# Projection onto Birkhoff Polytope

Suppose we would like to compute the Euclidean projection of an arbitrary matrix $A$ onto the Birkhoff polytope, the set of doubly-stochastic matrices.

1. Under some conditions on $A$, Sinkhorn's algorithm returns two diagonal matrices $D_1,D_2$ such that $D_1 A D_2$ is a doubly-stochastic matrix (which is unique). How does this matrix compare to the Euclidean projection, and if not, does this solution correspond to the projection under any particular metric?
2. Are there known methods for solving this efficiently besides just formulating the problem as a quadratic program and applying a generic QP algorithm?