  Optimization problems on probability measures in $\mathbb{R}^d$ are
considered where the cost functional involves multi-marginal optimal transport.
In a model of $N$ interacting particles, like in Density Functional Theory, the
interaction cost is repulsive and described by a two-point function $c(x,y)
=\ell(|x-y|)$ where $\ell: \mathbb{R}_+ \to [0,\infty]$ is decreasing to zero
at infinity. Due to a possible loss of mass at infinity, non existence may
occur and relaxing the initial problem over sub-probabilities becomes
necessary. In this paper we characterize the relaxed functional generalizing
the results of \cite{bouchitte2020relaxed} and present a duality method which
allows to compute the $\Gamma-$limit as $N\to\infty$ under very general
assumptions on the cost $\ell(r)$. We show that this limit coincides with the
convex hull of the so-called direct energy. Then we study the limit
optimization problem when a continuous external potential is applied.
Conditions are given with explicit examples under which minimizers are
probabilities or have a mass $<1$ . In a last part we study the case of a small
range interaction $\ell_N(r)=\ell (r/\varepsilon)$ ($\varepsilon\ll 1$) and we
show how the duality approach can be also used to determine the limit energy as
$\varepsilon\to 0$ of a very large number $N_\varepsilon$ of particles.
