I'm working on numerical optimization (linear programming), on probability distributions denoted P, Q. We want to find the minimal total variation distance and maximal Bhattacharyya coefficient between P and Q. We can quite easily find the minimal variation distance between P, Q already (via linear programming) but since the Bhattacharyya coefficient is a concave function, finding the optimal value there is harder.
We wonder, is it true that the P, Q pair that maximizes the Bhattacharyya coefficient also minimizes the total variation distance?
If this is true, this would mean that optimizing over the P, Q pairs that minimize the total variation distance should yield the P, Q pair that maximizes the Bhattacharyya coefficient. So we would have to optimize over a much smaller set making the optimization much easier.
I would really appreciate input / references.