Section: New Results

Improving Branch-and-Price Methods

We have made progress on stabilization techniques and math-heuristics that have become essential components for Branch-and-Price methods.

Smoothing and proximal methods based on penalizing the deviation from the incumbent dual solution have become standards of the domain. Interpreting column generation as cutting plane strategies in the dual problem, we analyze in [26] the mechanisms on which stabilization relies. In particular, the link is established between smoothing and in-out separation strategies to derive generic convergence properties. For penalty function methods as well as for smoothing, we describe proposals for parameter self-adjusting schemes. Such schemes make initial parameter tuning less of an issue as corrections are made dynamically. Such adjustments also allow to adapt the parameters to the phase of the algorithm. We provide extensive test reports that validate our self-adjusting parameter scheme and highlight their performances. Our results also show that using smoothing in combination with penalty function yields a cumulative effect on convergence speed-ups.

Effects of stabilization techniques can be seen in practice. Routing and logistics applications are often viewed as intractable for exact optimization tools. Although such problems are naturally suited for a decomposition approach, branch-and-price-and-cut algorithms of the literature typically do not scale to the size of real-life instances. Some recent progress in stabilization techniques amongst other advances (such as diving heuristics, strong branching, and the combination with cutting plane approaches) generate new ambitions for column generation approach in solving approximately very large scale instances. Let us for instance point to the new benchmarks for the Capacitated Vehicle Routing Problem (CVRP) in [62] . The paper [24] illustrates this trend, showing exact results for freight transportation instances of a scale never considered before. Our column generation algorithm yields dual bounds and serves as the core procedure for a primal heuristic. The overal procedure is quite competitive in great part due to the convergence speed-ups resulting from efficient stabilization schemes. It typically provides optimal solutions as primal and dual bounds tend to be equal. The very large scale freight transportation instances (with up to 1,025 stations, 5,300 demands, and 12,651 rail cars) were submitted to us by our Russian partner Freight-One.

Math-heuristics have become an essential component in mixed integer programming (MIP) solvers. Extending generic MIP heuristics, our study in [28] outlines generic procedures to build primal solutions in the context of a Branch-and-Price approach and reports on their performance. Rounding the linear relaxation solution of the Dantzig-Wolfe reformulation, which is typically tighter than that of the original compact formulation, sometimes produces better solutions than state-of-the-art specialised heuristics as revealed by our numerical experiments. We focus on the so-called diving methods and their combination with diversification-intensification paradigms such as Limited Discrepancy Search, sub-MIPing, relaxation induced neighbourhood search, local branching, and strong branching. The dynamic generation of variables inherent to a column generation approach requires specific adaptation of heuristic paradigms. Our contribution lies in proposing simple strategies to get around these technical issues. Our numerical results on Generalized Assignment, Cutting Stock, and Vertex Coloring problems sets new benchmarks, highlighting the performance of diving heuristics as generic procedures in a column generation context.