Feasibility Jump: an LP-free Lagrangian MIP heuristic

Abstract

We present Feasibility Jump (FJ), a primal heuristic for mixed-integer linear programs (MIP) using stochastic guided local search over a Lagrangian relaxation. The method is incomplete: it does not necessarily produce solutions to all feasible problems, the solutions it produces are not in general optimal, and it cannot detect infeasibility. It does, however, very quickly produce feasible solutions to many hard MIP problem instances. Starting from any variable assignment, Feasibility Jump repeatedly selects a variable and sets its value to minimize a weighted sum of constraint violations. These weights (which correspond to the Lagrangian multipliers) are adjusted for constraints that remain violated in local minima. Contrary to many other primal heuristics, Feasibility Jump does not require a solution of the continuous relaxation, which can be time-consuming for some problems. We compare FJ against FICO Xpress Solver 8.14 and we show that this heuristic is effective on a range of problems from the MIPLIB 2017 benchmark set, significantly improving the average time to find a first feasible solution. We also show that providing these quick solutions to Xpress produces a modest reduction in the average time to optimality in the same benchmark set. Our entry based on FJ to the MIP 2022 Computational Competition (which challenged participants to write LP-free MIP heuristics) won 1st place. Moreover, an implementation of Feasibility Jump now runs by default on FICO Xpress Solver 9.0, where similar results to the ones presented here could be observed.