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

Abstract

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.

Appendix: Generic Finiteness of Critical Varieties

Consider the polynomial optimization problem

$$\begin{aligned} \left\{ \begin{array}{r@{\quad }l} \min &{} f(x) \\ \text {s.t.} &{} h_i(x) = 0\,(i \in [m_1]), \, g_j(x) \ge 0\,( j\in [m_2+1]). \end{array} \right. \end{aligned}$$

(7.1)

For each $J \subseteq \{1,\ldots , m_2+1\}$, denote

$$\begin{aligned} {\mathcal {V}}_J := \{ x\in {\mathbb {C}}^n: \, h(x)=0, g_J(x)=0,\, \text{ rank }\, Jac(f,h,g_J)|_x \le m_1+|J| \}. \end{aligned}$$

The set of critical points of (7.1) with the active set $J$ is contained in ${\mathcal {V}}_J$, which is called a critical variety of (7.1). We show that if the coefficients of $f^{hom}, h^{hom}, g_J^{hom}$ satisfy some discriminantal inequalities, then ${\mathcal {V}}_J$ is finite. We refer to [33, Section 3]) for the definition of discriminants $\varDelta$.

Proposition 7.1

Let $f,h_i\,(i\in [m_1]), g_j \,(j\in [m_2+1]) \in {\mathbb {R}}[x]$, and ${\mathcal {V}}_J$ be defined as above. For any $J \subseteq [m_2+1]$, if

$$\begin{aligned} \varDelta (f^{hom}, h^{hom}, g_J^{hom}) \ne 0, \quad \varDelta (h^{hom}, g_J^{hom}) \ne 0, \end{aligned}$$

then ${\mathcal {V}}_J$ is finite.

Proof

Denote $\tilde{x}:=(x_0,x_1,\ldots ,x_n)$ and by $\tilde{p}$ the homogenization of a singleton or tuple of polynomials $p$. Let ${\mathcal {U}}_J$ be the projective variety in ${\mathbb {P}}^n$ (cf. [19]) defined as

$$\begin{aligned} \text{ rank }\, Jac(\tilde{f}, \tilde{h},\widetilde{g_J})|_x \le m_1 + |J|, \quad \widetilde{h}(\tilde{x}) =\widetilde{g_J}(\tilde{x}) =0. \end{aligned}$$

(7.2)

Clearly, if $u\in {\mathcal {V}}_J$, then $(1,u) \in {\mathcal {U}}_J$. Suppose otherwise ${\mathcal {V}}_J$ is infinite, then ${\mathcal {U}}_J$ is positively dimensional. By Bezout’s Theorem (cf. [19]), ${\mathcal {U}}_J$ must intersect the hyperplane $x_0=0$ in ${\mathbb {P}}^n$, i.e., (7.2) has a solution like $(0,v)$ with $0\ne v \in {\mathbb {C}}^n$. So, $v$ is a solution to the homogeneous polynomial system

$$\begin{aligned} \text{ rank }\, Jac(f^{hom}, h^{hom},g_J^{hom})|_x\le m_1 + |J|, \quad h^{hom}(x) =g_J^{hom}(x)=0. \end{aligned}$$

(7.3)

Since $\varDelta (h^{hom}, g_J^{hom}) \ne 0$, $\text{ rank }\, Jac(h^{hom},g_J^{hom})|_x = m_1 + |J|$ for all $0 \ne x \in V_{{\mathbb {C}}}(h^{hom},g_J^{hom})$. The rank condition in (7.3) implies that $f^{hom}(v)=0$ (cf. [33, Section 3]). Hence, $v$ is a nonzero singular solution to

$$\begin{aligned} f^{hom}(x) = h^{hom}(x) =g_J^{hom}(x)=0, \end{aligned}$$

which contradicts $\varDelta (f^{hom}, h^{hom}, g_J^{hom}) \ne 0$ (cf. [33]). So, ${\mathcal {V}}_J$ must be finite. $\square$