A generalized flat extension theorem for moment matrices

Abstract

Abstract. In this note we prove a generalization of the flat extension theorem of Curto and Fialkow (Memoirs of the American Mathematical Society, vol. 119. American Mathematical Society, Providence, 1996) for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators. Mathematics Subject Classification (2000). Primary 30E05; Secondary 12D10.

The Approach of Moments for Polynomial Equations

“…Then one can show that the ideal J = Ker M B + (Λ) is real radical, zerodimensional, and contained in R √ I, and thus V R (I) ⊆ V C (J); this result relies on a generalization of the flat extension theorem (Theorem 8) proved in [22,Thm. 1.4].…”

“…where u i ∈ R[x], σ j are sums of squares of polynomials with deg(u i h i ), deg(σ j g j ) ≤ t. Then, f * t ≤ f * for all t. Moreover, asymptotic convergence of (21) and (22) to the minimum f * of (19) can be shown when the feasible region of (19) is compact and satisfies some additional technical condition (see [15]). We now group some results showing finite convergence under certain rank condition, which can be seen as extensions of Theorems 11 and 12.…”

Border basis relaxation for polynomial optimization

“…In this section, we describe new criterion to check when the kernel of a truncated Hankel operator associated to an optimal linear form for f yields the generators of the minimizer ideal. It involves the flat extension theorem of (Laurent and Mourrain, 2009) and applies to polynomial optimization problems where the minimizer ideal I min is zero-dimensional. At the end of this section we verify the flat extension property in our running example.…”

“…We recall here a result from (Laurent and Mourrain, 2009), which gives a rank condition for the existence of a flat extension of a truncated Hankel operator 1 .…”

“…A new stopping criterion is given to detect when the relaxation sequence reaches the infimum, using a flat extension criterion from (Laurent and Mourrain, 2009). We also provide a new algorithm to reconstruct a finite sum of weighted Dirac measures from a truncated sequence of moments.…”