How to find a probability transfer matrix $P$ which closest turns two $P_1, P_2$ into $Q_{1},Q_{2}$?

Question

So I want to solve

$$P_o = \min_{P} \{\|PP_1 - Q_1\| + \|PP_2 - Q_2\|\}$$

This would be easy if using $L_2$ norm and I had no restrictions on $P$ and if we also had only one matrix to aim for, but now $P$ needs to be a stochastic matrix - non-negative entries and summing to 1.

But how to solve it in a more generic context with $L_p$ norm and the constraints of stochastic matrices.