Lift-and-Project Inequalities

The lift-and-project technique is a systematic way to generate valid inequalities for a mixed binary program. The technique is interesting both on the theoretical and on the practical point of view. On the theoretical side it allows you to construct the inequality description of the convex hull of all mixed-{0, 1} solutions of a binary MIP in n repeated applications of the technique, where n is the number of binary variables. On the practical side, a variant of the method allows you to derive some cutting planes from the simplex tableau rather efficiently. Lift-and-project inequalities were proposed in the early nineties as a way to strengthen the linear programming relaxation of a mixed-integer program. The initial idea was proposed by Lovász and Schrijver [1] when they obtain a strengthened formulation in an extended space by multiplying all constraints by all variables xi and their complement and to project back to the initial space of variables. A similar but slightly different approach was then later proposed by Sherali and Adams [2]. In this survey, we follow the approach of Balas, Ceria and Cornuéjols [3] who showed that a simplified version of the Lovász-Schrijver reformulation keeps its main theoretical property. They also showed how the approach can successfully be incorporated in a branch-and-cut solver [4]. The general idea behind lift-and-project is the following: from an initial formulation, we can create a quadratic equivalent formulation by multiplying all inequalities by a binary variable xj of the problem and its complement (1−xj). If we linearize this formulation by introducing a variable yi := xixj , and replacing xj = xj (since xj ∈ {0, 1}), we obtain an equivalent formulation in an extended space. This is the lifting phase. If we project this extended formulation onto the space of the initial variables, we obtain a strengthened formulation of the initial binary MIP. This is the projecting phase. This article is further subdivided into two parts. Section 1 presents the liftand-project technique and show that it encodes a simple convexification process. We then show that this convexification process can be used sequentially in order to generate the convex hull of all feasible points. Lift-and-project inequalities can be related to other classes of inequalities. We also make this link in Section 1. Section 2 focuses on different ways to generate lift-and-project inequalities ∗Montefiore Institute, University of Liège, Grande Traverse, 10, 4000 Liège, Belgium, email: [email protected]