I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my decision variables), I need to minimize some objective functions (OF) which are not directly related to my decision variables (the status of the pipes which is either open or close; 1 or 0 respectively). I have some constraints which bounded (such as minimum and maximum pressure) or linear (such as conservation of mass, or energy loss).
As an example, one of my objective functions is:
OF = Sum [q_i * delta(H_i)]
where delta(H_i) can be considered constant for each i, but q_i is affected by the decision variables (the status of the pipes) but it is not trivial to define a function to formulate their relationship. I can find the changes in q_i's by running a simulation with changing my decision variable.
I assume that this problem is somehow a binary integer programming (as my decision variables are binary and my constraints are bounded or linear). However, my objective functions are not a linear function of the decision variables, which changes the case (to my humble understanding).
Can anybody help me with this? Thanks.