Convex relaxation approaches for strictly correlated density functional theory

In this paper, we introduce methods from convex optimization to solve the multimarginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of $N$-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy. Numerical experiments demonstrate a gap of the order of $10^{-3}$ to $10^{-2}$ between the upper and lower bounds. The Kantorovich potential of the multi-marginal transport problem is also approximated with a similar accuracy.