90C06 Large-scale problems
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In linear optimization, matrix structure can often be exploited algorithmically. However, beneficial presolving reductions sometimes destroy the special structure of a given problem. In this article, we discuss structure-aware implementations of presolving as part of a parallel interior-point method to solve linear programs with block-diagonal structure, including both linking variables and linking constraints. While presolving reductions are often mathematically simple, their implementation in a high-performance computing environment is a complex endeavor. We report results on impact, performance, and scalability of the resulting presolving routines on real-world energy system models with up to 700 million nonzero entries in the constraint matrix.
In linear optimization, matrix structure can often be exploited algorithmically. However, beneficial presolving reductions sometimes destroy the special structure of a given problem. In this article, we discuss structure-aware implementations of presolving as part of a parallel interior-point method to solve linear programs with block-diagonal structure, including both linking variables and linking constraints. While presolving reductions are often mathematically simple, their implementation in a high-performance computing environment is a complex endeavor. We report results on impact, performance, and scalability of the resulting presolving routines on real-world energy system models with up to 700 million nonzero entries in the constraint matrix.
We present the problem of planning mobile tours of inspectors on German motorways to enforce the payment of the toll for heavy good trucks. This is a special type of vehicle routing problem with the objective to conduct as good inspections as possible on the complete network. In addition, the crews of the tours have to be scheduled. Thus, we developed a personalized crew rostering model. The planning of daily tours and the rostering are combined in a novel integrated approach and formulated as a complex and large scale Integer Program. The paper focuses first on different requirements for the rostering and how they can be modeled in detail. The second focus is on a bicriterion analysis of the planning problem to find the balance between the control quality and the roster acceptance. On the one hand the tour planning is a profit maximization problem and on the other hand the rostering should be made in a employee friendly way. Finally, computational results on real-world instances show the practicability of our method.
Given the steady increase in cores per CPU, it is only a matter of time
until supercomputers will have a million or more cores. In this article, we
investigate the opportunities and challenges that will arise when trying to
utilize this vast computing power to solve a single integer linear optimization
problem. We also raise the question of whether best practices in sequential
solution of ILPs will be effective in massively parallel environments.
Since the initial application of mathematical optimisation methods to mine planning in 1965, the Lerchs-Grossmann algorithm for computing the ultimate pit limit, operations researchers have worked on a variety of challenging problems in the area of open pit mining. This thesis focuses on the open pit mining production scheduling problem: Given the discretisation of an orebody as a block model, determine the sequence in which the blocks should be removed from the pit, over the lifespan of the mine, such that the net present value of the mining operation is maximised. In practise, when some material has been removed from the pit, it must be processed further in order to extract the valuable elements contained therein. If the concentration of valuable elements is not sufficiently high, the material is discarded as waste or stockpiled. Realistically-sized block models can contain hundreds of thousands of blocks. A common approach to render these problem instances computationally tractable is the aggregation of blocks to larger scheduling units. The thrust of this thesis is the investigation of a new mixed-integer programming formulation for the open pit mining production scheduling problem, which allows for processing decisions to be made at block level, while the actual mining schedule is still computed at aggregate level. A drawback of this model in its full form is the large number of additional variables needed to model the processing decisions. One main result of this thesis shows how these processing variables can be aggregated efficiently to reduce the problem size significantly, while practically incurring no loss in net present value. The second focus is on the application of lagrangean relaxation to the resource constraints. Using a result of Möhring et al. (2003) for project scheduling, the lagrangean relaxation can be solved efficiently via minimum cut computations in a weighted digraph. Experiments with a bundle algorithm implementation by Helmberg showed how the lagrangean dual can be solved within a small fraction of the time required by standard linear programming algorithms, while yielding practically the same dual bound. Finally, several problem-specific heuristics are presented together with computational results: two greedy sub-MIP start heuristics and a large neighbourhood search heuristic. A combination of a lagrangean-based start heuristic followed by a large neighbourhood search proved to be effective in generating solutions with objective values within a 0.05% gap of the optimum.
The Vehicle Positioning Problem (VPP) is a classical combinatorial optimization problem in public transport planning. A number of models and approaches have been suggested in the literature, which work for small problems, but not for large ones. We propose in this article a novel set partitioning model and an associated column generation solution approach for the VPP. The model provides a tight linear description of the problem. The pricing problem, and hence the LP relaxation itself, can be solved in polynomial resp. pseudo-polynomial time for some versions of the problems.