A multi-objective perspective on block-structured integer programs with one soft coupling constraint
This paper presents a multi-objective perspective on block-structured integer programs featuring a single soft coupling constraint. By interpreting the coupling constraint as a second objective, we transform the coupled single-objective problem into an additively-separable bi-objective optimization problem. To avoid the expensive computation of the full Pareto front, we introduce an algorithm, which uses a binary search to isolate a region of interest around the soft constraint limit. This algorithm provides provable bounds on the single-objective optimum. We further enhance this algorithm, by exploiting the block-structure, using a novel $\lambda$-lookup mechanism to skip repeated sub-problem calculations. Finally, for scenarios requiring all non-dominated solutions within the region of interest, we propose a new approach, that works its way from the middle of the region of interest outwards. This algorithm shows quick convergence in terms of representation. Computational studies demonstrate that our methods dramatically reduce integer programming calls, thereby outperforming traditional dichotomic search. For large instances the method works as a strong heuristic providing bounds on the gap to an optimal solution, providing trade-off information in addition to the solution.