A local branching heuristic for mixed‐integer programs with 2‐level variables, with an application to a telecommunication network design problem

Eective heuristic solution methods for general Mixed-Integer Programs (MIPs) are strongly required in many practical applications, and have been the subject of an intensive research eort in the recent years. Fischetti and Lodi [6] recently proposed an exact solution technique based on the use of branching conditions expressed through (invalid) linear inequalities called local branching cuts. In the concluding remarks of their paper, these authors anticipated the possibility their method be used to design a genuine MIP metaheuristic framework akin to Tabu Search (TS) or Variable Neighborhood Search (VNS), based on an external MIP solver. In the present paper we introduce and analyze computationally a specific implementation of the above idea. In particular, we address MIPs with binary variables, and propose a variant of the classical VNS scheme that we call Diversification, Refining, and Tight-refining (DRT). The new approach is intended to be of high generality, but exploits the specific structure of some MIPs where the set of binary variables partitions naturally into two levels, with the property that fixing the value of the first-level variables produces an easier-to-solve (but still hard) subproblem. This is often the case, e.g., in hard facility location problems arising in telecommunication network design. Our method detects automatically the presence in the MIP model of the first-level binary variables, if any, according to simple heuristic criteria. This information is then exploited during the intensification phase of the local search, so as to explore nested solution neighborhoods defined by local branching cuts aecting