Sparse facility location and network design problems

To further minimize the number of facilities even if it entails additional costs, policymakers often have this preference during facility location decisions. To cater to this preference, we introduce a sparsity-inducing term in this paper. This term generates sparse solutions for both the facility location model and the facility network design model, leading to the proposal of a sparse facility location model and a sparse facility network design model. These two sparse models are formulated as nonlinear mixed-integer programs, featuring objective functions that are non-Lipschitz continuous concerning continuous variables, making them highly challenging to solve. Consequently, we propose a continuous relaxation approach that converts these sparse discrete models into continuous nonlinear programs. We validate the efficacy of both the sparse discrete models and the relaxation method through two classic case studies.