We describe strong mixed-integer programming formulations for robust mixed 0-1 programming with uncertainty in the objective coefficients. In particular, we focus on an objective uncertainty set described as a polytope with a budget constraint. We show that for a robust 0-1 problem, there is a tight linear programming formulation with size polynomial in the size … Read more
Interior-point methods (IPMs), originally conceived in the context of linear programming have found a variety of applications in integer programming, and combinatorial optimization. This survey presents an up to date account of IPMs in solving NP-hard combinatorial optimization problems to optimality, and also in developing approximation algorithms for some of them. The surveyed approaches include … Read more
This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (l,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (l,S) inequalities to a general class of valid inequalities, called the (Q,S_Q) inequalities, and we … Read more
In the context of integer programming, we develop a polyhedral method for linearizing a product of a pair of real linear functions in 0/1 variables. As an example, by writing a pair of integer variables in binary expansion, we have a technique for linearizing their product. We give a complete linear description for the resulting … Read more
We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem is a relaxation of general mixed-integer programming. We show how strong inequalities valid for the semi-continuous knapsack polyhedron can … Read more
We consider the integer program P -> max {c’x | Ax=y; x\in N^n }. Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables … Read more
Recently, tools from algebraic geometry have been successfully applied to develop solution schemes for new classes of optimization problems. A central idea in these constructions is to express a polynomial that is positive on a given domain in terms of polynomials of higher degree so that its positivity is readily revealed. This resembles the “lifting” … Read more
We consider the problem of scheduling hypermedia documents with elastic times. The objects have variable durations characterized by ideal, minimum, and maximum values. A schedule is a set of tensions on the arcs of the precedence graph which also represents synchronization constraints. We consider the problem of deciding if there exists a schedule in which … Read more
We consider the problem of scheduling a set of tasks related by precedence constraints to a set of processors, so as to minimize their makespan. Each task has to be assigned to a unique processor and no preemption is allowed. A new integer programming formulation of the problem is given and strong valid inequalities are … Read more
The problem of automatic scheduling hypermedia documents consists in finding the optimal starting times and durations of objects to be presented, to ensure spatial and temporal consistency of a presentation while respecting limits on shrinking and stretching the ideal duration of each object. The combinatorial nature of the minimization of the number of objects whose … Read more