I have the following objective function. I want to approximate the third term with norm-2 in a way that I could decompose my problem by $j$ (I need $\sum_{j \in J}$ out of the square root). I am thinking about a linear approximation, maybe! $c_{ij}$ is a predefined matrix, and $y$ is a variable.
$\min_{y,\mu,\rho,\theta}\sum_{j\in J} f_{j}\mu_j+\sum_{j \in J}\sum_{i \in I}c_{ij}\xi^{nom}_iy_{ij}+r\|\sum_{j\in J}\sum_{i\in I}c_{ij}y_{ij}\|_2+tR\sum_{j \in J}\theta_j+t\sum_{j\in J}\rho_j$
I tried the Cauchy-Schwarz inequality, but I am not sure if it is correct in my case:
$\|\sum_{j\in J}\sum_{i\in I}c_{ij}y_{ij}\|_1= \sum_{j\in J}\sum_{i\in I}\mid c_{ij}y_{ij}\mid.1\le \big(\sum_{j\in J}\sum_{i\in I}\mid c_{ij}y_{ij} \mid^2\big)^{1/2}\big(\sum_{j\in J}\sum_{i\in I}1^2\big)^{1/2}\leq \sqrt{I \times J}\|\sum_{j\in J}\sum_{i\in I}c_{ij}y_{ij}\|_2$
I would appreciate it if anyone can help! Cheers