Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure. The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with constant capacity K for all plants. These facet inequalities depend on K and thus differ fundamentally from the valid inequalities for the uncapacitated version of the problem. We also introduce a second formulation for a model with indivisible cus-tomer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.

Lagrangean relaxation is largely used to solve combinatorial optimization problems. A known problem for Lagrangean relaxation application is the definition of convenient step size control in subgradient like methods. Even preserving theoretical convergence properties, a wrong defined control can reflect in performance and increase computational times, a critical point in large scale instances. We show in this work how to accelerate a classical subgradient method, using the local information of the surrogate constraints relaxed in the Lagrangean relaxation. It results in a one-dimensional search that corrects the step size and is independent of the step size control used. The application to Capacitated and Uncapacitated Facility Location problems is shown. Several computational tests confirm the superiority of this scheme. Key words: Location problems, Lagrangean relaxation, Subgradient method. 1. Introduction Facility location is the problem of locating a number of facilities from a s...