We present a generalized Farkas’ Lemma for an inequality system involving distributions. This lemma establishes an equivalence between an infinite-dimensional system of moment inequalities and a semi-definite system, assuming that the support for the distributions is a spectrahedron. To the best of our knowledge, it is the first extension of Farkas’ Lemma to the distributional paradigm. Applying the new Lemma, we then establish no-gap duality results between a class of moment optimization problems and numerically tractable semi-definite programs.

中文翻译：

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无间隙对偶半定规划的分布法卡斯引理和矩优化问题

我们提出了涉及分布的不平等系统的广义法卡斯引理。该引理在无限维矩不等式系统和半定系统之间建立了等价关系，假设分布的支持是谱面体。据我们所知，这是法卡斯引理首次扩展到分布范式。应用新的引理，我们在一类矩优化问题和数值上可处理的半定程序之间建立了无间隙对偶结果。