How to Solve Quadratic Matrix Programming with Matrix Inequality Constraint? Let $\mathbf{B} \in \mathbb{R}^{M \times M}$, $\mathbf{X} \in \mathbb{R}^{N \times M}$, and $\mathbf{A} \in \mathbb{R}^{M \times N}$. We intend to solve for $\mathbf{X}$ by solving the following optimization problem
\begin{align}
\arg \min_{\mathbf{X}} Tr( (\mathbf{A} \mathbf{X})^T \mathbf{B} ( \mathbf{A} \mathbf{X})) - 2Tr( \mathbf{B} \mathbf{A} \mathbf{X})
\end{align}
where $Tr()$ is the trace operator. The above problem can be rewritten as
\begin{align}
\arg \min_{\mathrm{vec}(\mathbf{AX})} \mathrm{vec}(\mathbf{AX})^T (\mathbf{B} \otimes \mathbf{I}) \mathrm{vec}(\mathbf{AX}) - 2 \mathrm{vec}(\mathbf{B} ) \mathrm{vec}(\mathbf{AX}).
\end{align}
The above optimization can be solved easily for $\mathrm{vec}(\mathbf{AX})$ as it is a quadratic program with no constraints. Suppose, we are given prior information that $\mathbf{X}_{ik}^{min}<\mathbf{X}_{ik}<\mathbf{X}_{ik}^{max}$. How do I solve it as an inequality constrained optimization problem for $\mathrm{vec}(\mathbf{X})$ not $\mathrm{vec}(\mathbf{AX})$? 

In case $ B $ is a Positive Definite Matrix then there is $ {C}^{T} C = B $ by the Cholesky Decomposition.
So the problem can be rewritten as:
$$\begin{aligned}
\arg \min_{X} \quad & \frac{1}{2} {\left\| A X C \right\|}_{F}^{2} - \operatorname{Tr} \left( D X \right) \\
\text{subject to} \quad & L \leq X \leq U \quad \text{Element wise}
\end{aligned}$$
Where $ D = B A $.
Then the gradient of the objective function is easy:
$$ {A}^{T} A X C {C}^{T} - {D}^{T} $$
Now, just use Projected Gradient Descent and you're done.
Remark
In case the matrix is Positive Semi Definite, then you can use the LDL Decomposition and build $ C $ from there in the same manner. If $ B $ is neither PSd nor PD then the problem is not convex. Then you can do the same but only local solution is guaranteed.
