A Difference-of-Convex Cutting Plane Algorithm for Mixed-Binary Linear Program

TL;DR

This paper introduces a novel cutting plane algorithm based on Difference-of-Convex programming to solve Mixed-Binary Linear Programs globally, combining DC cuts with classical global cuts for improved performance.

Contribution

The paper develops a new DC-based cutting plane algorithm (DCCUT) for MBLP, integrating DC cuts with classical cuts, and proves its convergence with enhanced variants and parallel implementations.

Findings

DCCUT outperforms classical algorithms on benchmark datasets.

DC cuts improve the lower bound in the cutting plane method.

Parallel DCCUT accelerates computation on large instances.

Abstract

In this paper, we propose a cutting plane algorithm based on DC (Difference-of-Convex) programming and DC cut for globally solving Mixed-Binary Linear Program (MBLP). We first use a classical DC programming formulation via the exact penalization to formulate MBLP as a DC program, which can be solved by DCA algorithm. Then, we focus on the construction of DC cuts, which serves either as a local cut (namely type-I DC cut) at feasible local minimizer of MBLP, or as a global cut (namely type-II DC cut) at infeasible local minimizer of MBLP if some particular assumptions are verified. Otherwise, the constructibility of DC cut is still unclear, and we propose to use classical global cuts (such as the Lift-and-Project cut) instead. Combining DC cut and classical global cuts, a cutting plane algorithm, namely DCCUT, is established for globally solving MBLP. The convergence theorem of DCCUT is proved. Restarting DCA in DCCUT helps to quickly update the upper bound solution and to introduce more DC cuts for lower bound improvement. A variant of DCCUT by introducing more classical global cuts in each iteration is proposed, and parallel versions of DCCUT and its variant are also designed which use the power of multiple processors for better performance. Numerical simulations of DCCUT type algorithms comparing with the classical cutting plane algorithm using Lift-and-Project cuts are reported. Tests on some specific samples and the MIPLIB 2017 benchmark dataset demonstrate the benefits of DC cut and good performance of DCCUT algorithms.