Interior point stabilization for column generation
Introduction
Column generation was introduced by Dantzig and Wolfe [3] to solve linear programs with decomposable structures. It has been applied to many problems with success and has become a leading optimization technique to solve routing and scheduling problems [5], [1]. However, some column generation methods often show very slow convergence partly due to heavy degeneracy problems. One of these problems arises when multiple dual solutions are associated with each primal solution. Choosing the dual solution then becomes a crucial part of the column generation algorithm as the subproblem solution often heavily depends on the dual values. This observation led to the introduction of several stabilization methods, which attempted to accelerate convergence by implementing diverse mechanisms to control the selection of the dual solution. The method proposed here tries to achieve the same goal by using a different approach.
The next section, which gives a brief overview of the column generation framework, is followed by a short description of some existing stabilization methods that we will use for comparison purposes. Section 3 introduces interior point stabilization and results are reported in Section 4.
Future work on this technique involves its application to different problems in order to clearly assess in which contexts it can be useful.
Section snippets
Column generation
Column generation is a general framework that can be applied to numerous problems. However, since an example is often useful to give a clear explanation, all results in this paper will be presented with respect to the vehicle routing problem with time windows (VRPTW). The description of IPS is done without loss of generality nor restrictions to deal with this particular problem. Although it cannot be said that the VRPTW is representative of all domain applications solved by column generation,
Stability problems
Column generation frameworks depend heavily on marginal costs to guide the search at the subproblem level. In some cases it is possible that, during the first iterations, the marginal costs associated with each customer are not appropriately estimated by the dual values. For instance, it is possible that in some routes some customers pick up most of the total dual values. If this is the case (as illustrated in Fig. 1), then in the subproblem a path that visits each of those overweighted
Interior point stabilization
The idea behind interior point stabilization is to generate a dual solution that is in the interior of the convex hull of optimal dual solutions to the master instead of using one of its extreme points. Bixby et al. [2] have proposed to achieve this by using an interior point method to solve (M). The authors note that, although the interior point method requires less iterations than a simplex based method, each iteration takes more time. They argue that each call to the subproblem generates a
Experimental results
In this section, we attempt to demonstrate the effectiveness of interior point stabilization with regard to two criteria. We first evaluate IPS as a simple (almost parameter free) stabilization method for column generation and then compare it to the stabilization techniques described in Section 2. The problem chosen to perform experiments is the vehicle routing problem with time windows (VRPTW).
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