Tight formulations for some simple mixed integer programs and convex objective integer programs

We study the polyhedral structure of simple mixed integer sets that generalize the two variable set {(s, z) ∈ IR1+×Z 1 : s ≥ b−z}. These sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs. For four such sets we give a complete description by valid inequalities and/or an integral extended formulation, and we also indicate what constraints can be added without destroying integrality. We apply these results to provide tight formulations for certain piecewise–linear convex objective integer programs, and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot-sizing problems. Work carried out at CORE, Université Catholique de Louvain, Belgium. Present address: Dept. of Industrial Engineering, University of Wisconsin-Madison, [email protected] CORE and INMA, Université Catholique de Louvain, Belgium, [email protected] This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors. This research was also supported by the European Commission GROWTH Programme, Research Project LISCOS, Large Scale Integrated Supply Chain Optimization Software Based on Branch–and–Cut and Constraint Programming Methods, Contract No. GRDI–1999–10056.