Mixed-integer Sets Research Articles

The Mixing Set with Flows

We consider here the mixing set with flows: s + xt ≥ bt, xt ≤ yt for 1 ≤ t ≤ n; s E IR, x E IR, y E Z. It models the flow version of the basic mixing set introduced and studied by Gunluk and Pochet, as well as the most simple stochastic lot-sizing problem with recourse, and more generally is a relaxation of certain mixed integer sets that arise in the study of production planning problems. We study the polyhedron obtained by convexifying the above set. Specifically we provide a system of inequalities that gives its external description and characterize its vertices and rays.

Valid inequalities based on simple mixed-integer sets

In this paper we use facets of simple mixed-integer sets with three variables to derive a parametric family of valid inequalities for general mixed-integer sets. We call these inequalities two-step MIR inequalities as they can be derived by applying the simple mixed-integer rounding (MIR) principle of Wolsey (1998) twice. The two-step MIR inequalities define facets of the master cyclic group polyhedron of Gomory (1969). In addition, they dominate the strong fractional cuts of Letchford and Lodi (2002).

Tight Mip Formulation for Multi-Item Discrete Lot-Sizing Problems

This paper discusses mixed-integer programming formulations of variants of the discrete lot-sizing problem. Our approach is to identify simple mixed-integer sets within these models and to apply tight formulations for these sets. This allows us to define integral linear programming formulations for the discrete lot-sizing problem in which backlogging and/or safety stocks are present, and to give extended formulations for other cases. The results help significantly to solve test cases arising from an industrial application motivating this research.

Strong formulations for mixed integer programs: valid inequalities and extended formulations

We examine progress over the last fifteen years in finding strong valid inequalities and tight extended formulations for simple mixed integer sets lying both on the ``easy'' and ``hard'' sides of the complexity frontier. Most progress has been made in studying sets arising from knapsack and single node flow sets, and a variety of sets motivated by different lot-sizing models. We conclude by citing briefly some of the more intriguing new avenues of research.

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)R 1 + ×ℤ1:s≥z−b}. 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.

On the facets of the mixed?integer knapsack polyhedron

We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.