International Journal of Operations Research Vol. 4, No. 3, 172-180 (2007) A Simple Stabilizing Method for Column Generation Heuristics: An Application to P-Median Location Problems Edson L. F. Senne 1, * , Luiz A. N. Lorena 2 , and Marcos A. Pereira 1 1 FEG/UNESP – São Paulo State University, 12516-410 – P.O. Box 205 – Guaratinguetá, SP – Brazil 2 LAC/INPE – Brazilian Space Research Institute, 12.201-970 – P.O. Box 515 – São José dos Campos, SP –Brazil Received May 2006; Revised September 2006; Accepted January 2007 AbstractThe Lagrangean/surrogate relaxation has been explored as a faster computational alternative to traditional Lagrangean heuristics. In this work the Lagrangean/surrogate relaxation and traditional column generation approaches are combined in order to accelerate and stabilize primal and dual bounds, through an improved reduced cost selection. The Lagrangean/surrogate multiplier modifies the reduced cost criterion, resulting in the selection of more productive columns for the p-median problem, which deals with the localization of p facilities (medians) on a network in order to minimize the sum of all the distances from each demand point to its nearest facility. Computational tests running p-median instances taken from the literature are presented. KeywordsP-median, Location, Column generation, Large-scale optimization, Integer programming * Corresponding author’ s email: elfsenne@feg.unesp.br 1. INTRODUCTION This work describes the use of the Lagrangean/ surrogate relaxation as a stabilizing method for the column generation process for linear programming problems. The Lagrangean/surrogate relaxation uses the local information of a surrogate constraint relaxed in the Lagrangean way, and has been used to accelerate subgradient-like methods. A local search is conducted at some initial iteration of subgradient methods, adjusting the step sizes. The reduction of computational times can be substantial for large-scale problems (Narciso and Lorena (1999), Senne and Lorena (2000)). Column generation is a powerful tool for solving large-scale linear programming problems that arise when the columns of the problem are not known in advance and a complete enumeration of all columns is not an option, or the problem is rewritten using Dantzig-Wolfe decomposition (Dantzig and Wolfe (1960)). Column generation is a natural choice in several applications, such as the well-known cutting-stock problem, vehicle routing and crew scheduling (Gilmore and Gomory (1961), Gilmore and Gomory (1963), Desrochers and Soumis (1989), Desrochers et al. (1992), Vance (1993), Vance et al. (1994), Day and Ryan (1997), Valério de Carvalho (1999)). In a classical column generation process, the algorithm iterates between a restricted master problem and a column generation subproblem. Solving the master problem yields a dual solution, which is used to update the cost coefficients for the subproblem that can produce new incoming columns. The equivalence between Dantzig-Wolfe decomposition, column generation and Lagrangean relaxation optimization is well known. Solving a linear programming by Dantzig-Wolfe decomposition is equivalent to solving the Lagrangean dual by Kelley’ s cutting plane method (Kelley (1960)). However, in many cases a straightforward application of column generation may result in slow convergence. This paper shows how to use the Lagrangean/surrogate relaxation to accelerate the column generation process, generating new productive sets of columns at each iteration of the algorithm. Other attempts to stabilize dual solutions have appeared before, like the Boxstep method (Marsten et al. (1975)), where the optimization in the dual space is explicitly restricted to a bounded region with the current dual solution as the central point. Bundle methods (Neame (1999)) define a trust region combined with penalties to prevent significant changes between consecutive dual solutions. The Analytic Center Cutting Plane method (du Merle et al. (1998)) considers the current analytic center of the dual function in the next iteration, instead of assuming the optimal dual solution, avoiding drastic oscillations on the dual multipliers. Other recent alternative methods to stabilize dual solutions have been considered in du Merle et al. (1999). See also Lübbecke and Desrosiers (2002) for selected topics in column generation. The search for p-median nodes on a network with n vertices is a classical location problem. The objective is to locate p < n facilities (medians) such that the sum of the distances from each demand point to its nearest facility is minimized. The problem is well known to be NP-hard and several heuristics have been developed for p-median 1813-713X Copyright © 2007 ORSTW International Journal of Operations Research