Library

Description: Convexity is an important property in nonlinear optimization since it allows to apply efficient local methods for finding global solutions. We propose to apply symbolic methods to prove or disprove convexity of rational functions over a polyhedral domain. Our algorithms reduce convexity questions to real quantifier elimination problems. Our methods are implemented and publicly available in the open source computer algebra system REDUCE. Our long term goal is to integrate REDUCE as a workhorse'' for symbolic computations into a numerical solver.

Description: Applications which need exclusive access to a shared resource in distributed systems require a fault-tolerant and scalable mechanism to coordinate this exclusive access. Examples of such applications include distributed file systems and master/slave data replication. We present Flease, an algorithm for decentralized and fault-tolerant lease coordination in distributed systems. Our algorithm allows the processes competing for a resource to coordinate exclusive access through leases among themselves without a central component. The resulting system easily scales with an increasing number of nodes and resources. We prove that Flease ensures exclusive access, i.e. guarantees that there is at most one valid lease at any time.

Description: We propose an efficient column generation method to minimize the probability of delay propagations along aircraft rotations. In this way, delay resistant schedules can be constructed. Computational results for large-scale real-world problems demonstrate substantial punctuality improvements. The method can be generalized to crew and integrated scheduling problems.

Description: In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and quasi-Newton methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy coding maintaining an error bound is sufficient.

Description: Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included are only known qualitatively as stimulatory or inhibitory effects, the model surprisingly well features quantitative observations made in reality. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences.

Description: Learning during search allows solvers for discrete optimization problems to remember parts of the search that they have already performed and avoid revisiting redundant parts. Learning approaches pioneered by the SAT and CP communities have been successfully incorporated into the SCIP constraint integer programming platform. In this paper we show that performing a heuristic constraint programming search during root node processing of a binary program can rapidly learn useful nogoods, bound changes, primal solutions, and branching statistics that improve the remaining IP search.

Description: We estimate potential energy savings in IP-over-WDM networks achieved by switching off router line cards in low-demand hours. We compare three approaches to react on dynamics in the IP traffic over time, FUFL, DUFL and DUDL. They provide different levels of freedom in adjusting the routing of lightpaths in the WDM layer and the routing of demands in the IP layer. Using MILP models based on three realistic network topologies as well as realistic demands, power, and cost values, we show that already a simple monitoring of the lightpath utilization in order to deactivate empty line cards (FUFL) brings substantial benefits. The most significant savings, however, are achieved by rerouting traffic in the IP layer (DUFL), which allows emptying and deactivating lightpaths together with the corresponding line cards. A sophisticated reoptimization of the virtual topologies and the routing in the optical domain for every demand scenario (DUDL) yields nearly no additional profits in the considered networks. These results are independent of the ratio between the demand and capacity granularities, the time scale and the network topology, and show little dependency on the demand structure.

Description: We propose a novel integer programming approach to transfer minimization for line planning problems in public transit. The idea is to incorporate penalties for transfers that are induced by “connection capacities” into the construction of the passenger paths. We show that such penalties can be dealt with by a combination of shortest and constrained shortest path algorithms such that the pricing problem for passenger paths can be solved efficiently. Connection capacity penalties (under)estimate the true transfer times. This error is, however, not a problem in practice. We show in a computational comparison with two standard models on a real-world scenario that our approach can be used to minimize passenger travel and transfer times for large-scale line planning problems with accurate results.

Description: The aim of this paper is to devise an adaptive timestep control in the contact--stabilized Newmark method (CONTACX) for dynamical contact problems between two viscoelastic bodies in the framework of Signorini's condition. In order to construct a comparative scheme of higher order accuracy, we extend extrapolation techniques. This approach demands a subtle theoretical investigation of an asymptotic error expansion of the contact--stabilized Newmark scheme. On the basis of theoretical insight and numerical observations, we suggest an error estimator and a timestep selection which also cover the presence of contact. Finally, we give a numerical example.

Description: Railway scheduling is based on the principle of the construction of a conflict-free timetable. This leads to a strict definition of capacity: in contrast with road transportation, it can be said in advance whether a given railway infrastructure can accommodate - at least in theory - a certain set of train requests. Consequently, auctions for railway capacity are modeled as auctions of discrete goods -- the train slots. We present estimates for the efficiency gain that may be generated by slot auctioning in comparison with list price allocation. We introduce a new class of allocation and auction problems, the feasible assignment problem, that is a proper generalization of the well-known combinatorial auction problem. The feasible assignment class was designed to cover the needs for an auction mechanism for railway slot auctions, but is of interest in its own right. As a practical instance to state and solve the railway slot allocation problem, we present an integer programming formulation, briefly the ACP, which turns out to be an instance of the feasible assignment problem and whose dual problem yields prices that can be applied to define a useful activity rule for the linearized version of the Ausubel Milgrom Proxy auction. We perform a simulation aiming to measure the impact on efficiency and convergence rate.

