I'm trying to model something that's somewhere between a max flow problem and a regular shortest path problem. The interesting twist is that the cost of traversing a specific edge E is a convex function related to the amount of flow traveling through that edge (and some constants, specific to that edge).
I've solved this for the "0 marginal cost" case, where I'm sending an infinitesimally small amount of flow through the graph. How would I solve this if, let's say, I wanted to maximize flow f through path P?
One saving grace of this problem is that the amount is a scalar variable, and I can assume all the output of one node goes into another node. I'm thinking of it as f4(f3(f2(f1(x)))), and am trying to figure out what the optimal path would be.
Thanks.