When I think of Benders Decomposition, I typically think of two-stage stochastic programs. However, I was wondering if there is any application to decomposing a large scale deterministic MILP with only one sub problem each iteration. A potential application could be say a large scale supply chain model where facility location decisions are binary and flow variables and inventory decisions are continuous. For illustration, assume we have an MILP problem split into a master/sub problem as in benders:
The master problem (MP)
$$\min_{x,\theta} c^T x + \theta\\ \text{s.t. } Ax \leq b\\ x \geq 0\\ \text{opt. cuts}\\ x \in \mathbb{Z}$$
The Sub problem (SP) where $(\hat{x_k},\hat{\theta_k})$ is the solution to MP at iteration $k$
$$\min_y d^Ty\\ \text{s.t. } Wy = h-T\hat{x_k}\\ y\geq 0 \\ y \in \mathbb{R}$$
We would then solve the MP and SP for $k = 0,1,2,\dots,n$ adding optimality cuts (assume the subproblem is always feasible for this example) to the master problem in the form $\theta \geq \pi_k^T(h-Tx)$ at each iteration where $\pi_k$ are the optimal dual vectors.
My questions are:
Does having only one subproblem each iteration (as opposed to a multi-cut Benders approach to stochastic programs) limit the effectiveness and number of cuts added to the MP?
For a very large MILP, would decomposing the problem in this way lead to any benefit or does the eventual large number of cuts added to the MP make it too difficult to solve?
Is there a name for this approach so that I can find any literature or published applications where I could learn more?
