TY - GEN A1 - Berthold, Timo A1 - Witzig, Jakob T1 - Conflict Analysis for MINLP N2 - The generalization of MIP techniques to deal with nonlinear, potentially non-convex, constraints have been a fruitful direction of research for computational MINLP in the last decade. In this paper, we follow that path in order to extend another essential subroutine of modern MIP solvers towards the case of nonlinear optimization: the analysis of infeasible subproblems for learning additional valid constraints. To this end, we derive two different strategies, geared towards two different solution approaches. These are using local dual proofs of infeasibility for LP-based branch-and-bound and the creation of nonlinear dual proofs for NLP-based branch-and-bound, respectively. We discuss implementation details of both approaches and present an extensive computational study, showing that both techniques can significantly enhance performance when solving MINLPs to global optimality. T3 - ZIB-Report - 20-20 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78964 SN - 1438-0064 ER - TY - CHAP A1 - Witzig, Jakob A1 - Berthold, Timo T1 - Conflict-Free Learning for Mixed Integer Programming T2 - Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2020 N2 - Conflict learning plays an important role in solving mixed integer programs (MIPs) and is implemented in most major MIP solvers. A major step for MIP conflict learning is to aggregate the LP relaxation of an infeasible subproblem to a single globally valid constraint, the dual proof, that proves infeasibility within the local bounds. Among others, one way of learning is to add these constraints to the problem formulation for the remainder of the search. We suggest to not restrict this procedure to infeasible subproblems, but to also use global proof constraints from subproblems that are not (yet) infeasible, but can be expected to be pruned soon. As a special case, we also consider learning from integer feasible LP solutions. First experiments of this conflict-free learning strategy show promising results on the MIPLIB2017 benchmark set. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-58942-4_34 IS - 12296 SP - 521 EP - 530 PB - Springer, Cham. ER - TY - GEN A1 - Gamrath, Gerald A1 - Anderson, Daniel A1 - Bestuzheva, Ksenia A1 - Chen, Wei-Kun A1 - Eifler, Leon A1 - Gasse, Maxime A1 - Gemander, Patrick A1 - Gleixner, Ambros A1 - Gottwald, Leona A1 - Halbig, Katrin A1 - Hendel, Gregor A1 - Hojny, Christopher A1 - Koch, Thorsten A1 - Le Bodic, Pierre A1 - Maher, Stephen J. A1 - Matter, Frederic A1 - Miltenberger, Matthias A1 - Mühmer, Erik A1 - Müller, Benjamin A1 - Pfetsch, Marc A1 - Schlösser, Franziska A1 - Serrano, Felipe A1 - Shinano, Yuji A1 - Tawfik, Christine A1 - Vigerske, Stefan A1 - Wegscheider, Fabian A1 - Weninger, Dieter A1 - Witzig, Jakob T1 - The SCIP Optimization Suite 7.0 N2 - The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming frame- work SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies mixed-integer linear op- timization problems and can be used stand-alone or integrated into SCIP via a presolver plugin. SCIP 7.0 provides additional support for decomposition algorithms. Besides im- provements in the Benders’ decomposition solver of SCIP, user-defined decomposition structures can be read, which are used by the automated Benders’ decomposition solver and two primal heuristics. Additionally, SCIP 7.0 comes with a tree size estimation that is used to predict the completion of the overall solving process and potentially trigger restarts. Moreover, substantial performance improvements of the MIP core were achieved by new developments in presolving, primal heuristics, branching rules, conflict analysis, and symmetry handling. Last, not least, the report presents updates to other components and extensions of the SCIP Optimization Suite, in particular, the LP solver SoPlex and the mixed-integer semidefinite programming solver SCIP-SDP. T3 - ZIB-Report - 20-10 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78023 SN - 1438-0064 ER - TY - GEN A1 - Rehfeldt, Daniel A1 - Koch, Thorsten T1 - On the exact solution of prize-collecting Steiner tree problems T3 - ZIB-Report - 20-11 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78174 SN - 1438-0064 ER - TY - GEN A1 - Hendel, Gregor A1 - Anderson, Daniel A1 - Le Bodic, Pierre A1 - Pfetsch, Marc T1 - Estimating the Size of Branch-And-Bound Trees N2 - This paper investigates the estimation of the size of Branch-and-Bound (B&B) trees for solving mixed-integer programs. We first prove that the size of the B&B tree cannot be approximated within a factor of~2 for general binary programs, unless P equals NP. Second, we review measures of the progress of the B&B search, such as the gap, and propose a new measure, which we call leaf frequency. We study two simple