation properties, which allow testing effectively when MoMexactness holds and recovering the minimizers.We show that the dual of the MoM hierarchy coincides with the SoS hierarchy extendedwith the real radical of the support of the defining quadratic module Q. We deduce thatflat truncation happens if and only if the support of the quadratic module associated withthe minimizers is of dimension zero. We also bound the order of the hierarchy at which flattruncation holds.As corollaries, we show that flat truncation and MoM exactness hold when regularityconditions, known as Boundary Hessian Conditions, hold (and thus that MoM exactness holdsgenerically); and when the support of the quadratic module Q is zero-dimensional. Effectivenumerical computations illustrate these flat truncation properties.