I am looking for a solution $X \in \mathbb{R}^{n \times n}$ which minimizes the following energy:
$$ \textrm{min}_X \| X A X^T - B\|^2 + \mu \|X \Lambda X^T - \Lambda \|^2 $$
where $A, B \in \mathbb{R}^{n \times n}$ are P.D., $\Lambda$ is diagonal with positive entries, $\mu \in \mathbb{R}_{> 0}$, and $ \| \cdot \|$ is the Frobenius norm. I am hoping to find an analytical solution for $X$.
The best I've been able to come up with so far is that the system can be re-written as $$ \textrm{min}_{Y} \| Y \textrm{vec}(A) - \textrm{vec}(B) \|^2 + \mu \| Y\textrm{vec}(\Lambda) - \textrm{vec}(\Lambda) \|^2, \qquad Y = (X \otimes X), \, X \in \mathbb{R}^{n \times n} $$ from which I could solve for some matrix $Y$ but am not sure how to enforce $Y = (X \otimes X)$. This also implies that $\textrm{vec}(\Lambda)$ should be close to an eigenvector of $(X \otimes X)$ with eigenvalue $1$.
How can I go about solving this?