The $${\mathcal {A}}$$ -Truncated $$K$$ -Moment Problem.

Let $${\mathcal {A}}\subseteq {\mathbb {N}}^n$$ be a finite set, and $$K\subseteq {\mathbb {R}}^n$$ be a compact semialgebraic set. An $${\mathcal {A}}$$ -truncated multisequence ( $${\mathcal {A}}$$ -tms) is a vector $$y=(y_{\alpha })$$ indexed by elements in $${\mathcal {A}}$$ . The $${\mathcal {A}}$$ -truncated $$K$$ -moment problem ( $${\mathcal {A}}$$ -TKMP) concerns whether or not a given $${\mathcal {A}}$$ -tms $$y$$ admits a $$K$$ -measure $$\mu $$ , i.e., $$\mu $$ is a nonnegative Borel measure supported in $$K$$ such that $$y_\alpha = \int _K x^\alpha \mathtt {d}\mu $$ for all $$\alpha \in {\mathcal {A}}$$ . This paper proposes a numerical algorithm for solving $${\mathcal {A}}$$ -TKMPs. It aims at finding a flat extension of $$y$$ by solving a hierarchy of semidefinite relaxations $$\{(\mathtt {SDR})_k\}_{k=1}^\infty $$ for a moment optimization problem, whose objective $$R$$ is generated in a certain randomized way. If $$y$$ admits no $$K$$ -measures and $${\mathbb {R}}[x]_{{\mathcal {A}}}$$ is $$K$$ -full (there exists $$p \in {\mathbb {R}}[x]_{{\mathcal {A}}}$$ that is positive on $$K$$ ), then $$(\mathtt {SDR})_k$$ 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 $$\{(\mathtt {SDR})_k\}_{k=1}^\infty $$ ; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of $$y$$ by solving $$(\mathtt {SDR})_k$$ for some $$k$$ ; iii) the obtained flat extensions admit a $$r$$ -atomic $$K$$ -measure with $$r\le |{\mathcal {A}}|$$ . 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 $${\mathcal {A}}$$ -TKMPs. They can be solved numerically by this algorithm. [ABSTRACT FROM AUTHOR]