I am trying to solve the below optimization problem
\begin{equation*} \begin{aligned} & \underset{{A}, {B}, {\Lambda}}{\text{minimize }} & & \|X - AB\Lambda C^TD^T|_F^2 \\ & \text{subject to} & & \Lambda \text{ and } B\Lambda C^T \text{ are diagonal matrices}\\ \end{aligned} \end{equation*} where $X \in R^{n\times n}, A\in R^{n\times k_1}, D\in R^{n\times k_1}, B \in R^{k_1\times k_2}, C \in R^{k_1\times k_2}, \Lambda \in R^{k_2\times k_2}$, $X$ is known and $n > k_1 > k_2$. I am not sure how to put a constraint on $B$, $C$ and $\Lambda$ such that $B\Lambda C^T$ is a diagonal matrix. I know I can solve for all unknown using alternative minimization but I am not sure how to include the constraint. I have asked a similar question here Diagonal constraint on product of matrices
Edit 1: $A, B, C$ and $D$ are sparse matrices i.e. $|B|_1 < \lambda$