Jean-Philippe P. Richard In this paper, we propose lifting techniques for generating strong cuts for nonlinear programs that are globally-valid. The theory is geometric and provides intuition into lifting-based cut generation procedures. As a special case, we find short proofs of earlier results on lifting techniques for mixed-integer programs. Using convex extensions, we obtain conditions that allow sequence-independent … Read more
In this paper, we introduce an operation that creates families of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. We call this family sequential-merge inequalities because they are produced by applying two group cuts one after the other and because the resultant inequality depends on the order of the … Read more
In this paper, we study the set of 0-1 integer solutions to a single knapsack constraint and a set of non-overlapping cardinality constraints (MCKP). This set is a generalization of the traditional 0-1 knapsack polytope and the 0-1 knapsack polytope with generalized upper bounds. We derive strong valid inequalities for the convex hull of its … 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
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
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