ways to transform these progress measures into B&B tree size estimates, either as a direct projection, or via double-exponential smoothing, a standard time-series forecasting technique. We then combine different progress measures and their trends into nontrivial estimates using Machine Learning techniques, which yields more precise estimates than any individual measure. The best method we have identified uses all individual measures as features of a random forest model. In a large computational study, we train and validate all methods on the publicly available MIPLIB and Coral general purpose benchmark sets. On average, the best method estimates B&B tree sizes within a factor of 3 on the set of unseen test instances even during the early stage of the search, and improves in accuracy as the search progresses. It also achieves a factor 2 over the entire search on each out of six additional sets of homogeneous instances we have tested. All techniques are available in version 7 of the branch-and-cut framework SCIP. T3 - ZIB-Report - 20-02 KW - mixed integer programming KW - machine learning KW - branch and bound KW - forecasting Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78144 SN - 1438-0064 ER - TY - JOUR A1 - Rössig, Ansgar A1 - Petkovic, Milena T1 - Advances in Verification of ReLU Neural Networks JF - Journal of Global Optimization N2 - We consider the problem of verifying linear properties of neural networks. Despite their success in many classification and prediction tasks, neural networks may return unexpected results for certain inputs. This is highly problematic with respect to the application of neural networks for safety-critical tasks, e.g. in autonomous driving. We provide an overview of algorithmic approaches that aim to provide formal guarantees on the behavior of neural networks. Moreover, we present new theoretical results with respect to the approximation of ReLU neural networks. On the other hand, we implement a solver for verification of ReLU neural networks which combines mixed integer programming (MIP) with specialized branching and approximation techniques. To evaluate its performance, we conduct an extensive computational study. For that we use test instances based on the ACAS Xu System and the MNIST handwritten digit data set. Our solver is publicly available and able to solve the verification problem for instances which do not have independent bounds for each input neuron. Y1 - 2020 U6 - https://doi.org/10.1007/s10898-020-00949-1 PB - Springer ER - TY - JOUR A1 - Gleixner, Ambros A1 - Steffy, Daniel T1 - Linear Programming using Limited-Precision Oracles JF - Mathematical Programming N2 - Since the elimination algorithm of Fourier and Motzkin, many different methods have been developed for solving linear programs. When analyzing the time complexity of LP algorithms, it is typically either assumed that calculations are performed exactly and bounds are derived on the number of elementary arithmetic operations necessary, or the cost of all arithmetic operations is considered through a bit-complexity analysis. Yet in practice, implementations typically use limited-precision arithmetic. In this paper we introduce the idea of a limited-precision LP oracle and study how such an oracle could be used within a larger framework to compute exact precision solutions to LPs. Under mild assumptions, it is shown that a polynomial number of calls to such an oracle and a polynomial number of bit operations, is sufficient to compute an exact solution to an LP. This work provides a foundation for understanding and analyzing the behavior of the methods that are currently most effective in practice for solving LPs exactly. Y1 - 2020 U6 - https://doi.org/10.1007/s10107-019-01444-6 VL - 183 IS - 1-2 SP - 525 EP - 554 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Hoppmann, Heide A1 - Karbstein, Marika A1 - Lindner, Niels T1 - Separation of cycle inequalities in periodic timetabling JF - Discrete Optimization N2 - Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem in public transport. We give the first pseudo-polynomial time separation algorithm for cycle inequalities, and we contribute a rigorous proof for the pseudo-polynomial time separability of the change-cycle inequalities. Moreover, we provide several NP-completeness results, indicating that pseudo-polynomial time is best possible. The efficiency of these cutting planes is demonstrated on real-world instances of the periodic timetabling problem. Y1 - 2020 U6 - https://doi.org/10.1016/j.disopt.2019.100552 IS - 35 SP - 100552 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Lindner, Niels A1 - Roth, Sarah T1 - A Concurrent Approach to the Periodic Event Scheduling Problem JF - Journal of Rail Transport Planning & Management N2 - We introduce a concurrent solver for the periodic event