Cutting Plane Algorithms for Integer Programming, Cutting Plane Algorithms

Cutting plane methods are exact algorithms for integer programming problems. They have proven to be very useful computationally in the last few years, especially when combined with a branch and bound algorithm in a branch and cut framework. These methods work by solving a sequence of linear programming relax-ations of the integer programming problem. The relaxations are gradually improved to give better approximations to the integer programming problem, at least in the neighborhood of the optimal solution. For hard instances that cannot be solved to optimality, cutting plane algorithms can produce approximations to the optimal solution in moderate computation times, with guarantees on the distance to optimality. Cutting plane algorithms have been used to solve many different integer programming problems , including the traveling salesman prob-contains a survey of applications of cutting plane methods, as well as a guide to the successful implementation of a cutting plane algorithm. The book [33] by G.L. Nemhauser and L. Wolsey provides an excellent and detailed description of cutting plane algorithms and the other material in this entry, as well as other aspects of integer programming. The book [35] by A. Schrijver and also the more recent article [36] are excellent sources of additional material. Cutting plane algorithms for general integer programming problems were first proposed by R.E. Gomory in [13, 14]. Unfortunately, the cutting planes proposed by Gomory did not appear to be very strong, leading to slow convergence of these algorithms, so the algorithms were neglected for many years. The development of polyhedral theory and the consequent introduction of strong, problem specific cutting planes led to a resurgence of cutting plane methods in the eighties, and cutting plane methods are now the method of choice for a variety of problems, including the traveling salesman problem. Recently , there has also been some research showing that the original cutting planes proposed by Gomory can actually be useful. There has also been research on other types of cutting planes for general integer programming problems. Current research is focused on developing cutting plane algorithms for a variety of hard combi-natorial optimization problems, and on solving large instances of integer programming problems using these methods. All of these issues are discussed below. A simple example. Consider, for example, the integer programming problem min −2x 1 − x 2 s.t. x 1 + 2x 2 ≤ 7 2x 1 − x 2 ≤ 3 x 1 , …