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We present a detailed and up-to-date survey of the literature on parallel branch-and-bound algorithms. We synthesize previous work in this area and propose a new classification of parallel branch-and-bound algorithms. This classification is used to analyze the methods proposed in the literature. To facilitate our analysis, we give a new characterization of branch-and-bound algorithms, which consists of isolating the performed operations without specifying any particular order for their execution.

This paper presents a comprehensive survey of models and algorithms for multicommodity capacitated network design problems, which are mostly encountered in telecommunications and transportation network planning. These problems are important not only due to the major relevance of their applications, but also because they pose considerable modeling and algorithmic challenges. We present a general arc-based model, describe useful alternative formulations and survey the literature on simplex-based cutting plane and Lagrangean relaxation approaches. We then focus on our own contributions that develop and compare several relaxation methods for a particular case of this model, the xed-charge problem. These methods are based on Lagrangean relaxation and nondi erentiable optimization techniques, namely, the subgradient and bundle approaches. Our experimental results, while very encouraging, indicate that solving e ciently these di cult problems requires a judicious combination of cutting planes, Lagrangean relaxation methods and sophisticated heuristics. In addition, due to their inherent decomposition properties, these techniques can beadapted to parallel computing environments, which is highly desirable in order to solve realistically sized instances.Key words : Multicommodity capacitated network design, cutting planes, Lagrangean relaxation, non-di erentiable optimization, parallel computing. R esum eCet article pr esente une revue de la litt erature sur les mod eles et les m ethodes de r esolution de probl emes de conception de r eseaux avec capacit es. Ces probl emes sont importants non seulement en raison de leurs applications en plani cation de r eseaux de transport et de t el ecommunications, mais egalement parce qu'ils posent des d e s consid erables. Nous pr esentons un mod ele g en eral, ainsi que d'autres formulations alternatives int eressantes, et nous passons en revue les travaux portant sur les m ethodes de coupes et de relaxation lagrangienne. Nous d ecrivons egalement nos propres contributions, dans lesquelles nous d eveloppons et comparons plusieurs m ethodes de relaxation pour un cas particulier, le probl eme avec coûts xes. Ces m ethodes sont bas ees sur la relaxation lagrangienne et l'optimisation non-di erentiable, en particulier les algorithmes de sous-gradients et de faisceaux. Nos r esultats exp erimentaux, bien qu'encourageants, sugg erent que les m ethodes les plus prometteuses consistent a combiner les m ethodes de coupes et de relaxation lagrangienne avec des heuristiques sophistiqu ees, et d'adapter ces m ethodes a des environnements parall eles, a n de r esoudre e cacement des exemplaires de grande taille.Mots-cl es : Conception de r eseaux avec capacit es, m ethodes de coupes, relaxation lagrangienne, optimisation non-di erentiable, calcul parall ele.ii

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.