Suppose we have a given matrix $B \in \mathbb{R}^{n \times n}$ that has non-negative entries. I'm interested in how to find saddle points of the function $$f(S, T)=tr(SBT^t)$$ where $S$ and $T$ are doubly stochastic matrices (rows and columns all sum to $1$). Specifically, I'm interested in points $(S, T)$ where $f$ is globally maximized holding $T$ constant, and globally minimized holding $S$ constant (or vice-versa).
Are there any general procedures for doing so? If it makes the problem simpler, you can assume $B$ is non-singular. I believe von Neumann's minimax theorem guarantees the existence of such a point (since $f$ is linear -- and hence both both concave and convex -- in $S$ and $T$ when holding the other constant).
Maybe there is a way to relate this function to extreme points (permutation matrices) in the convex set of doubly stochastic matrices?