scheduling problem (PESP). It combines mixed integer programming techniques, the modulo network simplex method, satisfiability approaches, and a new heuristic based on maximum cuts. Running these components in parallel speeds up the overall solution process. This enables us to significantly improve the current upper and lower bounds for all benchmark instances of the library PESPlib. Y1 - 2020 U6 - https://doi.org/10.1016/j.jrtpm.2019.100175 SN - 2210-9706 IS - 15 SP - 100175 ER - TY - CHAP A1 - Gleixner, Ambros A1 - Kempke, Nils-Christian A1 - Koch, Thorsten A1 - Rehfeldt, Daniel A1 - Uslu, Svenja T1 - First Experiments with Structure-Aware Presolving for a Parallel Interior-Point Method T2 - Operations Research Proceedings 2019 N2 - 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. KW - block structure KW - energy system models KW - interior-point method KW - high performance computing KW - linear programming KW - parallelization KW - presolving KW - preprocessing Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-48439-2_13 SP - 105 EP - 111 PB - Springer International Publishing ET - 1 ER - TY - JOUR A1 - Müller, Benjamin A1 - Serrano, Felipe A1 - Gleixner, Ambros T1 - Using two-dimensional Projections for Stronger Separation and Propagation of Bilinear Terms JF - SIAM Journal on Optimization N2 - One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over a strict, non-rectangular subset of the box. In order to exploit this in practice, we propose to compute valid linear inequalities for the projection of the feasible region onto the x-y-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for x, y, and xy over the available projections. Our computational evaluation using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly. Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-72767 VL - 30 IS - 2 SP - 1339 EP - 1365 ER - TY - CHAP A1 - Sofranac, Boro A1 - Gleixner, Ambros A1 - Pokutta, Sebastian T1 - Accelerating Domain Propagation: an Efficient GPU-Parallel Algorithm over Sparse Matrices T2 - 2020 IEEE/ACM 10th Workshop on Irregular Applications: Architectures and Algorithms (IA3) N2 - Fast domain propagation of linear constraints has become a crucial component of today's best algorithms and solvers for mixed integer programming and pseudo-boolean optimization to achieve peak solving performance. Irregularities in the form of dynamic algorithmic behaviour, dependency structures, and sparsity patterns in the input data make efficient implementations of domain propagation on GPUs and, more generally, on parallel architectures challenging. This is one of the main reasons why domain propagation in state-of-the-art solvers is single thread only. In this paper, we present a new algorithm for domain propagation which (a) avoids these problems and allows for an efficient implementation on GPUs, and is (b) capable of running propagation rounds entirely on the GPU, without any need for synchronization or communication with the CPU. We present extensive computational results which demonstrate the effectiveness of our approach and show that ample speedups are possible on practically relevant problems: on state-of-the-art GPUs, our geometric mean speed-up for reasonably-large instances is around 10x to 20x and can be as high as 195x on favorably-large instances. Y1 - 2020 U6 - https://doi.org/10.1109/IA351965.2020.00007 N1 - URL of the Slides: https://app.box.com/s/qy0pjmhtbm7shk2ypxjxlh2sj4nudvyu N1 - URL of the Abstract: http://www.pokutta.com/blog/research/2020/09/20/gpu-prob.html SP - 1 EP - 11 ER - TY - GEN A1 - Becker, Kai-Helge A1 - Hiller, Benjamin T1 - Efficient Enumeration of Acyclic Graph Orientations with Sources or Sinks Revisited N2 - In a recent paper, Conte et al. [CGMR2017] presented an algorithm for enumerating all acyclic orientations of a graph G=(V,E) with a single source (and related orientations) with delay O(|V||E|). In this paper we revisit the problem by going back to an early paper by de Fraysseix et al. [FMR1995], who proposed an algorithm for enumerating all bipolar orientations of a graph based on a recursion formula. We first formalize de Fraysseix et al.'s algorithm for bipolar orientations and determine that its delay is also O(|V||E|). We then apply their recursion formula to the case of Conte et al.'s enumeration problem and show that this yields a more efficient enumeration algorithm with delay O(\sqrt(|V|)|E|). Finally, a way to further streamline the algorithm that leads to a particularly simple implementation is suggested. T3 - ZIB-Report - 20-05 KW - acyclic orientations KW - enumeration algorithm KW - multiple sources and sinks KW - bipolar orientations Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-77684 SN - 1438-0064 ER - TY - GEN A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Olthoff, Mats T1 - A Solution