Given matrices $A,B\in\mathbb{R}^{n\times n}$ and $M,N\in\mathbb{R}^{v\times v}$. Find a nonzero matrix $P\in\mathbb{R}^{v\times n}$, where $v \leq n$, such that
$$\min_{P} \| PA - MP \| + \| P^T N P - B \|.$$
Can you please suggest me algorithms/methods to solve such a problem? What class of optimization problems does this problem belong to?