The Lagrangean/surrogate relaxation and the column generation: new bounds and new columns Luiz A. N. Lorena* Marcos A. Pereira# Silvely N. A. Salomªo+ *#Laboratrio Associado de Computaªo e MatemÆtica Aplicada - LAC. Instituto Nacional de Pesquisas Espaciais - INPE. Caixa Postal 515, 12201-970 Sªo JosØ dos Campos, SP. +FCT/UNESP Campus de Presidente Prudente Rua Roberto Simonsen 305, 19060-000 Presidente Prudente, SP Email: {*lorena, #marcos}@lac.inpe.br, +silvely@prudente.unesp.br Abstract The column generation and the Dantzig-Wolfe decomposition methods are well known as efficient methods for treatment of linear programming problems with huge number of variables. One restrict master problem is identified and new columns are generated by a subproblem. It is also well known that these methods presents stabilizing issues. In order to ease these problems, the norm of the dual variables is kept under control, in order to avoid great variations. The Lagrangean/surrogate relaxation was proposed recently for stabilizing subgradient methods. This work proposes the combination of the column generation method and Lagrangean/surrogate relaxation as a stabilization method. Some computational results are showed for the p-median problem and several applications are suggested. Some opening questions are arising for future investigation. 1. Introduction The recent computer science advances, with the construction of faster and more reliable equipments, provide robust systems for Mathematical Programming [3], allowing the resolution of problems with several constraints and/or variables. These tools allow that inherently complex problems can also be solved in acceptable computational time, by usage of combined techniques as, for example, the Column Generation Method applied to Integer Programming problems. Based on Dantzig and Wolfe [5], the first practical application of this technique was the determination of one- dimensional cutting patterns (Gilmore and Gomory [14, 15]) and, since then, its usage diffuses in an intensive way [2, 4, 6, 7, 8, 21, 26, 29, 31, 33, 43, 35]. The column generation technique can be employed for linear problems with huge dimensions, when all the columns are not known a priori, or when it is intended solve a problem using Dantzig- Wolfe decomposition, where the columns correspond to the extreme points of the convex set of feasible solutions of the problem. In this case, the resolution algorithm interchanges between a subproblem and a restrict master problem. Approaches based on the column generation technique appears in a large number of recent works, as an alternative to nonlinear methods based in Lagrangean relaxation (Bundle and subgradient methods) to solve huge integer problems [1]. A search for articles in “Web of Science on November 27 th , 2001, with the subject column generation, turned into 220 works, 93 only in the last three years. The straight application of the column generation method usually produces a large number of counterproductive columns, which difficults the convergence to the solution of the problem. In this case, the dual variables oscillates around the optimal dual solution, then methods that avoid this performance can accelerate the resolution of the problem. Among these are the Boxstep Method [23], that restricts the searching of the dual solutions to a limited region that contains the dual solution as center; the Analytic Center Cutting Plane Method [9], that uses the analytic center of a region of dual function as solution, instead of the optimal solution, not permitting strong changes between two dual solutions in two consecutive iterations; the Bundle Method [23], that combines trust regions and penalizations, so the dual solutions do not vary so much from one iteration to another. Others methods are described in Neame [26].