I am working on a two-stage facility location problem as I described in this question.

I am solving it with the L-shaped method (Benders decomposition). The cost value between each $(i,j)$ is a constant. Although the recourse is integers, I converted the $x$ and $w$ variables to a continuous variable so that I can consider the recourse as linear and can take the dual of the recourse as an approximation. However, if I consider a real road network, the cost is actually a function of the flow on the road. A common cost function for the road network is the BPR function which is expressed as $t(x)=t^0\left(1+\alpha\left(\dfrac xc\right)^\beta\right)$.

This makes the recourse function non-linear (still convex). Any idea how could this type of problem be solved in the L-shaped method?


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