# closed form solution for the optimization problem

I have some what complex minimization problem, where the objective function is of the form,

$$\lambda\sum\limits_{i=1}^m\|X_i\|_2 + \nu\|X\|_1 + \frac{\rho}{2}\|XP+Y\|^2_F$$

In the above, $P$ is a double stochastic matrix, $m$ is the number of columns of $X$, and $X$ is a symmetric matrix.

I found many papers discussing soft thresholding strategy for a similar optimization problem with term $\|X-Y\|^2_F$ instead of $\|XP+Y\|^2_F$. But introducing the double stochastic coefficient term complicates the derivation of soft-thresholding. Is there a closed form soft-thresholding based operator to solve the above optimization problem ?

• I'd guess not: this is like putting in a (doubly stochastic) design matrix into the group lasso, which is usually solved by iterative methods. – user795305 May 26 '17 at 14:28