@misc{RaackRaymondWerneretal., author = {Raack, Christian and Raymond, Annie and Werner, Axel and Schlechte, Thomas}, title = {Integer Programming and Sports Rankings}, issn = {1438-0064}, doi = {10.1515/jqas-2013-0111}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18068}, abstract = {Sports rankings are obtained by applying a system of rules to evaluate the performance of the participants in a competition. We consider rankings that result from assigning an ordinal rank to each competitor according to their performance. We develop an integer programming model for rankings that allows us to calculate the number of points needed to guarantee a team the ith position, as well as the minimum number of points that could yield the ith place. The model is very general and can thus be applied to many types of sports. We discuss examples coming from football (soccer), ice hockey, and Formula~1. We answer various questions and debunk a few myths along the way. Are 40 points enough to avoid relegation in the Bundesliga? Do 95 points guarantee the participation of a team in the NHL playoffs? Moreover, in the season restructuration currently under consideration in the NHL, will it be easier or harder to access the playoffs? Is it possible to win the Formula~1 World Championship without winning at least one race or without even climbing once on the podium? Finally, we observe that the optimal solutions of the aforementioned model are associated to extreme situations which are unlikely to happen. Thus, to get closer to realistic scenarios, we enhance the model by adding some constraints inferred from the results of the previous years.}, language = {en} } @misc{GleixnerBertholdMuelleretal., author = {Gleixner, Ambros and Berthold, Timo and M{\"u}ller, Benjamin and Weltge, Stefan}, title = {Three Enhancements for Optimization-Based Bound Tightening}, issn = {1438-0064}, doi = {10.1007/s10898-016-0450-4}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-57803}, abstract = {Optimization-based bound tightening (OBBT) is one of the most effective procedures to reduce variable domains of nonconvex mixed-integer nonlinear programs (MINLPs). At the same time it is one of the most expensive bound tightening procedures, since it solves auxiliary linear programs (LPs)—up to twice the number of variables many. The main goal of this paper is to discuss algorithmic techniques for an efficient implementation of OBBT. Most state-of-the-art MINLP solvers apply some restricted version of OBBT and it seems to be common belief that OBBT is beneficial if only one is able to keep its computational cost under control. To this end, we introduce three techniques to increase the efficiency of OBBT: filtering strategies to reduce the number of solved LPs, ordering heuristics to exploit simplex warm starts, and the generation of Lagrangian variable bounds (LVBs). The propagation of LVBs during tree search is a fast approximation to OBBT without the need to solve auxiliary LPs. We conduct extensive computational experiments on MINLPLib2. Our results indicate that OBBT is most beneficial on hard instances, for which we observe a speedup of 17\% to 19\% on average. Most importantly, more instances can be solved when using OBBT.}, language = {en} } @misc{MunguiaOxberryRajanetal., author = {Munguia, Lluis-Miquel and Oxberry, Geoffrey and Rajan, Deepak and Shinano, Yuji}, title = {Parallel PIPS-SBB: Multi-Level Parallelism For Stochastic Mixed-Integer Programs}, number = {ZIB-Report 17-58}, issn = {1438-0064}, doi = {10.1007/s10589-019-00074-0}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-65517}, abstract = {PIPS-SBB is a distributed-memory parallel solver with a scalable data distribution paradigm. It is designed to solve MIPs with a dual-block angular structure, which is characteristic of deterministic-equivalent Stochastic Mixed-Integer Programs (SMIPs). In this paper, we present two different parallelizations of Branch \& Bound (B\&B), implementing both as extensions of PIPS-SBB, thus adding an additional layer of parallelism. In the first of the proposed frameworks, PIPS-PSBB, the coordination and load-balancing of the different optimization workers is done in a decentralized fashion. This new framework is designed to ensure all available cores are processing the most promising parts of the B\&B tree. The second, ug[PIPS-SBB,MPI], is a parallel implementation using the Ubiquity Generator (UG), a universal framework for parallelizing B\&B tree search that has been successfully applied to other MIP solvers. We show the effects of leveraging multiple levels of parallelism in potentially improving scaling performance beyond thousands of cores.