Jean-Philippe P. Richard

Explicit convex and concave envelopes through polyhedral subdivisions

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.

Allocating security resources to a water supply network

This paper develops a method for allocating a security budget to a water supply network so as to maximize the networks resilience to physical attack. The method integrates max-min linear programming, hydraulic simulation, and genetic algorithms for constraint generation. The objective is to find a security allocation that maximizes an attackers marginal cost of inflicting damage through the destruction of network components. We illustrate the method on two example networks, one large and one small, and investigate its allocation effectiveness and computational characteristics.

Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms

Abstract.We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables. We develop a lifting theory for the continuous variables. In particular, we present a pseudo-polynomial algorithm for the sequential lifting of the continuous variables and we discuss its practical use.

On the extreme inequalities of infinite group problems

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Günlük (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous.

Facets of Two-Dimensional Infinite Group Problems

In this paper, we lay the foundation for the study of the two-dimensional mixed integer infinite group problem (2DMIIGP). We introduce tools to determine if a given continuous and piecewise linear function over the two-dimensional infinite group is subadditive and to determine whether it defines a facet of 2DMIIGP. We then present two different constructions that yield the first known families of facet-defining inequalities for 2DMIIGP. The first construction uses valid inequalities of the one-dimensional integer infinite group problem (1DIIGP) as building blocks for creating inequalities for the two-dimensional integer infinite group problem (2DIIGP). We prove that this construction yields all continuous piecewise linear facets of the two-dimensional group problem that have exactly two gradients. The second construction we present has three gradients and yields facet-defining inequalities of 2DMIIGP whose continuous coefficients are not dominated by those of facets of the one-dimensional mixed integer infinite group problem (1DMIIGP).

Lifted inequalities for 0-1 mixed integer programming: Superlinear lifting

Abstract.We study the mixed 0-1 knapsack polytope, which is defined by a single knapsack constraint that contains 0-1 and bounded continuous variables, through the lifting of continuous variables fixed at their upper bounds. We introduce the concept of a superlinear inequality and show that, in this case, lifting is significantly simpler than for general inequalities. We use the superlinearity theory, together with the traditional lifting of 0-1 variables, to describe families of facets of the mixed 0-1 knapsack polytope. Finally, we show that superlinearity results can be extended to nonsuperlinear inequalities when the coefficients of the variables fixed at their upper bounds are large.

Cook, Kannan and Schrijver’s example revisited

Abstract In 1990, Cook, Kannan and Schrijver [W. Cook, R. Kannan, A. Schrijver, Chvatal closures for mixed integer programming problems, Mathematical Programming 47 (1990) 155–174] proved that the split closure (the 1st 1-branch split closure) of a polyhedron is again a polyhedron. They also gave an example of a mixed-integer polytope in R 2 + 1 whose 1-branch split rank is infinite. We generalize this example to a family of high-dimensional polytopes and present a closed-form description of the k th 1-branch split closure of these polytopes for any k ≥ 1 . Despite the fact that the m -branch split rank of the ( m + 1 )-dimensional polytope in this family is 1, we show that the 2-branch split rank of the ( m + 1 )-dimensional polytope is infinite when m ≥ 3 . We conjecture that the t -branch split rank of the ( m + 1 )-dimensional polytope of the family is infinite for any 1 ≤ t ≤ m − 1 and m ≥ 2 .

An Optimization Model for Empty Freight Car Assignment at Union Pacific Railroad

Railroad companies face a difficult problem in assigning empty freight cars based on customer demand because these assignments depend on a variety of factors; these include the location of available empty cars, the urgency of the demand, and the possibilities of car substitution. In this paper, we present an optimization model implemented at Union Pacific Railroad (UP) to assign empty freight cars based on demand. The model seeks to reduce transportation costs, and improve delivery time and customer satisfaction. UP currently uses the model to make real-time assignments in a total car-management system. The model has helped UP to achieve significant reductions in its transportation costs, similar to the savings that our simulation study predicted. In addition, UP reduced the staff required for its demand fulfillment process, resulting in an ROI of 35 percent.

Valid inequalities for MIPs and group polyhedra from approximate liftings

In this paper, we present an approximate lifting scheme to derive valid inequalities for general mixed integer programs and for the group problem. This scheme uses superadditive functions as the building block of integer and continuous lifting procedures. It yields a simple derivation of new and known families of cuts that correspond to extreme inequalities for group problems. This new approximate lifting approach is constructive and potentially efficient in computation.

Strong valid inequalities for orthogonal disjunctions and bilinear covering sets

In this paper, we derive a closed-form characterization of the convex hull of a generic nonlinear set, when this convex hull is completely determined by orthogonal restrictions of the original set. Although the tools used in this construction include disjunctive programming and convex extensions, our characterization does not introduce additional variables. We develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed. We demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed-integer polyhedral sets and finding the convex hull of certain mixed/pure-integer bilinear sets. We then extend the utility of the convexification tool to relaxing nonconvex inequalities, which are not naturally disjunctive, by providing sufficient conditions for establishing the convex extension property over the non-negative orthant. We illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non-negative orthant. Although we illustrate our results primarily on bilinear covering sets, they also apply to more general polynomial covering sets for which they yield new tight relaxations.