Let $A \in \mathbb R^{n \times m}$, $B \in \mathbb R^{n\times r}$, $X \in \mathbb R^{m\times r}$, and $Y\in \mathbb{R}^{n\times m}$. Let $\|\cdot\|_F$ be the Frobenius norm of a matrix. How can we solve the following optimization problem in $X \in \mathbb R^{n \times n}$?
$$\begin{array}{ll} \min_{_{X,B}} & \|A-BX^\top\|^2_F\\ \text{subject to} & \text{Tr}(Y^\top(A-BX^\top))\leq 0\end{array}$$
Can this problem be solved in closed form?
Are there some references to solving the inequality constrained linear least Frobenius norm problems?
Thanks!