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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 $\mathrm{vec}(\mathbf{AX})$?

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  • $\begingroup$ Is $ B $ a PD Matrix? $\endgroup$
    – Royi
    May 5 '20 at 12:59
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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.

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