In this paper we discuss the generation of strong valid inequalities for (mixed) integer knapsack sets based on lifting of valid inequalities for basic knapsack sets with two integer variables (and one continuous variable). The description of the basic polyhedra can be made in polynomial time. We use superadditive valid functions in order to obtain sequence independent lifting.

We study valid inequalities for optimizationmodels that contain both binary indicator variables and separable concave constraints. Thesemodels reduce to amixedinteger linear program (MILP) when the concave constraints are ignored, or to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms designed to solve these problems to global optimality, cutting p...

In this paper we consider a semi-infinite relaxation of mixed integer linear programs. We show that minimal valid inequalities for this relaxation correspond to maximal latticefree convex sets, and that they arise from nonnegative, piecewise linear, positively homogeneous, convex functions.

We present a general time-indexed formulation that contains scheduling problems with unrelated parallel machines. We derive a class of basic valid inequalities for this formulation, and we show that a subset of these inequalities are facet-defining. We characterize all facet-defining inequalities with right-hand side 1. Further, we show how to efficiently separate these inequalities.

In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MIPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MIPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is that of generat...

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 a process which includes n consecutive app...

We study the interpolation procedure of Gomory and Johnson (1972), which generates cutting planes for general integer programs from facets of cyclic group polyhedra. This idea has recently been re-considered by Evans (2002) and Gomory, Johnson and Evans (2003). We compare inequalities generated by this procedure with mixed-integer rounding (MIR) based inequalities discussed in Dash and Gunluk (...

In this paper, we introduce an operation that creates families of facet-defining inequalities for highdimensional 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 ...

We present a scheme for generating new valid inequalities for mixed integer programs by taking pairwise combinations of existing valid inequalities. Our scheme is related to mixed integer rounding and mixing. The scheme is in general sequence-dependent and therefore leads to an exponential number of inequalities. For some important cases, we identify combination sequences that lead to a managea...