Let A\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {A}$$\end{document} be a finite subset of Nn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {N}^n$$\end{document} and R[x]A\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}[x]_{\mathcal {A}}$$\end{document} be the space spanned by monomials xα\documentclass[12pt]{minimal}
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\begin{document}$$x^\alpha $$\end{document} with α∈A\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in \mathcal {A}$$\end{document}. Let K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document} be a compact semialgebraic set of Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^n$$\end{document} such that a polynomial in R[x]A\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}[x]_{\mathcal {A}}$$\end{document} is positive on K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}. Denote by PA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{P}_{\mathcal {A}}(K)$$\end{document} the cone of polynomials in R[x]A\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}[x]_{\mathcal {A}}$$\end{document} that are nonnegative on K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}. The dual cone of PA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{P}_{\mathcal {A}}(K)$$\end{document} is RA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{R}_{\mathcal {A}}(K)$$\end{document}, the set of all truncated moment sequences in RA\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^{\mathcal {A}}$$\end{document} that admit representing measures supported in K\documentclass[12pt]{minimal}
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\begin{document}$$K$$\end{document}. First, we study geometric properties of the cones PA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{P}_{\mathcal {A}}(K)$$\end{document} and RA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{R}_{\mathcal {A}}(K)$$\end{document} (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 PA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{P}_{\mathcal {A}}(K)$$\end{document} and RA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{R}_{\mathcal {A}}(K)$$\end{document}, and prove its asymptotic and finite convergence. Third, we show how to check whether PA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{P}_{\mathcal {A}}(K)$$\end{document} and RA(K)\documentclass[12pt]{minimal}
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\begin{document}$$\fancyscript{R}_{\mathcal {A}}(K)$$\end{document} 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.
