Matrix diagonalization under a constraint

Question

Let $W_i \in M_{n \times n}(\mathbb{R})$ and $W = \begin{bmatrix} W_1 & W_2 &\cdots & W_v\end{bmatrix}^\top \in M_{vn \times n}(\mathbb{R})$. Let $S, D \in M_{vn \times vn}(\mathbb{R})$ be positive definite matrices. How to diagonalize $S^{-1} D$ under the constraint $W^\top S \, W = I_{n \times n}$?