Approach to the Vehicle Routing Problem with Perishable Goods N2 - This paper focuses on a special case of vehicle routing problem where perishable goods are considered. Deliveries have to be performed until a due date date, which may vary for different products. Storing products is prohibited. Since late deliveries have a direct impact on the revenues for these products, a precise demand prediction is important. In our practical case the product demands and vehicle driving times for the product delivery are dependent on weather conditions, i.e., temperatures, wind, and precipitation. In this paper the definition and a solution approach to the Vehicle Routing Problem with Perishable Goods is presented. The approach includes a procedure how historical weather data is used to predict demands and driving times. Its run time and solution quality is evaluated on different data sets given by the MOPTA Competition 2018. T3 - ZIB-Report - 20-03 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-77020 SN - 1438-0064 ER - TY - JOUR A1 - Witzig, Jakob A1 - Gleixner, Ambros T1 - Conflict-Driven Heuristics for Mixed Integer Programming JF - INFORMS Journal on Computing N2 - Two essential ingredients of modern mixed-integer programming (MIP) solvers are diving heuristics that simulate a partial depth-first search in a branch-and-bound search tree and conflict analysis of infeasible subproblems to learn valid constraints. So far, these techniques have mostly been studied independently: primal heuristics under the aspect of finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. Here, we combine both concepts in two different ways. First, we develop a diving heuristic that targets the generation of valid conflict constraints from the Farkas dual. We show that in the primal this is equivalent to the optimistic strategy of diving towards the best bound with respect to the objective function. Secondly, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classical coefficient diving. The computational performance of both methods is evaluated using an implementation in the source-open MIP solver SCIP. Experiments are carried out on publicly available test sets including Miplib 2010 and Cor@l. Y1 - 2020 U6 - https://doi.org/10.1287/ijoc.2020.0973 VL - 33 IS - 2 SP - 706 EP - 720 ER - TY - CHAP A1 - Serrano, Felipe A1 - Muñoz, Gonzalo T1 - Maximal Quadratic-Free Sets T2 - Integer Programming and Combinatorial Optimization: 21th International Conference, IPCO 2020 N2 - The intersection cut paradigm is a powerful framework that facilitates the generation of valid linear inequalities, or cutting planes, for a potentially complex set S. The key ingredients in this construction are a simplicial conic relaxation of S and an S-free set: a convex zone whose interior does not intersect S. Ideally, such S-free set would be maximal inclusion-wise, as it would generate a deeper cutting plane. However, maximality can be a challenging goal in general. In this work, we show how to construct maximal S-free sets when S is defined as a general quadratic inequality. Our maximal S-free sets are such that efficient separation of a vertex in LP-based approaches to quadratically constrained problems is guaranteed. To the best of our knowledge, this work is the first to provide maximal quadratic-free sets. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-45771-6_24 SP - 307 EP - 321 ER - TY - JOUR A1 - Sun, Yahui A1 - Rehfeldt, Daniel A1 - Brazil, Marcus A1 - Thomas, Doreen A1 - Halgamuge, Saman T1 - A Physarum-Inspired Algorithm for Minimum-Cost Relay Node Placement in Wireless Sensor Networks JF - IEEE/ACM Transactions on Networking Y1 - 2020 U6 - https://doi.org/10.1109/TNET.2020.2971770 ER - TY - CHAP A1 - Devriendt, Jo A1 - Gleixner, Ambros A1 - Nordström, Jakob T1 - Learn to Relax: Integrating 0-1 Integer Linear Programming with Pseudo-Boolean Conflict-Driven Search T2 - Integration of AI and OR Techniques in Constraint Programming. CPAIOR 2020 N2 - Conflict-driven Pseudo-Boolean (PB) solvers optimize 0-1 integer linear programs by extending the conflict-driven clause learning (CDCL) paradigm from SAT solving. Though PB solvers have the potential to be exponentially more efficient than CDCL solvers in theory, in practice they can sometimes get hopelessly stuck even when the linear program (LP) relaxation is infeasible over the reals. Inspired by mixed integer programming (MIP), we address this problem by interleaving incremental LP solving with cut generation within the conflict-driven PB search. This hybrid approach, which for the first time combines MIP techniques with full-blown conflict analysis over linear inequalities using the cutting planes method, significantly improves performance on a wide range of benchmarks, approaching a "best of two worlds" scenario between SAT-style conflict-driven