From an Interior Point to a Corner Point: Smart Crossover

Abstract: The crossover in solving linear programs is a procedure to recover an optimal corner/extreme point from an approximately optimal inner point generated by interior-point method or emerging first-order methods. Unfortunately it is often observed that the computation time of this procedure can be much longer than the time of the former stage. Our work shows that this bottleneck can be significantly improved if the procedure can smartly take advantage of the problem characteristics and implement customized strategies. For the problem with the network structure, our approach can even start from an inexact solution of interior-point method as well as other emerging first-order algorithms. It fully exploits the network structure to smartly evaluate columns' potential of forming the optimal basis and efficiently identifies a nearby basic feasible solution. For the problem with a large optimal face, we propose a perturbation crossover approach to find a corner point of the optimal face. The comparison experiments with state-of-art commercial LP solvers on classical linear programming problem benchmarks, network flow problem benchmarks and MINST datasets exhibit its considerable advantages in practice.