On the knapsack closure of 0-1 integer linear programs

Many inequalities for Mixed-Integer Linear Programs (MILPs) or pure Integer Linear Programs (ILPs) are derived from the Gomory corner relaxation, where all the nonbinding constraints at an optimal LP vertex are relaxed. Computational results show that the corner relaxation gives a good approximation of the integer hull for problems with general-integer variables, but the approximation is less satisfactory for problems with 0-1 variables only. A possible explanation is that, for 0-1 ILPs, even the non-binding variable bound constraints xj≥0 or xj≤1 play an important role, hence their relaxation produces weaker bounds.In this note we address a relaxation for 0-1 ILPs that explicitly takes all variable bound constraints into account. More specifically, we introduce the concept of knapsack closure as a tightening of the classical Chvátal-Gomory (CG) closure. The knapsack closure is obtained as follows: for all inequalities wTx≥w0 valid for the LP relaxation, add to the original system all the valid inequalities for the knapsack polytope conv{xε{0,1}n:wTx≥w0}. A MILP model for the corresponding separation problem is also introduced. © 2010 Elsevier B.V

New valid inequalities for the fixed-charge and single-node flow polytopes

The most effective software packages for solving mixed 0-1 linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions

Valid inequalities for mixed-integer programmes with fixed charges on sets of variables

We consider mixed 0-1 linear programs in which one is given a collection of (not necessarily disjoint) sets of variables and, for each set, a fixxed charge is incurred if and only if at least one of the variables in the set takes a positive value. We derive strong valid linear inequalities for these problems, and show that they generalise and dominate a subclass of the well-known flow cover inequalities for the classical fixed-charge problem

On Minimal Valid Inequalities for Mixed Integer Conic Programs

We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone