I have a system with $S$ service points. There are also $U$ users in the system.
We have $$U>S>G$$
One group can have maximum $M$ service points, but there is no restrictions on the number of users per group.
A user is served by all the service points belonging to the user's group/cluster. Therefore, if one user happens to be in a group of multiple service points, then it will be served by all the service points in that group. Also, the service points in one group negatively impact the service offered to a user in a different group.
There is a gain between a service point and a user, denoted by $h_{u,s}, u=1,2,\cdots,U, s=1,2,\cdots,S$.
The objective is to maximise the sum-quality of all the users in the system
$$\max\sum_{u=1}^Uq_u$$
How can I mathematically formulate this optimization problem? How to express the constraints mathematically?
Thoughts
A similar equation is given here. How to transform this problem with logarithmic objective function into an approximated convex optimization problem?
The solution suggested there does not help. It is helpful to convexity my objective may be. But my question is different here. I want to have a mathematical formulation. In the previous problem, we had only user and service points lets ay. But in this problem, we have grouping of service points and users, which make the formation difficult.
EDIT:
For linearising constraint (5) in the above formulation, let, $\nu_{u,s,g}=y_{u,g}z_{s,g}$. Then I follow what @RobPratt suggests.