Truncated K-moment problems in several variables

Abstract

Let $\beta \equiv {\beta }^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta )$, and let $r:=rank\mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure μ, which is r-atomic, with supp μ$equalto$\mathcal{V}(\mathcal{M}(n+1))$,thealgebraicvarietyof$\mathcal{M}(n+1)$.Further,\beta hasanr-atomic(minimal)representingmeasuresupportedinasemi-algebraicset$K_{\mathcal{Q}}$subordinatetoafamily$\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$ifandonlyif$\mathcal{M}(n)$ispositivesemidefiniteandadmitsarank-preservingextension$\mathcal{M}(n+1)$forwhichtheassociatedlocalizingmatrices$\mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$arepositivesemidefinite$(1\leq i\leq m)$;inthiscase,\mu (asabove)satisfiessupp\mu \subseteq K_{\mathcal{Q}}$, and μhas precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+°q_{i}}{2}])$atomsin$\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$,$1\leq i\leq m$.

33 pages; to appear in J. Operator Theory

Cited by in corpus (44)

• The A-truncated K-moment problem
• Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
• Moment/Sum-of-Squares Hierarchy for Complex Polynomial Optimization
• Separability discrimination and decomposition of $m$-partite quantum mixed states
• Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
• Non-extremal sextic moment problems
• Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables
• The strong truncated Hamburger moment problem with and without gaps
• Completely positive tensor decomposition
• The Truncated Moment Problem for Unital Commutative R-Algebras
• A Riesz-Haviland type result for truncated moment problems with solutions in $L^1$
• Semidefinite programming relaxations for linear semi-infinite polynomial programming
• Time-dependent moments from the heat equation and a transport equation
• The truncated moment problem on the union of parallel lines
• Generating Polynomials and Symmetric Tensor Decompositions
• Linear Optimization with Cones of Moments and Nonnegative Polynomials
• A Gröbner basis proof of the flat extension theorem for moment matrices
• The Division Algorithm in Sextic Truncated Moment Problems
• Exact Moment Representation in Polynomial Optimization
• The quintic complex moment problem
• Separability of Hermitian Tensors and PSD Decompositions
• The Saddle Point Problem of Polynomials
• Recursively determined representing measures for bivariate truncated moment sequences
• Positivstellensatz and flat functionals on path *-algebras
• The extremal truncated moment problem
• The Multi-Objective Polynomial Optimization
• Tight Relaxations for Polynomial Optimization and Lagrange Multiplier Expressions
• Estimating Mixture Models via Mixtures of Polynomials
• Interiors of completely positive cones
• The Hierarchy of Local Minimums in Polynomial Optimization
• Certifying Convergence of Lasserre's Hierarchy via Flat Truncation
• A Semidefinite Approach for Truncated K-Moment Problems
• Stochastic Polynomial Optimization
• Positivity of Riesz Functionals and Solutions of Quadratic and Quartic Moment Problems
• Regularization Methods for SDP Relaxations in Large Scale Polynomial Optimization
• On Solving a Class of Fractional Semi-infinite Polynomial Programming Problems
• The CP-matrix completion problem
• Distributionally Robust Optimization with Moment Ambiguity Sets
• Cubic column relations in truncated moment problems
• The Multivariable moment problems and recursive relations
• Tensor Eigenvalue Complementarity Problems
• On semi-infinite systems of convex polynomial inequalities and polynomial
• Positive Maps and Separable Matrices
• Real Eigenvalues of nonsymmetric tensors