Avoiding redundant columns by adding classical Benders cuts to column generation subproblems

This is the author accepted manuscript. The final version is available fro,m Elsevier via the DOI in this recordWhen solving the linear programming (LP) relaxation of a mixed-integer program (MIP) with column generation, columns might be generated that are not needed to express any integer optimal solution. Such columns are called strongly redundant and the dual bound obtained by solving the LP relaxation is potentially stronger if these columns are not generated. We introduce a sufficient condition for strong redundancy that can be checked by solving a compact LP. Using a dual solution of this compact LP we generate classical Benders cuts for the subproblem so that the generation of strongly redundant columns can be avoided. The potential of these cuts to improve the dual bound of the column generation master problem is evaluated computationally using an implementation in the branch-price-and-cut solver GCG. While their efficacy is limited on classical problems, the benefit of applying the cuts is demonstrated on structured models to which a temporal decomposition can be applied.Engineering and Physical Sciences Research Council (EPSRC

Shunting operations at flat yards : retrieving freight railcars from storage tracks

In this paper, we study the railcar retrieval problem (RRT) where specified numbers of certain types of railcars have to be withdrawn from the storage tracks of a flat yard. This task arises in the daily operations of workshop yards for railcar maintenance. The objective is to minimize the total cost of shunting via methods such as minimizing the usage of shunting engines. We describe the RRT formally, present a mixed-integer program formulation, and prove the general case to be NP-hard. For some special cases, exact algorithms with polynomial runtimes are proposed. We also analyze several intuitive heuristic solution approaches motivated by observed real-world planning routines. We evaluate their average performances in simulations with different scenarios and provide their worst-case performance guarantee. We show that although the analyzed heuristics result in much better solutions than the naive planning approach, they are still on average 30%-50% from the optimal objective value and may result in up to 14 times higher costs in the worst case. Therefore, we conclude that optimization should be implemented in practice in order to save valuable resources. Furthermore, we analyze the impacts of yard layout and the widespread organizational routine of presorting on the railcar retrieval cost

Large-scale optimization with the primal-dual column generation method

The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.Comment: 28 pages, 1 figure, minor revision, scaled CPU time