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 $\alpha = \max_{i} [\mathbf{X}\mathbf{X}^{\mathrm{H}}]_{i,i}$.
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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:
answered Mar 10, 2020 at 15:58