138 research outputs found
Decomposition Methods for Network Design
AbstractNetwork design applications are prevalent in transportation and logistics. We consider the multicommodity capacitated fixed-charge network design problem (MCND), a generic model that captures three important features of network design applications: the interplay between investment and operational costs, the multicommodity aspect, and the presence of capacity constraints. We focus on mathematical programming approaches for the MCND and present three classes of methods that have been used to solve large-scale instances of the MCND: a cutting-plane method, a Benders decomposition algorithm, and Lagrangian relaxation approaches
Lagrangian-based methods for single and multi-layer multicommodity capacitated network design
Le problème de conception de réseau avec coûts fixes et capacités (MCFND) et le problème
de conception de réseau multicouches (MLND) sont parmi les problèmes de
conception de réseau les plus importants. Dans le problème MCFND monocouche, plusieurs
produits doivent être acheminés entre des paires origine-destination différentes
d’un réseau potentiel donné. Des liaisons doivent être ouvertes pour acheminer les produits,
chaque liaison ayant une capacité donnée. Le problème est de trouver la conception
du réseau à coût minimum de sorte que les demandes soient satisfaites et que les capacités
soient respectées. Dans le problème MLND, il existe plusieurs réseaux potentiels,
chacun correspondant à une couche donnée. Dans chaque couche, les demandes pour un
ensemble de produits doivent être satisfaites. Pour ouvrir un lien dans une couche particulière,
une chaîne de liens de support dans une autre couche doit être ouverte. Nous
abordons le problème de conception de réseau multiproduits multicouches à flot unique
avec coûts fixes et capacités (MSMCFND), où les produits doivent être acheminés uniquement
dans l’une des couches.
Les algorithmes basés sur la relaxation lagrangienne sont l’une des méthodes de résolution
les plus efficaces pour résoudre les problèmes de conception de réseau. Nous
présentons de nouvelles relaxations à base de noeuds, où le sous-problème résultant se
décompose par noeud. Nous montrons que la décomposition lagrangienne améliore significativement
les limites des relaxations traditionnelles.
Les problèmes de conception du réseau ont été étudiés dans la littérature. Cependant,
ces dernières années, des applications intéressantes des problèmes MLND sont apparues,
qui ne sont pas couvertes dans ces études. Nous présentons un examen des problèmes de
MLND et proposons une formulation générale pour le MLND. Nous proposons également
une formulation générale et une méthodologie de relaxation lagrangienne efficace
pour le problème MMCFND. La méthode est compétitive avec un logiciel commercial
de programmation en nombres entiers, et donne généralement de meilleurs résultats.The multicommodity capacitated fixed-charge network design problem (MCFND) and
the multilayer network design problem (MLND) are among the most important network
design problems. In the single-layer MCFND problem, several commodities have to
be routed between different origin-destination pairs of a given potential network. Appropriate
capacitated links have to be opened to route the commodities. The problem
is to find the minimum cost design and routing such that the demands are satisfied and
the capacities are respected. In the MLND, there are several potential networks, each
at a given layer. In each network, the flow requirements for a set of commodities must
be satisfied. However, the selection of the links is interdependent. To open a link in a
particular layer, a chain of supporting links in another layer has to be opened. We address
the multilayer single flow-type multicommodity capacitated fixed-charge network
design problem (MSMCFND), where commodities are routed only in one of the layers.
Lagrangian-based algorithms are one of the most effective solution methods to solve
network design problems. The traditional Lagrangian relaxations for the MCFND problem
are the flow and knapsack relaxations, where the resulting Lagrangian subproblems
decompose by commodity and by arc, respectively. We present new node-based
relaxations, where the resulting subproblem decomposes by node. We show that the
Lagrangian dual bound improves significantly upon the bounds of the traditional relaxations.
We also propose a Lagrangian-based algorithm to obtain upper bounds.
