We propose a duality theory for multi-marginal repulsive cost that appears in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the N-marginals relaxed cost in terms of a stratification formula which takes into account all k interactions with $k\le N$. We then develop a duality framework involving continuous functions vanishing at infinity and deduce primal-dual necessary and sufficient optimality conditions. Next we prove the existence and the regularity of an optimal dual potential under very mild assumptions. In the last part of the paper, we apply our results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for optimal solutions.

我们提出了一种多边际排斥成本的对偶理论，该对偶理论出现在密度泛函理论中出现的最优运输问题中。相关的优化问题涉及整个空间上的概率，并且由于最小化序列可能会在无穷远处失去质量，因此很自然地期望得到放松的解决方案，即子概率。我们首先表征Ñ在一个分层式其中考虑到所有的方面-marginals轻松成本ķ与相互作用$ķ\le ñ$。然后，我们建立一个包含连续函数的无穷二元性框架，该连续性在无穷大时消失，并推论出原始对偶必要和充分的最优性条件。接下来，我们在非常温和的假设下证明最优对偶势的存在性和规律性。在本文的最后一部分，我们将结果应用于涉及给定连续电位的最小化问题，并给出了最佳解决方案的质量量化效果的证据。