}, language = {en} } @misc{FujiiKimKojimaetal., author = {Fujii, Koichi and Kim, Sunyoung and Kojima, Masakazu and Mittelmann, Hans D. and Shinano, Yuji}, title = {The Largest Unsolved QAP Instance Tai256c Can Be Converted into A 256-dimensional Simple BQOP with A Single Cardinality Constraint}, issn = {1438-0064}, doi = {10.12752/8808}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-88086}, abstract = {Tai256c is the largest unsolved quadratic assignment problem (QAP) instance in QAPLIB; a 1.48\% gap remains between the best known feasible objective value and lower bound of the unknown optimal value. This paper shows that the instance 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.The converted BQOP is much simpler than the original QAP tai256c and it also inherits some of the symmetry properties. However, it is still very difficult to solve. We present an efficient branch and bound method for improving the lower bound effectively. A new lower bound with 1.36\% gap is also provided.}, language = {en} } @misc{GottwaldMaherShinano, author = {Gottwald, Robert Lion and Maher, Stephen J. and Shinano, Yuji}, title = {Distributed domain propagation}, issn = {1438-0064}, doi = {10.4230/LIPIcs.SEA.2017.6}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61380}, abstract = {Portfolio parallelization is an approach that runs several solver instances in parallel and terminates when one of them succeeds in solving the problem. Despite it's simplicity portfolio parallelization has been shown to perform well for modern mixed-integer programming (MIP) and boolean satisfiability problem (SAT) solvers. Domain propagation has also been shown to be a simple technique in modern MIP and SAT solvers that effectively finds additional domain reductions after a variables domain has been reduced. This paper investigates the impact of distributed domain propagation in modern MIP solvers that employ portfolio parallelization. Computational experiments were conducted for two implementations of this parallelization approach. While both share global variable bounds and solutions they communicate differently. In one implementation the communication is performed only at designated points in the solving process and in the other it is performed completely asynchronously. Computational experiments show a positive performance impact of communicating global variable bounds and provide valuable insights in communication strategies for parallel solvers.}, language = {en} } @misc{GriewankStreubelLehmannetal., author = {Griewank, Andreas and Streubel, Tom and Lehmann, Lutz and Hasenfelder, Richard and Radons, Manuel}, title = {Piecewise linear secant approximation via Algorithmic Piecewise Differentiation}, issn = {1438-0064}, doi = {10.1080/10556788.2017.1387256}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61642}, abstract = {It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x - x̌|| ||x - x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.}, language = {en} } @misc{FackeldeyRoeblitzScharkoietal., author = {Fackeldey, Konstantin and R{\"o}blitz, Susanna and Scharkoi, Olga and Weber, Marcus}, title = {Soft Versus Hard Metastable Conformations in Molecular Simulations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13189}, number = {11-27}, abstract = {Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide.}, language = {en} } @misc{AndersonHiller, author = {Anderson, Lovis and Hiller, Benjamin}, title = {A Sweep-Plane Algorithm for the Computation of the Volume of a Union of Polytopes}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69489}, abstract = {Optimization models often feature disjunctions of polytopes as submodels. Such a disjunctive set is initially (at best) relaxed to its convex hull, which is then refined by branching. To measure the error of the convex relaxation, the (relative) difference between the volume of the convex hull and the volume of the disjunctive set may be used. This requires a method to compute the volume of the disjunctive set. Naively, this can be done via inclusion/exclusion and leveraging the existing code for the volume of polytopes. However, this is often inefficient. We propose a revised variant of an old algorithm by Bieri and Nef (1983) for this purpose. The algorithm uses a sweep-plane to incrementally calculate the volume of the disjunctive set as a function of the offset parameter of the sweep-plane.}, language = {en} } @misc{Anderson, type = {Master Thesis}, author = {Anderson, Lovis}, title = {The Computation of the Volume of the Union of Polytopes via a Sweep-Plane Algorithm}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-71829}, pages = {52}, abstract = {Das Thema dieser Arbeit ist ein Volumen-Algorithmus f{\"u}r die Vereinigung von Polytopen. Der Algorithmus basiert auf der Arbeit von Bieri und Nef. Er berechnet das Volumen der Vereinigung von Polytopen mit einem Sweep-Verfahren. Dabei wird eine Hyperebene im Raum verschoben und das Volumen auf der einen Seite der Hyperebene berechnet. Umso weiter die Hyperebene verschobe wird, desto gr{\"o}ßer ist auch der Halbraum. Unser Algorithmus berechnet das Volumen einer Vereinigung von Polytopen geschnitten mit dem Halbraum der Sweep-Ebene als eine Funktion abh{\"a}ngig von der Veschiebung. Ab einem gewissen Punkt liegt der