# A close-form solution for a simple quadratic optimization problem

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 $$m\leq n$$. 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 $$\alpha = \max_{i} [\mathbf{X}\mathbf{X}^{\mathrm{H}}]_{i,i}$$.

The existence of a closed-form solution appears highly unlikely. Even in the case when $$m=1$$ and $$n=2$$, Mathematica has been working for over 20 min now, with no result:
Even with specific integer entries in $$\mathbf A$$ and $$\mathbf B$$, and still with $$m=1$$ and $$n=2$$, Mathematica takes 2 sec, a huge amount of time for a computer, to give the answer: