On optimizing over lift-and-project closures

The lift-and-project closure is the relaxation obtained by computing all lift-and-project cuts from the initial formulation of a mixed integer linear program or equivalently by computing all mixed integer Gomory cuts read from all tableau's corresponding to feasible and infeasible bases. In this paper, we present an algorithm for approximating the value of the lift-and-project closure. The originality of our method is that it is based on a very simple cut generation linear programming problem which is obtained from the original linear relaxation by simply modifying the bounds on the variables and constraints. This separation LP can also be seen as the dual of the cut generation LP used in disjunctive programming procedures with a particular normalization. We study some properties of this separation LP in particular relating it to the equivalence between lift-and-project cuts and Gomory cuts shown by Balas and Perregaard. Finally, we present some computational experiments and comparisons with recent related works

En-Route Optimal Flight Planning Constrained to Pass Through Waypoints using MINLP

Abstract: In this paper we study the en-route strategic flight planning of a commercial aircraft constrained to pass through a set of waypoints whose sequence is not predefined. This problem has been solved as an hybrid optimal control problem in which, given the dynamic model of the aircraft, the initial and final states, the path constraints constituting the envelope of flight, and a set of waypoints in the European air space, one has to find the control inputs, the switching times, the optimal sequence of waypoints and the corresponding trajectory of the aircraft that minimize the direct operating cost during the flight. The complete layout of waypoints in the European airspace is reduced and waypoints are gathered into a small number of clusters. The aircraft is constrained to pass through one waypoint inside every cluster of waypoints. The presence of multi point constraints makes the optimal control problem particularly difficult to solve. The hybrid optimal control problem is converted into a mixed integer non linear programming problem first making the unknown switching times part of the state, then introducing binary variable to enforce the constraint of passing through one waypoint inside every cluster, and finally applying a direct collocation method. The resulting mixed integer non linear programming problem has been solved using a branch and bound algorithm. The cases studied and the numerical results show the effectiveness, efficiency and applicability of this method for enroute strategic flight plans definition

The Logarithmic Sobolev Constant of The Lamplighter

We give estimates on the logarithmic Sobolev constant of some finite lamplighter graphs in terms of the spectral gap of the underlying base. Also, we give examples of application

Flot maximum avec contrainte de délai proportionnel

International audienceÉtant donnés un réseau et un ensemble de paires source-destination (connexions), nous considérons le problème de maximiser la somme des flots avec des contraintes de délai proportionnel. Dans cet article, le délai pour traverser un lien est proportionnel au flot total supporté par ce lien. Si une connexion supporte un flot non nul, alors la somme des délais tout au long du chemin correspondant à cette connexion doit être plus petite qu'une certaine borne donnée. Ces contraintes de délai sont des contraintes de type "on-off" car si une connexion supporte un flot nul, alors il n'y a pas de contrainte pour cette connexion. La difficulté de ce problème réside dans le choix des connexions supportant un flot non nul. Nous prouvons une approximation générale en utilisant la programmation linéaire. Nous montrons ensuite un schéma d'approximation polynomial lorsque le graphe d'intersection des chemins a une largeur arborescente bornée. Nous prouvons enfin que le problème est NP-difficile même lorsque le réseau est un arbre

Maximum flow under proportional delay constraint

Given a network and a set of source destination pairs (connections), we con- sider the problem of maximizing the sum of the flow under proportional delay constraints. In this paper, the delay for crossing a link is proportional to the total flow crossing this link. If a connection supports non-zero flow, then the sum of the delay along any path corresponding to that connection must be lower than a given bound. The constraints of delay are on-off constraints because if a connection does not support non-zero flow, then there is no constraint for that connection. The difficulty of the problem comes from the choice of the connections supporting non-zero flow. We first prove a general approximation ratio using linear programming for a variant of the problem. We then prove a Polynomial Time Approximation Scheme when the graph of intersection of the paths has bounded treewidth. We finally prove that the problem is NP-hard even when the network is a tree

Plénière: Programmation non-linéaire en variables mixtes et applications

International audienceLa programmation non-linéaire en variable mixtes consiste à minimiser une fonction non-linéaire sur un ensemble de solutions décrit par des fonctions non-linéaires et des contraintes d'intégrité portant sur tout ou partie des variables. Depuis une dizaine d'année de nombreux travaux se sont intéressé aux aspects théoriques et algorithmiques de la résolution de ces problèmes. Ces travaux ont donné lieu à la réalisation de plusieurs solveurs open-source ou commerciaux. Dans cette présentation, nous décrirons les principales classes de problèmes et les idées algorithmiques implémentées dans les logiciels de résolutions actuels. Nous nous intéresserons en particulier au cas où la relaxation continue naturelle du problème est un problème convexe et au cas ou seule la fonction objective est non-linéaire et est quadratique. Nous nous appuierons en particulier sur les exemples du solveur Bonmin et du solveur global de programmation quadratique non-convexe récemment introduit dans le logiciel CPLEX.</p

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

This paper provides a survey of recent progress and software for solving mixed integer nonlinear programs (MINLP) wherein the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in very years. By exploiting analogies to the case of well-known techniques for solving mixed integer linear programs and incorporating these techniques into the software, significant improvements have been made in our ability to solve the problems

A note on the MIR closure

In 1988, Nemhauser and Wolsey introduced the concept of MIR inequality for mixed integer linear programs. In 1998, Wolsey defined MIR inequalities differently. In some sense these definitions are equivalent. However, this note points out that the natural concepts of MIR closures derived from these two definitions are distinct. Dash, Günlük and Lodi made the same observation independently