The A-Truncated K-Moment Problem

• Published in: Foundations of computational mathematics Vol. 14; no. 6; pp. 1243 - 1276
• Main Author: Nie, Jiawang
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2014; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Computer Science; Economics; Fuzzy sets; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Set theory; United States; Semidefinite program; A-truncated multisequence; Completely positive matrix; 47A57; 90C22; Sums of even powers; 90C90; Localizing matrix; 44A60; Flat extension
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-014-9225-9

Let A ⊆ N n be a finite set, and K ⊆ R n be a compact semialgebraic set. An A -truncated multisequence ( A -tms) is a vector y = ( y α ) indexed by elements in A . The A -truncated K -moment problem ( A -TKMP) concerns whether or not a given A -tms y admits a K -measure μ , i.e., μ is a nonnegative Borel measure supported in K such that y α = ∫ K x α d μ for all α ∈ A . This paper proposes a numerical algorithm for solving A -TKMPs. It aims at finding a flat extension of y by solving a hierarchy of semidefinite relaxations { ( SDR ) k } k = 1 ∞ for a moment optimization problem, whose objective R is generated in a certain randomized way. If y admits no K -measures and R [ x ] A is K -full (there exists p ∈ R [ x ] A that is positive on K ), then ( SDR ) k is infeasible for all k big enough, which gives a certificate for the nonexistence of representing measures. If y admits a K -measure, then for almost all generated R , this algorithm has the following properties: i) we can asymptotically get a flat extension of y by solving the hierarchy { ( SDR ) k } k = 1 ∞ ; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of y by solving ( SDR ) k for some k ; iii) the obtained flat extensions admit a r -atomic K -measure with r ≤ | A | . The decomposition problems for completely positive matrices and sums of even powers of real linear forms, and the standard truncated K -moment problems, are special cases of A -TKMPs. They can be solved numerically by this algorithm.