Multi-marginal optimal transport, density functional theory, and semidefinite programming

We introduce methods from convex optimization to solve the multimarginal transport problem arising in the context of strictly correlated electron 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 based on tensor decomposition to obtain upper bounds to the energy. Numerical experiments demonstrate a gap of order $10^{-3}$ to $10^{-2}$ between the upper and lower bounds. 

This is joint work with Lin Lin, Michael Lindsey, and Lexing Ying.

Yuehaw Khoo is a post-doctoral scholar at Stanford University, working with Lexing Ying. Previously, he did his Ph.D. study at Princeton with Amit Singer. He is interested in the application of optimization and machine learning techniques in biological and physical applications.