Projection and Lifting in Combinatorial Optimization

This is an overview of the significance and main uses of projection and lifting in integer programming and combinatorial optimization. Its first four sections deal with those basic properties of projection that make it such an effiective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like the integrality-preserving property of projection, the dimension of projected polyhedra, when do facets of a polyhedron project into facets of its projection, and so on. They also describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations. The next section of the survey deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Finally, the last two sections review the recent developments in disjunctive programming, namely lift-and-project cuts, the construction of cut generating linear programs, techniques for lifting and for strengthening disjunctive cuts, and the embedding of these cuts into a branch-and-cut framework, along with a brief outline of computational results.