A Semidefinite Approach for Truncated K-Moment Problems

TL;DR

This paper introduces a semidefinite programming approach to solve the truncated K-moment problem, enabling the verification of measure existence and construction of representing measures for truncated moment sequences.

Contribution

It develops a practical SDP-based method to determine the existence of measures and find finitely atomic representing measures when they exist, with theoretical guarantees.

Findings

SDP approach effectively checks for K-measures

Method provides certificates of non-existence of measures

Numerical experiments confirm practical efficiency

Abstract

A truncated moment sequence (tms) of degree d is a vector indexed by monomials whose degree is at most d. Let K be a semialgebraic set.The truncated K-moment problem (TKMP) is: when does a tms y admit a positive Borel measure supported? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms y of degree d admits a K-measure or notcan be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate will be given; when y admits a K-measure, a representing measure for y would be obtained from solving the SDP 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 finds a finitely atomic representing measure for a tms when it admits one.