Closed form solution for quadratic optimization

Question

Is there any closed form solution for the following optimization problem: \begin{align} &\min_{\mathbf{X},\alpha} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\mathrm{H}}] + \alpha\\ &\mathrm{s.t.}\qquad [\mathbf{X}\mathbf{X}^{\mathrm{H}}]_{i,i}\leq \alpha, \qquad\forall i, \end{align} where $\mathbf{X}_{n\times m}$, $\mathbf{B}_{m\times n}$, $\mathbf{A}_{m\times m}$ are complex matrices with $m\leq n$, $[.]_{i,i}$ denotes the $i$th diagonal entries of matrix. This problem could be considered as the following unconstrained optimization problem \begin{align} \min_{\mathbf{X}} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\mathrm{H}}] + \max_{i} [\mathbf{X}\mathbf{X}^{\mathrm{H}}]_{i,i}, \end{align} and $\alpha = \max_{i} [\mathbf{X}\mathbf{X}^{\mathrm{H}}]_{i,i}$.