90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
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The current cut selection algorithm used in mixed-integer programming solvers has remained largely unchanged since its creation. In this paper, we propose a set of new cut scoring measures, cut filtering techniques, and stopping criteria, extending the current state-of-the-art algorithm and obtaining a 5\% performance improvement for SCIP over the MIPLIB 2017 benchmark set.
About 20% of the German energy demand is supplied by natural gas. Ad-
ditionally, for about twice the amount Germany serves as a transit country. Thereby, the German network represents a central hub in the European natural gas transport network. The transport infrastructure is operated by so-called transmissions system operators or TSOs. The number one priority of the TSOs is to ensure security of supply. However, the TSOs have no knowledge of the intentions and planned actions of the shippers (traders).
Open Grid Europe (OGE), one of Germany’s largest TSO, operates a high-
pressure transport network of about 12.000 km length. Since flexibility and security of supply is of utmost importance to the German Energy Transition (“Energiewende”) especially with the introduction of peak-load gas power stations, being able to predict in- and out-flow of the network is of great importance.
In this paper we introduce a new hybrid forecast method applied to gas
flows at the boundary nodes of a transport network. The new method employs optimized feature minimization and selection. We use a combination of an FAR, LSTM DNN and mathematical programming to achieve robust high quality forecasts on real world data for different types of network nodes.
Keywords: Gas Forecast, Time series, Hybrid Method, FAR, LSTM, Mathematical Optimisation
Consolidation of commodities and coordination of vehicle routes are fundamental features of supply chain management problems. While locations for consolidation and coordination are typically known a priori, in adaptive transportation networks this is not the case. The identification of such consolidation locations forms part of the decision making process. Supply chain management problems integrating the designation of consolidation locations with the coordination of long haul and local vehicle routing is not only challenging to solve, but also very difficult to formulate mathematically. In this paper, the first mathematical model integrating location clustering with long haul and local vehicle routing is proposed. This mathematical formulation is used to develop algorithms to find high quality solutions. A novel parallel framework is developed that combines exact and heuristic methods to improve the search for high quality solutions and provide valid bounds. The results demonstrate that using exact methods to guide heuristic search is an effective approach to find high quality solutions for difficult supply chain management problems.
Cut selection is a subroutine used in all modern mixed-integer linear programming solvers with the goal of selecting a subset of generated cuts that induce optimal solver performance. These solvers have millions of parameter combinations, and so are excellent candidates for parameter tuning. Cut selection scoring rules are usually weighted sums of different measurements, where the weights are parameters. We present a parametric family of mixed-integer linear programs together with infinitely many family-wide valid cuts. Some of these cuts can induce integer optimal solutions directly after being applied, while others fail to do so even if an infinite amount are applied. We show for a specific cut selection rule, that any finite grid search of the parameter space will always miss all parameter values, which select integer optimal inducing cuts in an infinite amount of our problems. We propose a variation on the design of existing graph convolutional neural networks, adapting them to learn cut selection rule parameters. We present a reinforcement learning framework for selecting cuts, and train our design using said framework over MIPLIB 2017. Our framework and design show that adaptive cut selection does substantially improve performance over a diverse set of instances, but that finding a single function describing such a rule is difficult. Code for reproducing all experiments is available at https://github.com/Opt-Mucca/Adaptive-Cutsel-MILP.
Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB. It is known that QAP tai256c can be converted into a 256 dimensional binary quadratic optimization problem (BQOP) with a single cardinality constraint which requires the sum of the binary variables to be 92. As the BQOP is much simpler than the original QAP, the conversion increases the possibility to solve the QAP. Solving exactly the BQOP, however, is still very difficult. Indeed, a 1.48% gap remains between the best known upper bound (UB) and lower bound (LB) of the unknown optimal value. This paper shows that the BQOP admits a nontrivial symmetry, a property that makes the BQOP very hard to solve. The symmetry induces equivalent subproblems in branch and bound (BB) methods. To effectively improve the LB, we propose an efficient BB method that incorporates a doubly nonnegative relaxation, the standard orbit branching and a technique to prune equivalent subproblems. With this BB method, a new LB with 1.25% gap is successfully obtained, and computing an LB with 1.0% gap is shown to be still quite difficult.
Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for some cutting planes. We relate branching decisions and cutting planes to each other through the underlying disjunctions that they are based on, with a focus on Gomory mixed-integer cuts and their corresponding split disjunctions. We show that selecting branching decisions based on quality measures of Gomory mixed-integer cuts leads to relatively small branch-and-bound trees, and that the result improves when using cuts that more accurately represent the branching decisions. Finally, we show how the history of previously computed Gomory mixed-integer cuts can be used to improve the performance of the state-of-the-art hybrid branching rule of SCIP. Our results show a 4% decrease in solve time, and an 8% decrease in number of nodes over affected instances of MIPLIB 2017.