Network design problems have been the object of extensive literature reviews. However,
in recent years, interesting applications of multilayer problems have appeared that
are not covered in these surveys. We present a review of multilayer problems and propose
a general formulation for the MLND. We also propose a general formulation and
an efficient Lagrangian-based solution methodology for the MMCFND problem. The
method is competitive with (and often significantly better than) a state-of-the-art mixedinteger
programming solver on a large set of randomly generated instances
Multicommodity capacitated network design
Network design models have wide applications in telecommunications and transportation planning; see, for example, the survey articles by Magnanti and Wong (1984), Minoux (1989), Chapter 16 of the book by Ahuja, Magnanti and Orlin (1993), Section 13 of Ahuja et al. (1995). In particular, Gavish (1991) and Balakrishnan et al. (1991) present reviews of important applications in telecommunications. In many of these applications, it is required to send flows (which may be fractional) to satisfy demands given arcs with existing capacities, or to install, in discrete amounts, additional facilities with fixed capacities. In doing so, one pays a price not only for routing flows, but also for using an arc or installing additional facilities. The objective is then to determine the optimal amounts of flows to be routed and the facilities to be installed.
Document type: Part of book or chapter of boo
Node-based Lagrangian relaxations for multicommodity capacitated fixed-charge network design
Classical Lagrangian relaxations for the multicommodity capacitated fixed-charge network design problem are the so-called flow and knapsack relaxations, where the resulting Lagrangian subproblems decompose by commodities and by arcs, respectively. We introduce node-based Lagrangian relaxations, where the resulting Lagrangian subproblem decomposes by nodes. We show that the Lagrangian dual bounds of these relaxations improve upon the linear programming relaxation bound, known to be equal to the Lagrangian dual bounds for the flow and knapsack relaxations. We also develop a Lagrangian matheuristic to compute upper bounds. The computational results on a set of benchmark instances show that the Lagrangian matheuristic is competitive with the state-of-the-art heuristics from the literature
A Stabilized Structured Dantzig-Wolfe Decomposition Method
We discuss an algorithmic scheme, which we call the stabilized structured Dantzig-Wolfe decomposition method, for solving large-scale structured linear programs. It can be applied when the subproblem of the standard Dantzig-Wolfe approach admits an alternative master model amenable to column generation, other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron. Stabilization is achieved by the same techniques developed for the standard Dantzig-Wolfe approach and it is equally useful to improve the performance, as shown by computational results obtained on an application to the multicommodity capacitated network design problem
Hybrid Statistical Data Mining Framework for Multi-Commodity Fixed Charge Network Flow Problem
This paper presents a new approach to analyze the network structure in multi-commodity fixed charge network flow problems (MCFCNF). This methodology uses historical data produced from repeatedly solving the traditional MCFCNF mathematical model as input for the machine-learning framework. Further, we reshape the problem as a binary classification problem and employ machine-learning algorithms to predict network structure. This predicted network structure is further used as an initial solution for our mathematical model. The quality of the initial solution generated is judged on the basis of predictive accuracy, feasibility and reduction in solving time
Dynamic smoothness parameter for fast gradient methods
We present and computationally evaluate a variant of the fast gradient method by Nesterov that is capable of exploiting information, even if approximate, about the optimal value of the problem. This information is available in some applications, among which the computation of bounds for hard integer programs. We show that dynamically changing the smoothness parameter of the algorithm using this information results in a better convergence profile of the algorithm in practice
On the Computational Efficiency of Subgradient Methods: a Case Study with Lagrangian Bounds
Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different Lagrangian duals of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs
Dynamic Smoothness Parameter for Fast Gradient Methods
We present and computationally evaluate a variant of the fast gradient method by Nesterov that is capable of exploiting information, even if approximate, about the optimal value of the problem. This information is available in some applications, among which the computation of bounds for hard integer programs. We show that dynamically changing the smoothness parameter of the algorithm using this information results in a better convergence profile of the algorithm in practice
Métaheuristiques de recherche avec tabous pour le problème de synthèse de réseau multiproduits avec capacités
Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal
- …