In this paper, we study 0-1 mixed-integer bilinear covering sets. We derive several families of facet-defining inequalities via sequence-independent lifting techniques. We then show that these sets have polyhedral structures that are similar to those of certain fixed-charge single-node flow sets. As a result, we obtain new facet-defining inequalities for these sets that generalize well-known … Read more
Admission control problems have been studied extensively in the past. In a typical setting, resources like bandwidth have to be distributed to the different customers according to their demands maximizing the profit of the company. Yet, in real-world applications those demands are deviating and in order to satisfy their service requirements often a robust approach … Read more
In this paper, we investigate the recoverable robust knapsack problem, where the uncertainty of the item weights follows the approach of Bertsimas and Sim (2003,2004). In contrast to the robust approach, a limited recovery action is allowed, i.e., upto k items may be removed when the actual weights are known. This problem is motivated by … Read more
We describe a procedure to reduce variable bounds in Mixed Integer Nonlinear Programming (MINLP) as well as Mixed Integer Linear Programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction technique extends the implied bounds procedure used in MINLP and MILP and … Read more
Mathematical modelling languages such as AMPL, GAMS, and Xpress-MP enable mathematical models such as mixed-integer linear programmes (MILPs) to be expressed clearly for solution in solvers such as CPLEX, MINOS and Gurobi. However some models are sufficiently difficult that they cannot be solved using “out-of-the-box” solvers, and customisation of the solver framework to exploit model-specific … Read more
Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In … Read more
Given an undirected graph and a positive integer k, the maximum k-colorable subgraph problem consists of selecting a k-colorable induced subgraph of maximum cardinality. The natural integer programming formulation for this problem exhibits two kinds of symmetry: arbitrarily permuting the color classes and/or applying a non-trivial graph automorphism gives equivalent solutions. It is well known … Read more
This paper is concerned with the improved constraints handling in mixed-integer optimization problems. The novel element is the reduction of the number of binary variables used for expressing the complement of a convex (polytopic) region. As a generalization, the problem of representing the complement of a possibly non-connected union of such convex sets is detailed. … Read more
The behavior of enumeration-based programs for solving MINLPs depends at least in part on the quality of the bounds on the optimal value and of the feasible solutions found. We consider MINLP problems with linear constraints. The convex hull relaxation (CHR) is a special case of the primal relaxation (Guignard 1994, 2007) that is very … Read more
The paper describes a new primal relaxation (PR) for computing bounds on nonlinear integer programming (NLIP) problems. It is a natural extension to NLIP problems of the geometric interpretation of Lagrangean relaxation presented by Geoffrion (1974) for linear problems, and it is based on the same assumption that some constraints are complicating and are treated … Read more