We study the optimal power flow (OPF) problem in electrical distribution networks. The objective is to minimize total line losses subject to the physical constraints governing the network. This optimization problem is inherently non-convex, making it challenging to solve in practice. In particular, classical solution methods may fail to converge to a global minimum.

A common strategy to address this difficulty is to consider a convex relaxation, where the original non-convex feasible set is replaced by a larger convex set. In this work, we focus on a semidefinite programming (SDP) relaxation obtained by dropping a rank constraint on certain matrix variables.

The main goal of this talk is to show that this relaxation is in fact exact, meaning that it yields the same solution as the original OPF problem. To establish this result, we rely on two key ingredients: the characterization of the Pareto front of the feasible sets and the exploitation of the tree structure of the network.