Hi Guys,
So when I formulate a problem I am trying to solve for work, the above (please see attached figure) optimization framework results. I am not too familiar with optimization techniques (apart from linear and simple convex techniques). Can anybody tell me if this is problem has a solution, and if so, how I would go about acheiving it? Any references or leads would be appreciated as well.
As an engineer, I am interested in a numeric solution (I understand that this problem can have multiple solutions), but if this can be reduced to some analytical form to make it resemble some other standard optimization problem, I would be interested in that too.
Thanks a lot in advance for taking the time to read through this!
Assuming the subsets $X_i,E_i$ are given and their indices are denoted as subsets $S_i$, I guess you just need an integer programming formulation in which you can use a binary variable $f_i$ in place of the function:
$ \max \sum_t f_i$
$ k - \sum_{i \in S_i} (x_i+e_i) \leq M_i f_i \quad i=1,\ldots,m$
$ \sum_{i \in S_i} (x_i+e_i) -k \leq M_i (1-f_i) \quad i=1,\ldots,m$
$ lb\leq e_i \leq ub \quad i=1,\ldots,n$
$\sum_{i=1}^n e_i x_i= 0 $
$ f_i\in \{0,1\} \quad i=1,\ldots,m$
where $M$ is an upper bound to $|k - \sum_{i \in S_i} x_i+e_i |$, the so-called big$-M$. The tighter $M_i$, the better. Please check I got the right logical implication. But the overall idea should be OK.