SDP-based Benders decomposition for solving p-median quadratic facility location problems

The $p$-median facility location problem involves selecting the locations for $p$ facilities from a set of potential locations, to balance the trade-off between facility establishment costs and distance to end users. In this paper, we introduce an extension, named the $p$-median quadratic facility location problem, that also considers inter-facility distance—important in applications where the facilities serve as intermediate hubs linking sites in different regions. This creates a quadratic term in the objective that makes the problem nonlinear. We develop a Benders decomposition method that uses a tight semi-definite programming relaxation of the Benders master problem, instead of solving the nonlinear master problem directly. We incorporate this decomposition strategy into a branch and bound framework to ensure convergence. Our numerical results show that this method outperforms the linear reformulation and classical Benders decomposition techniques, working independently or in tandem.