I have a set of groups $G$. Each group $G_i$ is composed of values $[g_{i0}...g_{ij}]$ where $g_{ij} \in \mathbb{R}$ . I also have a set of values $C_0 ... C_k$ where $C_k \in \mathbb{R}$.
In this problem there are also conceptual "subgroups" $S_{il}$ where $i$ is the group index and $l$ is a subgroup index. $S_{il} \subset G_i$ and $S_{il} \cap S_{im} = \emptyset \ \forall l, m$.
I want to construct subgroups and subgroup matchings to minimize $$\sum_0^k [C_k - \sum S_{il}]$$
In words, I want to create a sum of non-intersecting subgroups closest to each $C_k$
I've been trying to wrap my head around this problem for a while now, and haven't really gotten anywhere productive. The subgroup thing is really throwing me for a loop. I was thinking that a possible max-flow solution could work, but never really got the right construction of the graph down. I also have been thinking that I could try to solve this with linear programming, but my chops are a little rusty.
Would love help with an answer, but if anyone has resources to point me towards I'm always down for a little bit of reading.
Thanks.