K{\"o}rper dabei komplett im Halbraum der Sweep-Ebene und das Volumen bleibt konstant. Unser Algorithmus unterscheidet sich in zwei Punkten von dem Algorithmus von Bieri und Nef. Erstens funktioniert er nur auf der Vereinigung von Polytopen, wohingegen der Algorithmus von Bieri und Nef f{\"u}r Nef-Polyeder funktioniert. Diese sind eine Verallgemeinerung von Polyedern, die auch die Klasse der Vereinigung von Polytopen umfasst. F{\"u}r uns ist das allerdings kein Nachteil, da unsere Datens{\"a}tze zu Vereinigungen von Polytopen f{\"u}hren. Zweitens ist unser Algorithmus in zwei Teile aufgeteilt. Im ersten Teil wird eine Datenstruktur entwickelt, aus der im zweiten Teil zusammen mit einer Richtung die Sweep-Ebenen-Volumenfunktion berechnet wird. Der Großteil der Komplexit{\"a}t liegt im ersten Teil des Algorithmus. Das hat den Vorteil, dass wir die Volumenfunktionen f{\"u}r viele verschiedene Richtungen berechnen k{\"o}nnen. So k{\"o}nnen Einblicke in die Struktur des K{\"o}rpers gewonnen werden. Der Algorithmus beruht auf zwei verschiedenen Zerlegungsans{\"a}tzen. Zuerst k{\"o}nnen wir mit Hilfe von Anordnungen von Hyperebenen eine Vereinigung von Polytopen in ihre Zellen zerlegen. Dabei berufen wir uns auf die Arbeit von Gerstner und Holtz, in der das Konzept der Positionsvektoren eingef{\"u}hrt wird. Diese nutzen wir um die Ecken und ihre benachbarten Zellen zu bestimmen. So erhalten wir eine Zerlegung unserer Vereinigung in Zellen, deren paarweise Schnitte kein Volumen haben. Das zweite Zerlegungskonzept ist die konische Zerlegung, wie sie von Lawrence eingef{\"u}hrt wurde. Mit Hilfe dieser k{\"o}nnen wir die Indikatorfunktion eines Polytops als die Summe der Indikatorfunktionen seiner Vorw{\"a}rtskegel schreiben. Die Sweep-Ebenen Volumenfunktionen k{\"o}nnen dann leicht mit Hilfe einer altbekannten Formel f{\"u}r das Volumen von Simplices berechnet werden.}, language = {en} } @misc{MasingLindnerEbert, author = {Masing, Berenike and Lindner, Niels and Ebert, Patricia}, title = {Forward and Line-Based Cycle Bases for Periodic Timetabling}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-89731}, abstract = {The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks, and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.}, language = {en} } @phdthesis{Miltenberger, author = {Miltenberger, Matthias}, title = {Linear Programming in MILP Solving - A Computational Perspective}, publisher = {Verlag Dr. Hut GmbH}, isbn = {9783843953238}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-91873}, pages = {237}, abstract = {Mixed-integer linear programming (MILP) plays a crucial role in the field of mathematical optimization and is especially relevant for practical applications due to the broad range of problems that can be modeled in that fashion. The vast majority of MILP solvers employ the LP-based branch-and-cut approach. As the name suggests, the linear programming (LP) subproblems that need to be solved therein influence their behavior and performance significantly. This thesis explores the impact of various LP solvers as well as LP solving techniques on the constraint integer programming framework SCIP Optimization Suite. SCIP allows for comparisons between academic and open-source LP solvers like Clp and SoPlex, as well as commercially developed, high-end codes like CPLEX, Gurobi, and Xpress. We investigate how the overall performance and stability of an MILP solver can be improved by new algorithmic enhancements like LP solution polishing and persistent scaling that we have implemented in the LP solver SoPlex. The former decreases the fractionality of LP solutions by selecting another vertex on the optimal hyperplane of the LP relaxation, exploiting degeneracy. The latter provides better numerical properties for the LP solver throughout the MILP solving process by preserving and extending the initial scaling factors, effectively also improving the overall performance of SCIP. Both enhancement techniques are activated by default in the SCIP Optimization Suite. Additionally, we provide an analysis of numerical conditions in SCIP through the lens of the LP solver by comparing different measures and how these evolve during the different stages of the solving process. A side effect of our work on this topic was the development of TreeD: a new and convenient way of presenting the search tree interactively and animated in the three-dimensional space. This visualization technique facilitates a better understanding of the MILP solving process of SCIP. Furthermore, this thesis presents the various algorithmic techniques like the row representation and iterative refinement that are implemented in SoPlex and that distinguish the solver from other simplex-based codes. Although it is often not as performant as its competitors, SoPlex demonstrates the ongoing research efforts in the field of linear programming with the simplex method. Aside from that, we demonstrate the rapid prototyping of algorithmic ideas and modeling approaches via PySCIPOpt, the Python interface to the SCIP Optimization Suite. This tool allows for convenient access to SCIP's internal data structures from the user-friendly Python programming language to implement custom algorithms and extensions without any prior knowledge of SCIP's programming language C. TreeD is one such example, demonstrating the use of several Python libraries on top of SCIP. PySCIPOpt also provides an intuitive modeling layer to formulate problems directly in the code without having to utilize another modeling language or framework. All contributions presented in this thesis are readily accessible in source code in SCIP Optimization Suite or as separate projects on the public code-sharing platform GitHub.}, language = {en} } @misc{LieFackeldeyWeber, author = {Lie, Han Cheng and Fackeldey, Konstantin and Weber, Marcus}, title = {A square root approximation of transition rates for a Markov State Model}, issn = {1438-0064}, doi = {10.1137/120899959}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42195}, abstract = {Trajectory- or mesh-based methods for analyzing the dynamical behavior of large molecules tend to be impractical due to the curse of dimensionality - their computational cost increases exponentially with the size of the molecule. We propose a method to break the curse by a novel square root approximation of transition rates, Monte Carlo quadrature and a discretization approach based on solving linear programs. With randomly sampled points on the molecular energy landscape and randomly generated discretizations of the molecular configuration space as our initial data, we construct a matrix describing the transition rates between adjacent discretization regions. This transition rate matrix yields a Markov State Model of the molecular dynamics. We use Perron cluster analysis and coarse-graining techniques in order to identify metastable sets in configuration space and approximate the transition rates between the metastable sets. Application of our method to a simple energy landscape on a two-dimensional configuration space provides proof of concept and an example for which we compare the performance of different discretizations. We show that the computational cost of our method grows only polynomially with the size of the molecule. However, finding discretizations of higher-dimensional configuration spaces in which metastable sets can be identified remains a challenge.}, language = {en} } @misc{BorndoerferHoppmannKarbsteinetal., author = {Bornd{\"o}rfer, Ralf and Hoppmann, Heide and Karbstein, Marika and Lindner, Niels}, title = {Separation of Cycle Inequalities in Periodic Timetabling}, issn = {1438-0064}, doi = {https://doi.org/10.1016/j.disopt.2019.100552}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-69746}, abstract = {Cycle inequalities play an important role in the polyhedral study of the periodic timetabling problem. We give the first pseudo-polynomial time separation algo- rithm for cycle inequalities, and we give 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.}, language = {en} } @misc{TateiwaShinanoYasudaetal., author = {Tateiwa, Nariaki and Shinano, Yuji and Yasuda, Masaya and Kaji, Shizuo and Yamamura, Keiichiro and Fujisawa, Katsuki}, title = {Massively parallel sharing lattice basis reduction}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-85209}, abstract = {For cryptanalysis in lattice-based schemes, the performance evaluation of lattice basis reduction using high-performance computers is becoming increasingly important for the determination of the security level. We propose a distributed and asynchronous parallel reduction algorithm based on randomization and DeepBKZ, which is an improved variant of the block Korkine-Zolotarev (BKZ) reduction algorithm. Randomized copies of a lattice basis are distributed to up to 103,680 cores and independently reduced in parallel, while some basis vectors are shared asynchronously among all processes via MPI. There is a trade-off between randomization and information sharing; if a substantial amount of information is shared, all processes will work on the same problem, thereby diminishing the benefit of parallelization. To monitor this balance between randomness and sharing, we propose a metric to quantify the variety of lattice bases. We empirically find an optimal parameter of sharing for high-dimensional lattices. We demonstrate the efficacy of our proposed parallel algorithm and implementation with respect to both performance and scalability through our experiments.}, language = {en} } @misc{FujiiKimKojimaetal., author = {Fujii, Koichi and Kim, Sunyoung and Kojima, Masakazu and Mittelmann, Hans D. and Shinano, Yuji}, title = {An Exceptionally Difficult Binary Quadratic Optimization Problem with Symmetry: a Challenge for The Largest Unsolved QAP Instance Tai256c}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-93072}, abstract = {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.}, language = {en} }