$$\begin{equation}\begin{split}\mathbf{W}^{(k+1)}&=\text{argmin}_{\mathbf{W}}\left\|\mathbf{F}-\mathbf{A\mathbf{W}}^{(k)}\right\|_F^2+\frac{\alpha_1}{2}\left\|\mathbf{B}\mathbf{W}^{(k)}-\mathbf{W}_\text{Air}+\frac{1}{\alpha_1}\mathbf{\Lambda}\right\|_F^2\\&+\frac{\alpha_2}{2}\left(\left\|\max\left(\frac{\Phi_1}{\alpha_2}+\left(\mathbf{W}^{(k)}-\mathbf{E}\right),0\right)\right\|_F^2-\frac{\left\|\Phi_1\right\|_F^2}{\alpha_2^2}\right)\\&+\frac{\alpha_3}{2}\left(\left\|\max\left(\mathbf{W}^{(k)}-\frac{\Phi_2}{\alpha_3},0\right)\right\|_F^2-\frac{\left\|\Phi_2\right\|_F^2}{\alpha_3^2}\right)\end{split}\end{equation}\label{1}\tag{1}$$ where
- $\mathbf{W}$, $\mathbf{F}$,$\mathbf{A}$,$\Phi_{1}$,$\Phi_2,\mathbf{E}, \mathbf{\Lambda},\mathbf{B},\mathbf{W}_\text{Air}$ are matrices ($\mathbf{W}_\text{Air}$ has nothing to do with $\mathbf{W}$, despite the similarity of their symbols), and
- $\alpha_1,\alpha_2 $ and $\alpha_3$ are scalars.
I need to determine $\mathbf{W}^{(k+1)}$ in such a way that Formula \eqref{1} above reach the local minimum. I am inclined to directly differentiate the formula respect to $\mathbf{W}^{(k+1)}$ and then equate the found derivative to 0, but I am at a loss for the $\max$ sign in the formula. How should I derive this formula? How should I Find the local minimum of $\mathbf{W}^*$?