Abstract

Let A be a finite subset of N^n, and K be a compact semialgebraic set. An A-tms is a vector y indexed by elements in A. The A-Truncated K-Moment Problem (A-TKMP) studies whether a given A-tms y admits a K-measure

(i.e., a Borel measure supported in K) or not. We propose a numerical algorithm for solving A-TKMPs. It is based on finding a flat extension of y by solving a hierarchy of semidefinite relaxations, whose objective R is generated in a certain randomized way. If y admits no K-measures and R[x]_A is K-full, we can get a certificate for the nonexistence of representing measures. If y admits a K-measure, then for almost all generated R, we prove

that: i) we can asymptotically get a flat extension of y; ii) under a general condition that is almost sufficient and necessary, we can get a flat

extension of y. The complete positive matrix decomposition and sum of even powers of linear forms decomposition problems can be solved as an A-TKMP.