Lift-and-project for Mixed 0-1 programming: recent progress

This article reviews the disjunctive programming or lift-and-project approach to 0-1 programming, with an emphasis on recent developments. Disjunctive programming is optimization over unions of polyhedra. The first three sections of the paper define basic concepts and introduce the two fundamental results underlying the approach. Thus, section 2 describes the compact higher dimensional representation of the convex hull of a union of polyhedra, and its projection on the original space; whereas section 3 is devoted to the sequential convexifiability of facial disjunctive programs, which include mixed 0-1 programs. While these results originate in Balas' work in the early- to mid-seventies, some new results are also included: it is shown that on the higher dimensional polyhedron representing the convex hull of a union of polyhedra, the maximum edge-distance between any two vertices in 2. Also, it is shown that in the process of sequential convexification of a 0-1 program, fractional intermediate values of the variables can occur only under very special circumstances. The next section relates the above results to the matrix-cone approach of Lovasz and Schrijver and of Sherali and Adams. Section 5 introduces the lift-and-project cuts of Balas, Ceria and Cornuejols from the early nineties, and discusses the cut generating linear program (CGLP), cut lifting and cut strengthening. The next section briefly outlines the branch and cut framework in which the lift-and-project cuts turned out to be computationally useful, while section 7 discusses some crucial aspects of the cut generating procedure: alternative normalizations of (CGLP), complementarity of the solution components, size reduction of (CGLP), and ways of deriving multiple cuts from a disjunction. Finally, section 8 discusses computational results in branch-and-cut mode as well as in cut-and-branch mode.