Linear optimization with cones of moments and nonnegative polynomials

Abstract

Let $\mathcal {A}$ be a finite subset of $\mathbb {N}^n$ and $\mathbb {R}[x]_{\mathcal {A}}$ be the space spanned by monomials $x^\alpha$ with $\alpha \in \mathcal {A}$. Let $K$ be a compact semialgebraic set of $\mathbb {R}^n$ such that a polynomial in $\mathbb {R}[x]_{\mathcal {A}}$ is positive on $K$. Denote by $\fancyscript{P}_{\mathcal {A}}(K)$ the cone of polynomials in $\mathbb {R}[x]_{\mathcal {A}}$ that are nonnegative on $K$. The dual cone of $\fancyscript{P}_{\mathcal {A}}(K)$ is $\fancyscript{R}_{\mathcal {A}}(K)$, the set of all truncated moment sequences in $\mathbb {R}^{\mathcal {A}}$ that admit representing measures supported in $K$. First, we study geometric properties of the cones $\fancyscript{P}_{\mathcal {A}}(K)$ and $\fancyscript{R}_{\mathcal {A}}(K)$ (like interiors, closeness, duality, memberships), and construct a convergent hierarchy of semidefinite relaxations for each of them. Second, we propose a semidefinite algorithm for solving linear optimization problems with the cones $\fancyscript{P}_{\mathcal {A}}(K)$ and $\fancyscript{R}_{\mathcal {A}}(K)$, and prove its asymptotic and finite convergence. Third, we show how to check whether $\fancyscript{P}_{\mathcal {A}}(K)$ and $\fancyscript{R}_{\mathcal {A}}(K)$ intersect affine subspaces; if they do, we show how to get a point in the intersections; if they do not, we prove certificates for the empty intersection.

Appendix: Checking \(K\)-fullness

Recall that $\mathbb {R}[x]_{\mathcal {A}}$ is $K$-full if there exists $p \in \mathbb {R}[x]_{\mathcal {A}}$ that is positive on $K$. We can check whether $\mathbb {R}[x]_{\mathcal {A}}$ is $K$-full or not as follows. Clearly, $\mathbb {R}[x]_{\mathcal {A}}$ is $K$-full if and only if there exists $\lambda \in \mathbb {R}^{\mathcal {A}}$ such that

$$\begin{aligned} \sum _{\alpha \in \mathcal {A}} \lambda _\alpha x^\alpha - 1 \in \fancyscript{P}_{\mathcal {A}^\prime }(K), \end{aligned}$$

(6.1)

where $\mathcal {A}^\prime = \mathcal {A}\cup \{ 0 \}$. Since $1\in \fancyscript{P}_{\mathcal {A}^\prime }(K)$, $\mathbb {R}[x]_{\mathcal {A}^\prime }$ is always $K$-full. Thus, checking $K$-fullness is reduced to solving a feasibility/infeasiblity issue. This question was discussed in Sect. 5. Suppose $K$ is a compact semialgebraic set as in (1.2). If $\mathbb {R}[x]_{\mathcal {A}}$ is $K$-full, we can get a $\lambda$ satisfying (6.1) (cf. Sect. 5.1). If $\mathbb {R}[x]_{\mathcal {A}}$ is not $K$-full, we can get a certificate for nonexistence of such $\lambda$, under a general assumption (cf. Lemma 5.5).