Enhancements of Lift-and-Project Cuts

In a recent paper [4] Balas, Ceria and Cornuejoels provided computational results that their Lift-and-Project method [3] successfully solved a series of pure and mixed 0-1 benchmark programs and performed on the average better than several commercial general mixed integer optimization packages existing at the time. The Lift-and-Project approach uses a linear programming formulation derived by imposing the 0/1 condition on a single variable. By solving a cut generating linear program (CGLP) derived from this formulation a ”deepest cut” is obtained. The size of this CGLP is roughly twice the size of the LP formulation underlying the mixed 0-1 program. Hence it requires a lot of effort to find such a ”deepest cut”. Our interest in this project is to examine the possibility of exploiting the CGLP after the first expensive cut has been found, to cheaply generate additional cuts. We are looking at sets of cuts which will have the property that they are essential to consolidate an optimal solution with the 0/1 condition imposed on just the one variable. E. Balas [2] proposes to find such a set of cuts using pivots in the CGLP simplex tableau. In this paper we examine experimentally the potential of this approach as well as an alternative iterative method. We will also exploit the concept of working in a subspace, as used in the Lift-and-Project method, as a relaxation to improve computational times. Furthermore we will look into CGLP’s obtained by imposing the 0/1 condition on more than one variable. Preliminary results though suggest that the reduction in the number of Branch-and-Cut nodes, bounded by generating these cut families, is countered by the added expense to find the cuts. Our computational tests on MIPLIB problem instances show comparable solution times between the Lift-and-Project cuts and our proposed cut families.