The A-truncated K-moment problem

TL;DR

This paper introduces a numerical algorithm based on semidefinite relaxations to determine whether a given truncated moment sequence admits a representing measure on a semialgebraic set, with applications to matrix decomposition and polynomial sum-of-squares problems.

Contribution

It proposes a hierarchy of semidefinite relaxations for solving the A-truncated K-moment problem, providing certificates of nonexistence and methods to find representing measures.

Findings

The algorithm can asymptotically find flat extensions for measures if they exist.

It provides certificates for nonexistence of measures when no solutions are found.

Numerical experiments confirm the effectiveness of the method in various cases.

Abstract

Let A be a finite subset of N^n, and K be a compact semialgebraic set in R^n. 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 or not. This paper proposes a numerical algorithm for solving A-TKMPs. It is based on finding a flat extension of y by solving a hierarchy of semidefinite relaxations {(SDR)_k} 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, 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, we prove that: i) we can asymptotically get a flat extension of y by solving the hierarchy {(SDR)_k\}; 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; this occurred in all our numerical experiments; 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, and hence can be solved numerically by this algorithm.