A Convex-Analysis Perspective on Disjunctive Cuts

An updated version of this paper has appeared in Math. Program., Ser. A 106, pp 567-586 (2006), DOI 10.1007/s10107-005-0670-8We treat the general problem of cutting planes with tools from convex analysis. We emphasize the case of disjunctive polyhedra and the generation of facets. We conclude with some considerations on the design of disjunctive cut generators

A Class of variable metric bundle methods

Projet PROMATHTo minimize a convex function [??], we state a class of penalty-type bundle algorithms, where the penalty uses a variable metric. This metric is updated according to quasi-Newton formulae based on Moreau-Yosida approximations of [??]. In particular, we introduce a "reversal" quasi-Newton formula, specially suited for our purpose. We consider several variants in the algorithm and discuss their respective merits. Furthermore, we accept a degenerate penalty term in the Moreau-Yosida regularization

Méthode de faisceaux désagrégée avec gradients creux

Projet PROMATHL'optimisation de la production d'électricité, résolue par relaxation lagrangienne, résulte pour la phase de coordination en un problème non différentiable degrande taille. Les temps de calcul impliqués nécessitent un algorithme de coordination très performant, qui ne doit pas requérir trop de résolutions des problèmes locaux. Pour cela, on tire parti de la structure additive de la fonction duale: chaque agent local peut donner lieu à sa propre linéarisation par plans sécants, ce qui raffine l'approximati- on de la fonction duale. Le présent rapport décrit l'implémentation de cette technique dans une méthode de faisceaux récente. Les performances du code résultant sont illustrées sur divers problèmes de gestion de la production. Ce travail a fait l'objet d'un contrat entre EdF et l'Inria

Application de la méthode de faisceaux à la gestion journalière de la production

Projet PROMATHL'optimisation de la production journalière d'électricité est un problème hétérogène de grande taille (quelque 200 groupes de production dans le réseau français). Il se prête particulièrement bien à la décomposition lagrangienne, ce qui donne lieu à un problème d'optimisation non différentiable. Le présent rapport rend compte de la résolution de ce problème par application de la méthode de faisceaux, suivant différents schémas de dualisation. Ce travail a constitué un contrat entre EdF et l'Inria

Proximal Convexification Procedures in Combinatorial Optimization

Final version has appeared under the title "On a primal-proximal heuristic in discrete optimization", in Math. Program., Ser. A 104, pp 105-128 (2005), DOI 10.1007/s10107-004-0571-2Lagrangian relaxation is useful to bound the optimal value of a given optimization problem, and also to obtain relaxed solutions. To obtain primal solutions, it is conceivable to use a convexification procedure suggested by D.P. Bertsekas in 1979, based on the proximal algorithm in the primal space. The present paper studies the theory assessing the approach in the framework of combinatorial optimization. Our results indicate that very little can be expected in theory, even though fairly good practical results have been obtained for the unit-commitment problem

Cut-generating functions and S-free sets

International audienceWe consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts

A Primal-Proximal Heuristic Applied to the Unit-Commitment Problem

An updated version of this paper has appeared in Math. Program., Ser. A 104, pp 129-151 (2005), DOI 10.1007/s10107-005-0593-4This paper is devoted to the numerical resolution of unit-commitment problems. More precisely we present the French model optimizing the daily production of electricity. Its resolution is done is two phases: first a Lagrangian relaxation solves the dual to find a lower bound; it also gives a primal relaxed solution. The latter is used in the second phase for a heuristic resolution based on a primal proximal algorithm. This second step comes as an alternative to an earlier approach based on augmented Lagrangian (i.e. a dual proximal algorithm). We illustrate the method with some real-life numerical results. A companion paper is devoted to a theoretical study of the heuristic in the second phase

On the Equivalence Between Complementarity Systems, Projected Systems and Unilateral Differential Inclusions

An updated version of this paper has appeared in Systems & Control Letters, 55, (2006), pp 45-51, DOI 10.1016/j.sysconle.2005.04.015In this note we prove the equivalence, under appropriate conditions, between several dynamical formalisms: projected dynamical systems, two types of unilateral differential inclusions, and a class of complementarity dynamical systems. Each of these dynamical systems can also be considered as a hybrid dynamical system. This work is of interest since it both generalises some previous results and sheds new light on the relationship between known formalisms

A Family of variable metric proximal methods

An updated version of this paper has appeared in Mathematical Programming, no 68 (1995), pp. 15-47, DOI 10.1007/BF01585756Nous considérons des méthodes conceptuelles d'optimisation combinant deux idéees : La régularisation de Moreau-Yosida en analyse convexe et les approximations quasi-Newtoniennes des fonctions régulières