search and MIP-style branch-and-cut. Y1 - 2020 VL - 12296 SP - xxiv EP - xxvi PB - Springer ER - TY - CHAP A1 - Müller, Benjamin A1 - Muñoz, Gonzalo A1 - Gasse, Maxime A1 - Gleixner, Ambros A1 - Lodi, Andrea A1 - Serrano, Felipe T1 - On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming T2 - Integer Programming and Combinatorial Optimization: 21th International Conference, IPCO 2020 N2 - The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-45771-6_25 SP - 322 EP - 337 ER - TY - JOUR A1 - Chen, Ying A1 - Xu, Xiuqin A1 - Koch, Thorsten T1 - Day-ahead high-resolution forecasting of natural gas demand and supply in Germany with a hybrid model JF - Applied Energy N2 - As the natural gas market is moving towards short-term planning, accurate and robust short-term forecasts of the demand and supply of natural gas is of fundamental importance for a stable energy supply, a natural gas control schedule, and transport operation on a daily basis. We propose a hybrid forecast model, Functional AutoRegressive and Convolutional Neural Network model, based on state-of-the-art statistical modeling and artificial neural networks. We conduct short-term forecasting of the hourly natural gas flows of 92 distribution nodes in the German high-pressure gas pipeline network, showing that the proposed model provides nice and stable accuracy for different types of nodes. It outperforms all the alternative models, with an improved relative accuracy up to twofold for plant nodes and up to fourfold for municipal nodes. For the border nodes with rather flat gas flows, it has an accuracy that is comparable to the best performing alternative model. KW - Natural gas flow forecasting KW - Neural network KW - Hybrid model KW - Functional autoregressive Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1016/j.apenergy.2019.114486 VL - 262 IS - 114486 ER - TY - JOUR A1 - Tesch, Alexander T1 - A Polyhedral Study of Event-Based Models for the Resource-Constrained Project Scheduling Problem JF - Journal of Scheduling N2 - We consider event-based Mixed-Integer Programming (MIP) models for the Resource-Constrained Project Scheduling Problem (RCPSP) that represent an alternative to the common time-indexed model (DDT) of Pritsker et al. (1969) for the case where the underlying time horizon is large or job processing times are subject to huge variations. In contrast to the time-indexed model, the size of event-based models does not depend on the time horizon. For two event-based formulations OOE and SEE of Koné et al. (2011) we present new valid inequalities that dominate the original formulation. Additionally, we introduce a new event-based model: the Interval Event-Based Model (IEE). We deduce linear transformations between all three models that yield the strict domination order IEE > SEE > OOE for their linear programming (LP) relaxations, meaning that IEE has the strongest linear relaxation among the event-based models. We further show that the popular DDT formulation can be retrieved from IEE by certain polyhedral operations, thus giving a unifying view on a complete branch of MIP formulations for the RCPSP. In addition, we analyze the computational performance of all presented models on test instances of the PSPLIB (Kolisch and Sprecher 1997). Y1 - 2020 ER - TY - JOUR A1 - Streubel, Tom A1 - Tischendorf, Caren A1 - Griewank, Andreas T1 - Piecewise Polynomial Taylor Expansions - The Generalization of Faà di Bruno's Formula JF - Modeling, Simulation and Optimization of Complex Processes HPSC 2018 N2 - We present an extension of Taylor's Theorem for the piecewise polynomial expansion of non-smooth evaluation procedures involving absolute value operations. Evaluation procedures are computer programs of mathematical functions in closed form expression and allow a different treatment of smooth operations or calls to the absolute value function. The well known classical Theorem of Taylor defines polynomial approximations of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential- or differential-algebraic equations, for the construction of solvers for continuous non-linear optimization of finite dimensional objective functions and for root solving of non-linear systems of equations. The long term goal is the stabilization and acceleration of already known methods and the derivation of new methods by incorporating piecewise polynomial Taylor expansions. The herein provided proof of the higher order approximation quality of the new generalized expansions is constructive and allows efficiently designed algorithms for the execution and computation of the piecewise polynomial expansions. As a demonstration towards the ultimate goal we will derive a prototype of a {\$}{\$}k{\$}{\$}k-step method on the basis of polynomial interpolation and the proposed generalized expansions. Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-55240-4_3 IS - Modeling, Simulation and Optimization of Complex Processes HPSC 2018 SP - 63 EP - 82 PB - Springer International Publishing ER - TY - JOUR A1 - Duarte, Belmiro A1 - Sagnol, Guillaume T1 - Approximate and exact optimal designs for 2^k factorial experiments for generalized linear models via second order cone programming JF - Statistical Papers N2 - Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for 2𝑘 factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684–1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A- and log det D-optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary log–log models and consider full and fractional factorial designs. Y1 - 2020 U6 - https://doi.org/10.1007/s00362-018-01075-7 VL - 61 SP - 2737 EP - 2767 ER - TY - JOUR A1 - Griewank, Andreas A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - On the abs-polynomial expansion of piecewise smooth functions JF - Optimization Methods and Software N2 - Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\bar d-1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|...|$ apart from arithmetic operations and $\bar d$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of $\mathcal O({\bar d}^2)$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in $\bar d$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains. Y1 - 2020 U6 - https://doi.org/10.1080/10556788.2020.1817448 PB - Taylor & Francis ER - TY - GEN A1 - Shinano, Yuji A1 - Achterberg, Tobias A1 - Berthold, Timo A1 - Heinz, Stefan A1 - Koch, Thorsten A1 - Winkler, Michael T1 - Solving Previously Unsolved MIP Instances with ParaSCIP on Supercomputers by using up to 80,000 Cores N2 - Mixed-integer programming (MIP) problem is arguably among the hardest classes of optimization problems. This paper describes how we solved 21 previously unsolved MIP instances from the MIPLIB benchmark sets. To achieve these results we used an enhanced version of ParaSCIP, setting a new record for the largest scale MIP computation: up to 80,000 cores in parallel on the Titan supercomputer. In this paper, we describe the basic parallelization mechanism of ParaSCIP, improvements of the dynamic load balancing and novel techniques to exploit the power of parallelization for MIP solving. We give a detailed overview of computing times and statistics for solving open MIPLIB instances. T3 - ZIB-Report - 20-16 KW - Mixed Integer Programming, Parallel processing, Node merging, Racing, ParaSCIP, Ubiquity Generator Framework, MIPLIB Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78393 SN - 1438-0064 ER - TY - GEN A1 - Griewank, Andreas A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - On the abs-polynomial expansion of piecewise smooth functions N2 - Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\bar d-1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|...|$ apart from arithmetic operations and $\bar d$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of $\mathcal O({\bar d}^2)$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in $\bar d$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains. T3 - ZIB-Report - 20-14 KW - Nonsmooth Taylor polynomial/series KW - forward mode propagation KW - abs-normal form KW - abs-linear form KW - absolute convergence KW - Moore recurrences KW - quadratic complexity Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78448 SN - 1438-0064 ER - TY - JOUR A1 - Serrano, Felipe A1 - Schwarz, Robert A1 - Gleixner, Ambros T1 - On the relation between the extended supporting hyperplane algorithm and Kelley’s cutting plane algorithm JF - Journal of Global Optimization N2 - Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) and demonstrated its computational benefits for solving convex mixed integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley’s cutting plane algorithm (J Soc Ind Appl Math 8(4):703–712, 1960) applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by a class of general, not necessarily convex nor differentiable, functions. Y1 - 2020 U6 - https://doi.org/10.1007/s10898-020-00906-y VL - 78 SP - 161 EP - 179 ER - TY - GEN A1 - Lindner, Niels A1 - Reisch, Julian T1 - Parameterized Complexity of Periodic Timetabling N2 - Public transportation networks are typically operated with a periodic timetable. The Periodic Event Scheduling Problem (PESP) is the standard mathematical modelling tool for periodic timetabling. Since PESP can be solved in linear time on trees, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. To complete the picture, we present two pseudo-polynomial-time dynamic programming algorithms solving PESP on input networks with bounded tree- or branchwidth. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special -- but standard -- structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib. T3 - ZIB-Report - 20-15 KW - Parameterized complexity KW - Periodic timetabling KW - Treewidth KW - Branchwidth KW - Carvingwidth KW - Periodic Event Scheduling Problem Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-78314 SN - 1438-0064 ER - TY - GEN A1 - Grimm, Boris A1 - Borndörfer, Ralf A1 - Reuther, Markus A1 - Schlechte, Thomas T1 - A Cut Separation Approach for the Rolling Stock Rotation Problem with Vehicle Maintenance T2 - 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019) N2 - For providing railway services the company’s railway rolling stock is one if not the most important ingredient. It decides about the number of passenger or cargo trips the company can offer, about the quality a passenger experiences the train ride and it is often related to the image of the company itself. Thus, it is highly desired to have the available rolling stock in the best shape possible. Moreover, in many countries, as Germany where our industrial partner DB Fernverkehr AG (DBF) is located, laws enforce regular vehicle inspections to ensure the safety of the passengers. This leads to rolling stock optimization problems with complex rules for vehicle maintenance. This problem is well studied in the literature for example see Maroti and Kroon 2005, or Cordeau et. al. 2001 for applications including vehicle maintenance. The contribution of this paper is a new algorithmic approach to solve the Rolling Stock Rotation Problem for the ICE high speed train fleet of DBF with included vehicle maintenance. It is based on a relaxation of a mixed integer linear programming model with an iterative cut generation to enforce the feasibility of a solution of the relaxation in the solution space of the original problem. The resulting mixed integer linear programming model is based on a hypergraph approach presented in Borndörfer et. al. 2015. The new approach is tested on real world instances modeling different scenarios for the ICE high speed train network in Germany and compared to the approaches of Reuther 2017 that are in operation at DB Fernverkehr AG. The approach shows a significant reduction of the run time to produce solutions with comparable or even better objective function values. T3 - ZIB-Report - 19-61 Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-75501 SN - 1438-0064 ER - TY - JOUR A1 - Gleixner, Ambros A1 - Maher, Stephen A1 - Müller, Benjamin A1 - Pedroso, João Pedro T1 - Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition JF - Annals of Operations Research N2 - Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig–Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a “price-and-verify” algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature. Y1 - 2018 U6 - https://doi.org/10.1007/s10479-018-3115-5 VL - 284 IS - 2 SP - 527 EP - 555 PB - Springer US ER - TY - JOUR A1 - Sahin, Guvenc A1 - Ahmadi, Amin A1 - Borndörfer, Ralf A1 - Schlechte, Thomas T1 - Multi-period line planning with resource transfers JF - Transportation Research Part C: Emerging Technologies N2 - Urban transportation systems are subject to a high level of variation and fluctuation in demand over the day. When this variation and fluctuation are observed in both time and space, it is crucial to develop line plans that are responsive to demand. A multi-period line planning approach that considers a changing demand during the planning horizon is proposed. If such systems are also subject to limitations of resources, a dynamic transfer of resources from one line to another throughout the planning horizon should also be considered. A mathematical modelling framework is developed to solve the line planning problem with a cost-oriented approach considering transfer of resources during a finite length planning horizon of multiple periods. We use real-life public transportation network data for our computational results. We analyze whether or not multi-period solutions outperform single period solutions in terms of feasibility and relevant costs. The importance of demand variation on multi-period solutions is investigated. We evaluate the impact of resource transfer constraints on the effectiveness of solutions. We also study the effect of period lengths along with the problem parameters that are significant for and sensitive to the optimality of solutions. Y1 - 2020 U6 - https://doi.org/10.1016/j.trc.2020.102726 VL - 119 SP - 102726 ER - TY - THES A1 - Löbel, Fabian T1 - Implementing the Network Simplex for Hypergraphs N2 - The well-known network simplex algorithm is a powerful tool to solve flow problems on graphs. Based on a recent dissertation by Isabel Beckenbach, we develop the necessary theory to extend the network simplex to capacitated flow problems on hypergraphs and implement this new variant. We then attempt to solve instances arising from real-life vehicle rotation planning problems. Y1 - 2020 ER -