Originally posted by: angelo_carlino

Hello, I'm wrting a MILP problem about water resource optimization. I have one variable (ReleaseFromReservoir) that has an upper and lower bound. These two bounds have to be computed using a function depending on two variables (ReservoirLevel, ReservoirInflow). I have a grid of points and the value for each of these points. Is it possible to implement that as a SOS constraint in CPLEX? Just to let you know, I am writing the problem in Pyomo, but I would like to know just if that is feasible in CPLEX.

Best,

Angelo

I don't think there is a direct way to do this. One thing that comes to mind is the following. Assign a continuous variable w_ij with domain [0,1] to the point in R² formed by the combination of the i-th grid value for ReservoirLevel and the j-th value of ReservoirInFlow. Constrain the w variables to sum to 1, so that you are taking a convex combination of grid points. Approximate ReleaseFromReservoir with the sum of w_ij times the function value at the point.

Now comes the tricky part. To make the approximation as accurate as possible, you really want to assign nonzero weights only to points that are corners of a minimal rectangle (i.e., formed from adjacent grid values on each axis). To do this, add binary variables x_i for each grid value of ReservoirLevel and y_j for each value of ReservoirInflow. They will signal which values form the lower left corner of the rectangle being used. The x variables should sum to 1, and so should the y variables. Finally, constrain w_ij to be <= x_i-1+x_i and also <= y_j-1+y_j.