Mixed-Integer Linear Programming (MILP) is a ubiquitous and practical modelling paradigm that is essential for optimising a broad range of real-world systems. The backbone of all modern MILP solvers is the branch-and-cut algorithm, which is a hybrid of the branch-and-bound and cutting planes algorithms. Cutting planes (cuts) are linear inequalities that tighten the relaxation of a MILP. While a lot of research has gone into deriving valid cuts for MILPs, less emphasis has been put on determining which cuts to select. Cuts in general are generated in rounds, and a subset of the generated cuts must be added to the relaxation. The decision on which subset of cuts to add is called cut selection. This is a crucial task since adding too many cuts makes the relaxation large and slow to optimise over. Conversely, adding too few cuts results in an insufficiently tightened relaxation, and more relaxations need to be enumerated. To further emphasise the difficulty, the effectiveness of an applied cut is both dependent on the other applied cuts, and the state of the MILP solver. In this thesis, we present theoretical results on the importance and difficulty of cut selection, as well as practical results that use cut selection to improve general MILP solver performance. Improving general MILP solver performance is of great importance for practitioners and has many runoff effects. Reducing the solve time of currently solved systems can directly improve efficiency within the application area. In addition, improved performance enables larger systems to be modelled and optimised, and MILP to be used in areas where it was previously impractical due to time restrictions.
Each chapter of this thesis corresponds to a publication on cut selection, where the contributions of this thesis can naturally be divided into four components. The first two components are motivated by instance-dependent performance. In practice, for each subroutine, including cut selection, MILP solvers have adjustable parameters with hard-coded default values. It is ultimately unrealistic to expect these default values to perform well for every instance. Rather, it would be ideal if the parameters were dependent on the given instance. To show this motivation is well founded, we first introduce a family of parametric MILP instances and cuts to showcase worst-case performance of cut selection for any fixed parameter value. We then introduce a graph neural network architecture and reinforcement learning framework for learning instance-dependent cut scoring parameters. In the following component, we formalise language for determining if a cut has theoretical usefulness from a polyhedral point of view in relation to other cuts. In addition, to overcome issues of infeasible projections and dual degeneracy, we introduce analytic center based distance measures. We then construct a lightweight multi-output regression model that predicts relative solver performance of an instance for a set of distance measures. The final two components are motivated by general MILP solver improvement via cut selection. Such improvement was shown to be possible, albeit difficult to achieve, by the first half of this thesis. We relate branch-and-bound and cuts through their underlying disjunctions. Using a history of previously computed Gomory mixed-integer cuts, we reduce the solve time of SCIP over the 67% of affected MIPLIB 2017 instances by 4%. In the final component, we introduce new cut scoring measures and filtering methods based on information from other MILP solving processes. The new cut selection techniques reduce the solve time of SCIP over the 97% of affected MIPLIB 2017 instances by 5%.
Cutting planes are a crucial component of state-of-the-art mixed-integer programming solvers, with the choice of which subset of cuts to add being vital for solver performance. We propose new distance-based measures to qualify the value of a cut by quantifying the extent to which it separates relevant parts of the relaxed feasible set. For this purpose, we use the analytic centers of the relaxation polytope or of its optimal face, as well as alternative optimal solutions of the linear programming relaxation. We assess the impact of the choice of distance measure on root node performance and throughout the whole branch-and-bound tree, comparing our measures against those prevalent in the literature. Finally, by a multi-output regression, we predict the relative performance of each measure, using static features readily available before the separation process. Our results indicate that analytic center-based methods help to significantly reduce the number of branch-and-bound nodes needed to explore the search space and that our multiregression approach can further improve on any individual method.
A decision support system relies on frequent re-solving of similar problem instances. While the general structure remains the same in corresponding applications, the input parameters are updated on a regular basis. We propose a generative neural network design for learning integer decision variables of mixed-integer linear programming (MILP) formulations of these problems. We utilise a deep neural network discriminator and a MILP solver as our oracle to train our generative neural network. In this article, we present the results of our design applied to the transient gas optimisation problem. With the trained network we produce a feasible solution in 2.5s, use it as a warm-start solution, and thereby decrease global optimal solution solve time by 60.5%.
This work presents an innovative short to mid-term forecasting model that analyzes nonlinear complex spatial and temporal dynamics in energy networks under demand and supply balance constraints using Network Nonlinear Time Series (TS) and Mathematical Programming (MP) approach. We address three challenges simultaneously, namely, the adjacency matrix is unknown; the total amount in the network has to be balanced; dependence is unnecessarily linear. We use a nonparametric approach to handle the nonlinearity and estimate the adjacency matrix under the sparsity assumption. The estimation is conducted with the Mathematical Optimisation method.
We illustrate the accuracy and effectiveness of the model on the example of the natural gas transmission network of one of the largest transmission system operators (TSOs) in Germany, Open Grid Europe. The obtained results show that, especially for shorter forecasting horizons, the proposed method outperforms all considered benchmark models, improving the average nMAPE for 5.1% and average RMSE for 79.6% compared to the second-best model. The model is capable of capturing the nonlinear dependencies in the complex spatial-temporal network dynamics and benefits from both sparsity assumption and the demand and supply balance constraint.