How to find the minimizer of the following optimization problem?
$$\min_P \sum_{i,j=1}^{m,n}\frac{r}{2} \|P_{ij}- Z_{ij}\|^2_2 + \frac{\mu}{2} (\|P_{ij}\|_{1}-1)^2$$
Can we take a simple partial derivative to find closed form solution? or can we use soft thresholding scheme?
Please tell me any suitable method to get the closed form solution of the above optimization problem. Thanks.
Note that there are some other sub-problems in my project, for which I already got the solution.
The given sub-problem can be written in continuous setting as: $$\min_P\frac{r}{2} \int_\Omega|P-Z|^2dx+\frac{\mu}{2}\int_\Omega(|P|-1)^2dx,$$ where $\Omega= [1,N]\times[1,M]$ be a set of $N \times M$ points in $\mathbb{R}^2$. However my question is still that how to the find the closed form solution of the problem with respect to variable $P$?