Gérard Cornuéjols

A lift-and-project cutting plane algorithm for mixed 0---1 programs

We propose a cutting plane algorithm for mixed 0–1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LPs needed to generate the cuts. An additional strengthening step suggested by Balas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cone relaxations of Lovász and Schrijver and the hierarchy of relaxations of Sherali and Adams.

An algorithmic framework for convex mixed integer nonlinear programs

This paper is motivated by the fact that mixed integer nonlinear programming is an important and difficult area for which there is a need for developing new methods and software for solving large-scale problems. Moreover, both fundamental building blocks, namely mixed integer linear programming and nonlinear programming, have seen considerable and steady progress in recent years. Wishing to exploit expertise in these areas as well as on previous work in mixed integer nonlinear programming, this work represents the first step in an ongoing and ambitious project within an open-source environment. COIN-OR is our chosen environment for the development of the optimization software. A class of hybrid algorithms, of which branch-and-bound and polyhedral outer approximation are the two extreme cases, are proposed and implemented. Computational results that demonstrate the effectiveness of this framework are reported. Both the library of mixed integer nonlinear problems that exhibit convex continuous relaxations, on which the experiments are carried out, and a version of the software used are publicly available.

The Traveling Salesman Problem on a Graph and Some Related Integer Polyhedra.

Given a graphG = (N, E) and a length functionl: E → ℝ, the Graphical Traveling Salesman Problem is that of finding a minimum length cycle goingat least once through each node ofG. This formulation has advantages over the traditional formulation where each node must be visited exactly once. We give some facet inducing inequalities of the convex hull of the solutions to that problem. In particular, the so-called comb inequalities of Grötschel and Padberg are generalized. Some related integer polyhedra are also investigated. Finally, an efficient algorithm is given whenG is a series-parallel graph.

A comparison of heuristics and relaxations for the capacitated plant location problem

Abstract Approaches proposed in the literature for the Capacitated Plant Location Problem are compared. The comparison is based on new theoretical and computational results. The main emphasis is on relaxations. In particular, dominance relations among the various relaxations found in the literature are identified. In the computational study, the relaxations are compared as a function of various characteristics of the test problems. Several of these relaxations can be used to generate heuristic feasible solutions that are better than the classical greedy or interchange heuristics, both in computing time and in the quality of the solutions found.

A Column Generation Approach for Large-Scale Aircrew Rostering Problems

This article describes a method for solving the crew rostering problem in air transportation. This problem consists of constructing personalized schedules that assign pairings, days off, and other activities to airline crew members. A generalized set partitioning model and a method using column generation have been used. This method has been adapted in a number of ways to take advantage of the nature of the problem and to accelerate solution. Numerical tests on problems from Air France have demonstrated that this method is capable of solving very large scale problems with thousands of constraints and hundreds of subproblems. The tests have also shown that these adaptations are capable of reducing solution time by a factor of about a thousand. Finally, results from this method are compared with those obtained with the method currently used at Air France.

Combinatorial Optimization: Packing and Covering

Preface 1. Clutters 2. T-Cuts and T-Joins 3. Perfect Graphs and Matrices 4. Ideal Matrices 5. Odd Cycles in Graphs 6. 0,+1 Matrices and Integral Polyhedra 7. Signing 0,1 Matrices to Be Totally Unimodular or Balanced 8. Decomposition by k-Sum 9. Decomposition of Balanced Matrices 10. Decomposition of Perfect Graphs Bibliography Index.

Valid inequalities for mixed integer linear programs

This tutorial presents a theory of valid inequalities for mixed integer linear sets. It introduces the necessary tools from polyhedral theory and gives a geometric understanding of several classical families of valid inequalities such as lift-and-project cuts, Gomory mixed integer cuts, mixed integer rounding cuts, split cuts and intersection cuts, and it reveals the relationships between these families. The tutorial also discusses computational aspects of generating the cuts and their strength.

Submodular set functions, matroids and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem

Abstract For the problem maxlcub; Z ( S ): S is an independent set in the matroid X rcub;, it is well-known that the greedy algorithm finds an optimal solution when Z is an additive set function (Rado-Edmonds theorem). Fisher, Nemhauser and Wolsey have shown that, when Z is a nondecreasing submodular set function satisfying Z (∅)=0, the greedy algorithm finds a solution with value at least half the optimum value. In this paper we show that it finds a solution with value at least 1/(1 + α) times the optimum value, where α is a parameter which represents the ‘total curvature’ of Z . This parameter satisfies 0≤α≤1 and α=0 if and only if the set function Z is additive. Thus the theorems of Rado-Edmonds and Fisher-Nemhauser-Wolsey are both contained in the bound 1/(1 + α). We show that this bound is best possible in terms of α. Another bound which generalizes the Rado-Edmonds theorem is given in terms of a ‘greedy curvature’ of the set function. Unlike the first bound, this bound can prove the optimality of the greedy algorithm even in instances where Z is not additive. A third bound, in terms of the rank and the girth of X , unifies and generalizes the bounds (e−1)/e known for uniform matroids and 1 2 for general matroids. We also analyze the performance of the greedy algorithm when X is an independence system instead of a matroid. Then we derive two bounds, both tight: The first one is [1−(1−α/ K ) k ]/α where K and k are the sizes of the largest and smallest maximal independent sets in X respectively; the second one is 1/( p +α) where p is the minimum number of matroids that must be intersected to obtain X .

Polyhedral study of the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP) considered in this paper occurs when goods must be delivered from a central depot to clients with known demands, usingk vehicles of fixed capacity. Each client must be assigned to exactly one of the vehicles. The set of clients assigned to each vehicle must satisfy the capacity constraint. The goal is to minimize the total distance traveled. When the capacity of the vehicles is large enough, this problem reduces to the famous traveling salesman problem (TSP). A variant of the problem in which each client is visited by at least one vehicle, called the graphical vehicle routing problem (GVRP), is also considered in this paper and used as a relaxation of CVRP. Our approach for CVRP and GVRP is to extend the polyhedral results known for TSP. For example, the subtour elimination constraints can be generalized to facets of both CVRP and GVRP. Interesting classes of facets arise as a generalization of the comb inequalities, depending on whether the depot is in a handle, a tooth, both or neither. We report on the optimal solution of two problem instances by a cutting plane algorithm that only uses inequalities from the above classes.

Halin graphs and the travelling salesman problem

A Halin graphH=T∪C is obtained by embedding a treeT having no nodes of degree 2 in the plane, and then adding a cycleC to join the leaves ofT in such a way that the resulting graph is planar. These graphs are edge minimal 3-connected, hamiltonian, and in general have large numbers of hamilton cycles. We show that for arbitrary real edge costs the travelling salesman problem can be polynomially solved for such a graph, and we give an explicit linear description of the travelling salesman polytope (the convex hull of the incidence vectors of the hamilton cycles) for such a graph.