We analyze two integer programming reformulations of the n-dimensional knapsack feasibility problem without assuming any structure on the weight vector $a.$ Both reformulations have a constraint matrix in which the columns form a reduced basis in the sense of Lenstra, Lenstra, and Lov\’asz. The nullspace reformulation of Aardal, Hurkens and Lenstra has n-1 variables, and … Read more
Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory … Read more
Stochastic dominance constraints allow a decision-maker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first and second-order stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, … Read more
This paper describes a new primal relaxation for nonlinear integer programming problems with linear constraints. This relaxation, contrary to the standard Lagrangean relaxation, can be solved efficiently. It requires the solution of a nonlinear penalized problem whose linear constraint set is known only implicitly, but whose solution is made possible by the use of a … Read more
A finite test set for an integer maximization problem enables us to verify whether a feasible point attains the global maximum. We establish in this paper several general results that apply to integer maximization problems with nonlinear objective functions. CitationOperations Research Letters 36 (2008) 439–443ArticleDownload View PDF
Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or 1/2, such cuts are known as {0, 1/2}-cuts. It has been proven by Caprara and Fischetti (1996) that separation of {0, 1/2}-cuts is NP-hard. In this paper, we study … Read more
We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b’ ≤ Ax ≤ b, x ∈ Zn with b’ ≤ (AU)y ≤ b, y ∈ Zn, where U is a unimodular matrix computed via basis reduction, to make the … Read more
Valid inequalities for 0-1 knapsack polytopes often prove useful when tackling hard 0-1 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the so-called extended cover and weight inequalities can be solved exactly in O(nb) … Read more
The Quadratic Knapsack Problem can be formulated, using an idea of Glover, as a mixed 0-1 linear program with only 2n variables. We present a simple method for strengthening that formulation, which gives good results when the profit matrix is dense and non-negative. CitationWorking Paper, Department of Management Science, Lancaster University, May 2007.ArticleDownload View PDF
In this paper we study capacitated network design problems, differentiating directed, bidirected and undirected link capacity models. We complement existing polyhedral results for the three variants by new classes of facet-defining valid inequalities and unified lifting results. For this, we study the restriction of the problems to a cut of the network. First, we show … Read more