A stabilized structured Dantzig–Wolfe decomposition method

Abstract

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.

Models and algorithms for network design problems

“…Second, it is in general impossible to project out integer variables, and thus we can not apply that methodology to problems with integer routing. Dantzig-Wolfe Decomposition has been used for (ND) by Frangioni and Gendron (2010), among others, and for the closely related multi-commodity ow problem by Barnhart et al (2000), among others. Finally, Lagrangian decomposition has been used by Crainic et al (2001), among others, together with a bundle algorithm.…”

Column Generation for Extended Formulations

“…Our paper establishes the validity of the column-and-row generation algorithm in a form that encompasses all special cases of the literature, completing the presentations of [8,9,10,11,14,15,17,23]. We identify what explains the faster convergence: the recombination of previously generated subproblem solutions.…”

“…where G denotes the set of "feasible packings" satisfying (10)(11)(12) and vector x g defines the associated solution g ∈ G. When solving its linear relaxation by column generation, the pricing problem takes the form:…”

“…Moreover, there are recent efforts to explore this approach further. In a study developed concomitantly to ours, Frangioni and Gendron [11] present a "structured Dantzig-Wolfe decomposition" for which they adapt column generation stabilization techniques (from linear penalties to the bundle method). Compared to [11], our generic presentation, relying on the definition of an extended formulation, assumes slightly less restrictive assumptions and extends to approximate extended formulation.…”

“…In a study developed concomitantly to ours, Frangioni and Gendron [11] present a "structured Dantzig-Wolfe decomposition" for which they adapt column generation stabilization techniques (from linear penalties to the bundle method). Compared to [11], our generic presentation, relying on the definition of an extended formulation, assumes slightly less restrictive assumptions and extends to approximate extended formulation. In [8], Gendreau et al present a branch-and-price-and-cut algorithm where columns and cuts are generated simultaneously; their presentation considers approximate extended formulation but with a weaker termination condition.…”