Solving a Class of Cut-Generating Linear Programs via Machine Learning

Cut-generating linear programs (CGLPs) play a key role as a separation oracle to produce valid inequalities for the feasible region of mixed-integer programs. When incorporated inside branch-and-bound, the cutting planes obtained from CGLPs help to tighten relaxations and improve dual bounds. However, running the CGLPs at the nodes of the branch-and-bound tree is computationally cumbersome due to the large number of node candidates and the lack of a priori knowledge on which nodes admit useful cutting planes. As a result, CGLPs are often avoided at default settings of branch-and-cut algorithms despite their potential impact on improving dual bounds. In this paper, we propose a novel framework based on machine learning to approximate the optimal value of a CGLP class that determines whether a cutting plane can be generated at a node of the branch-and-bound tree. Translating the CGLP as an indicator function of the objective function vector, we show that it can be approximated through conventional data classification techniques. We provide a systematic procedure to efficiently generate training data sets for the corresponding classification problem based on the CGLP structure. We conduct computational experiments on benchmark instances using classification methods such as logistic regression. These results suggest that the approximate CGLP obtained from classification can improve the solution time compared to that of conventional cutting plane methods. Our proposed framework can be efficiently applied to a large number of nodes in the branch-and-bound tree to identify the best candidates for adding a cut.

Consistency for 0-1 Programming

Concepts of consistency have long played a key role in constraint programming but never developed in integer programming (IP). Consistency nonetheless plays a role in IP as well. For example, cutting planes can reduce backtracking by achieving various forms of consistency as well as by tightening the linear programming (LP) relaxation. We introduce a type of consistency that is particularly suited for 0-1 programming and develop the associated theory. We define a 0-1 constraint set as LP-consistent when any partial assignment that is consistent with its linear programming relaxation is consistent with the original 0-1 constraint set. We prove basic properties of LP-consistency, including its relationship with Chvatal-Gomory cuts and the integer hull. We show that a weak form of LP-consistency can reduce or eliminate backtracking in a way analogous to k-consistency but is easier to achieve. In so doing, we identify a class of valid inequalities that can be more effective than traditional cutting planes at cutting off infeasible 0-1 partial assignments.