A Semidefinite Approach for Truncated K-Moment Problems

Abstract

A truncated moment sequence (tms) in n variables and of degree d is a finite sequence y=(yź) indexed by nonnegative integer vectors ź:=(ź1,ź,źn) such that ź1+ź+źn≤d. Let K⊆źn be a semialgebraic set. The truncated K-moment problem (TKMP) is: How can one check if a tms y admits a K-measure μ (a nonnegative Borel measure supported in K) such that $y_{\alpha}= \int_{K} x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\,\mathrm{d}\mu$ for every ź? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms admits a K-measure or not can be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate for the nonexistence can be found; when y admits one, a representing measure for y can be obtained from solving the SDP problems under some necessary and some sufficient conditions. Moreover, we also propose a practical SDP method for finding flat extensions, which in our numerical experiments always found a finitely atomic representing measure when it exists.