Weighted Norm Minimization

Question

I have a minimization problem of the form $min (w_1 \|x\|+w_2\|y\|)$ subject to constraints $A_1x=b_1$, $A_2y=b_2$, $0< l_1 \leq x \leq u_1$ and $0< l_2 \leq y \leq u_2$ where $l_1,l_2,u_1,u_2\in \mathbb{R}^n $ are vectors of lower and upper bounds and $x,y,b_1,b_2\in \mathbb{R}^n$. The weights $w_1,w_2\in \mathbb{R}$ are constant. . How can I solve this problem?

Answer 1

You can split your problem into $$ \min \|x\| \quad {s.t.} A_1 x = b_1, \ l_1\le x\le u_1, $$ and $$ \min \|y\| \quad {s.t.} A_2 y = b_2, \ l_2\le y\le u_2. $$ There is no coupling between both optimization variables.