We present new families of valid inequalities for (mixed) integer programming (MIP) problems. These valid inequalities are based on a generalization of the 2-step mixed integer rounding (MIR), proposed by Dash and Günlük (2006). We prove that for any positive integer n, n facets of a certain (n+1)-dimensional mixed integer set can be obtained through … Read more
Gomory mixed-integer (GMI) cuts generated from optimal simplex tableaus are known to be useful in solving mixed-integer programs. Further, it is well-known that GMI cuts can be derived from facets of Gomory’s master cyclic group polyhedron and its mixed-integer extension studied by Gomory and Johnson. In this paper we examine why cutting planes derived from … Read more
We propose an approximate lifting procedure for general integer programs. This lifting procedure uses information from multiple constraints of the problem formulation and can be used to strengthen formulations and cuts for mixed integer programs. In particular we demonstrate how it can be applied to improve Gomory’s fractional cut which is central to Glover’s primal … Read more
An explicit description of the convex hull of solutions to the uncapacitated lot-sizing problem with backlogging, in its natural space of production, setup, inventory and backlogging variables, has been an open question for many years. In this paper, we identify facet-defining inequalities that subsume all previously known valid inequalities for this problem. We show that … Read more
This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to $q$-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arborescence probem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. … Read more
In this paper, we derive new families of piecewise linear facet-defining inequalities for the finite group problem and extreme inequalities for the infinite group problem using approximate lifting. The new valid inequalities for the finite group problem are two- and three-slope facet-defining inequalities as well as the first family of four-slope facet-defining inequalities. The new … Read more
In this paper we derive new properties of extreme inequalities for infinite group problems. We develop tools to prove that given valid inequalities for the infinite group problem are extreme. These results show that integer infinite group problems have discontinuous extreme inequalities. These inequalities are strong when compared to related classes of continuous extreme inequalities. … Read more
We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal sub ject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain … Read more
In this paper, we lay the foundation for the study of the two-dimensional mixed integer infinite group problem (2DMIIGP). We introduce tools to determine if a given continuous and piecewise linear function over the two-dimensional infinite group is subadditive and to determine whether it defines a facet. We then present two different constructions that yield … Read more
Given a complete graph K = (V , E), with edge weight ce on each edge, we consider the problem of partitioning the vertices of graph K into subcliques that each have at least S vertices, so as to minimize the total weight of the edges that have both endpoints in the same subclique. It … Read more