The importance of Relaxations and Benders Cuts in Decomposition Techniques: Two Case Studies

When solving combinatorial optimization problems it can happen that using a single technique is not efficient enough. In this case, simplifying assumptions can transform a huge and hard to solve problem in a manageable one, but they can widen the gap between the real world and the model. Heuristic approaches can quickly lead to solutions that can be far from optimality. For some problems, that show a particular structure, it is possible to use decomposition techniques that produce manageable subproblems and solve them with different approaches. Benders Decomposition [1] is one of such approaches applicable to Integer Linear Programming. The subproblem should be a Linear Problem. This restriction has been relaxed in [4] where the technique has been extended to solvers of any kind and called Logic-Based Benders Decomposition (LBBD). The general technique is to find a solution to the first problem (called Master Problem (MP)) and than search for a solution to the second problem (Sub-problem (SP)) constraining it to comply with the solution found by the MP. The two solvers are interleaved and they converge to the optimal solution (if any) for the problem overall. When solving problems with a Benders Decomposition based technique, a number of project choices arises: