A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems

Abstract

We present a Cost Decomposition approach for the linear Multicommodity Min-Cost Flow problem, where the mutual capacity constraints are dualized and the resulting Lagrangean Dual is solved with a dual-ascent algorithm belonging to the class of Bundle methods. Although decomposition approaches to block-structured Linear Programs have been reported not to be competitive with general-purpose software, our extensive computational comparison shows that, when carefully implemented, a decomposition algorithm can outperform several other approaches, especially on problems where the number of commodities is "large" with respect to the size of the graph. Our specialized Bundle algorithm is characterized by a new heuristic for the trust region parameter handling, and embeds a specialized Quadratic Program solver that allows the efficient implementation of strategies for reducing the number of active Lagrangean variables. We also exploit the structural properties of the single-commodity Min-Cost Flow subproblems to reduce the overall computational cost. The proposed approach can be easily extended to handle variants of the problem. Keywords: Multicommodity Flows, Bundle methods, Decomposition Methods, Lagrangean Dual IntroductionThe Multicommodity Min-Cost Flow problem (MMCF), i.e. the problem of shipping flows of different nature (commodities) at minimal cost on a network, where different commodities compete for the resources represented by the arc capacities, has been widely addressed in the literature since it models a wide variety of transportation and scheduling problems [28,58,68,3,2,6,11] . MMCF is a structured Linear Program (LP), but the instances arising from practical applications are often huge and the usual solution techniques are not efficient enough: this is especially true if the solution is required to be integral, since then the problem is NP-hard and most of the solution techniques From an algorithmic viewpoint, MMCF has motivated many important ideas that have later found broader application: examples are the column generation approach [22] and the Dantzig-Wolfe decomposition algorithm [21] . This "pushing" effect is still continuing, as demonstrated by a number of interesting recent developments [59,31,32,33] .MMCF problems also arise in finding approximate solutions to several hard graph problems [10,43] . Recently, some ε-approximation approaches have been developed for the problem [36,60,64] , making MMCF one of the few LPs for which approximations algorithms of practical interest are known [52,34] .

Multicommodity Flow Approximation Used for Exact Graph Partitioning

“…The reason for this at first surprising fact is that large multicommodity flow instances have to be solved in many application areas and in combination with a whole variety of discrete optimization problems such as network design or graph bisection. And standard simplex or interior point LP-solvers, even specialized software based on primal basis partitioning [4,5] or Lagrangian relaxation based resource-or cost-decompositions [3,9] are simply not fast enough to tackle real-size instances of these problems very efficiently. Therefore, there is a big interest in the development of algorithms that provide near optimal solutions more quickly.…”

Solving Difficult Multicommodity Problems with a Specialized Interior-Point Algorithm

“…The algorithm of Frangioni and Gallo (1999) provides excellent results for the Mnetgen instances, but this good behaviour is not observed in general for the PDS ones (these problems are described in section 4). However, and through an indirect comparison of the results of Frangioni and Gallo (1999) with those of this paper, CPLEX 6.5 seems to provide similar performances to that of the bundlemethod-based algorithm. The excellent computational results of Goffin et al (1996) are difficult to evaluate, since they are obtained with an ad-hoc nonlinear multicommodity generator, and no results are reported with either the standard Mnetgen or PDS instances.…”

“…The first one relies on bundle-methods (Frangioni and Gallo, 1999) while the second applies the analytic center cutting plane method (ACCPM) (Goffin et al, 1996), both for the maximization of the non-differentiable dual function. The algorithm of Frangioni and Gallo (1999) provides excellent results for the Mnetgen instances, but this good behaviour is not observed in general for the PDS ones (these problems are described in section 4). However, and through an indirect comparison of the results of Frangioni and Gallo (1999) with those of this paper, CPLEX 6.5 seems to provide similar performances to that of the bundlemethod-based algorithm.…”

Solving Large-Scale Linear Multicommodity Flow Problems with an Active Set Strategy and Proximal-ACCPM

“…The bundle method [12] can also be used to solve the Lagrangian dual. A. Fragioni and G. Gallo [7] implement a bundle type approach. In a recent contribution, Larsson and Yuang [11] apply an augmented Lagrangian algorithm that combines Lagrangian relaxation and nonlinear penalty technique.…”

“…The use of an active set strategy in solving the general multicommodity flow problem is not new. It has been implemented within the framework of bundle method to solve the Lagrangian dual [7] and in a primal partitioning method [15]. Both papers report significant speed-ups.…”