I want to optimize the following objective function subjected to the constraint as given below: $$ min \hspace{2mm} ||x||_1+j$$ $$s.t. \hspace{2mm} ||Ax-\frac{1}{j}(b_1+b_2+\ldots +b_k)||_2 \leq \epsilon \\ 1\leq j \leq k\\$$
$\text{where}, \\ A=m \times n \hspace{2mm} \text{matrix and is known.}\\ x=n \times 1 \hspace{2mm} \text{vector}\\ b_i, i=1,2, \ldots k \hspace{2mm} \hspace{2mm} \text{all} \hspace{2mm} m \times 1 \hspace{2mm} \text{vector and are known.}\\ j=\text{total numbers of $b_i$}$
where, the $k$ is fixed integer value and $j$ is in between $1$ and $k$. Can anyone please suggest me how can I proceed this further. I want average of $j$ $b_i'\text{s}, i=1,2 \ldots k$ and $x$ which minimizes the given objective function. Any general method to solve the given optimization problem?
For each fixed $j$, this is a straight forward problem that can be solved by iterative soft threshholding (ISTA), FISTA, ADMM, IRLS, etc.
Unless the $b_{i}$ vectors have some special property that you haven't told us about, there's no reason to believe that you do better than solving the problem for $j=1$, $2$, $\ldots$, $k$ and then pick the best objective value you found.