Suppose function $f$ is non-convex and non-linear, $P$ is a permutation matrix and $x$ is input data. In this case, we consider a convex relaxation of $P$ by letting $P$ be in the set of doubly stochastic matrices,. How can one solve the following?
$$\begin{array}{ll} \underset{P}{\text{minimize}} & f (P x)\\ \text{subject to} & \displaystyle\sum_i p_{ij} = \displaystyle\sum_j p_{ij} = 1\\ & p_{ij} \geq 0\end{array}$$
Is there a name for theses types of permutation optimization problems?