On generalized branching methods for mixed integer programming
Friday, July 29, 2005 - 9:00am - 8:45pm
Sanjay Mehrotra (Northwestern University)
We present a restructuring of the computations in Lenstra's methods for solving mixed integer linear programs. A concept of Adjoint lattices is introduced and used for this purpose. This allows us to develop branching on hyperplane algorithms in the space of original variables, without requiring any dimension reduction for pure or mixed integer programs. Reduced lattice basis are also computed in the space of original variables. Based on these results we give a new natural heuristic way of generating branching hyperplanes, and discuss its relationship with recent reformulation techniques of Aardal and Lenstra. We show that the reduced basis available at the root node has useful information on the branching hyperplanes for the generalized branch-and-bound tree. Our development allows us to give algorithms for mixed convex integer programs with properties similar to those for the mixed integer linear programs. Implementation of this algorithm in a prototype software system IMPACT is described. Computational results are presented on difficult knapsack and market split problems using this approach.