Description: We consider a variant of the classical lot-sizing problem in which the capacity in each period is an integer multiple of some basic batch size. Pochet and Wolsey (Math. Oper. Res. 18, 1993) presented an O(n^2 min{n,C}) algorithm to solve this problem and a linear program with O(n^3) variables and inequalities, where n is the number of periods and C the batch size. We provide a linear program of size O(n^2 min{n,C}), that is, in case that C 〈 n, our formulation is smaller.

Description: Providing realistic performance indicators of online algorithms for a given online optimization problem is a difficult task in general. Due to significant drawbacks of other concepts like competitive analysis, Markov decision problems (MDPs) may yield an attractive alternative whenever reasonable stochastic information about future requests is available. However, the number of states in MDPs emerging from real applications is usually exponential in the original input parameters. Therefore, the standard methods for analyzing policies, i.e., online algorithms in our context, are infeasible. In this thesis we propose a new computational tool to evaluate the behavior of policies for discounted MDPs locally, i.e., depending on a particular initial state. The method is based on a column generation algorithm for approximating the total expected discounted cost of an unknown optimal policy, a concrete policy, or a single action (which assumes actions at other states to be made according to an optimal policy). The algorithm determines an $\varepsilon$-approximation by inspecting only relatively small local parts of the total state space. We prove that the number of states required for providing the approximation is independent of the total number of states, which underlines the practicability of the algorithm. The approximations obtained by our algorithm are typically much better than the theoretical bounds obtained by other approaches. We investigate the pricing problem and the structure of the linear programs encountered in the column generation. Moreover, we propose and analyze different extensions of the basic algorithm in order to achieve good approximations fast. The potential of our analysis tool is exemplified for discounted MDPs emerging from different online optimization problems, namely online bin coloring, online target date assignment, and online elevator control. The results of the experiments are quite encouraging: our method is mostly capable to provide performance indicators for online algorithms that much better reflect observations made in simulations than competitive analysis does. Moreover, the analysis allows to reveal weaknesses of the considered online algorithms. This way, we developed a new online algorithm for the online bin coloring problem that outperforms existing ones in our analyses and simulations.

Description: This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.

Description: In this paper a bottom-up approach of automatic simplification of a railway network is presented. Starting from a very detailed, microscopic level, as it is used in railway simulation, the network is transformed by an algorithm to a less detailed level (macroscopic network), that is sufficient for long-term planning and optimization. In addition running and headway times are rounded to a pre-chosen time discretization by a special cumulative method, which we will present and analyse in this paper. After the transformation we fill the network with given train requests to compute an optimal slot allocation. Then the optimized schedule is re-transformed into the microscopic level and can be simulated without any conflicts occuring between the slots. The algorithm is used to transform the network of the very dense Simplon corridor between Swiss and Italy. With our aggregation it is possible for the first time to generate a profit maximal and conflict free timetable for the corridor across a day by a simultaneously optimization run.

Description: The paper considers linear elliptic equations with regular Borel measures as inhomogeneity. Such equations frequently appear in state-constrained optimal control problems. By a counter-example of Serrin, it is known that, in the presence of non-smooth data, a standard weak formulation does not ensure uniqueness for such equations. Therefore several notions of solution have been developed that guarantee uniqueness. In this note, we compare different definitions of solutions, namely the ones of Stampacchia and the two notions of solutions of Casas and Alibert-Raymond, and show that they are the same. As side results, we reformulate the solution in the sense of Stampacchia, and prove the existence and uniqueness of solutions in in case of mixed boundary conditions.

Description: This cumulative thesis collects the following six papers for obtaining the habilitation at the Technische Universität Berlin, Fakultät II – Mathematik und Naturwissenschaften: (1) Set packing relaxations of some integer programs. (2) Combinatorial packing problems. (3) Decomposing matrices into blocks. (4) A bundle method for integrated multi-depot vehicle and duty scheduling in public transit. (5) Models for railway track allocation. (6) A column-generation approach to line planning in public transport. Some changes were made to the papers compared to the published versions. These pertain to layout unifications, i.e., common numbering, figure, table, and chapter head layout. There were no changes with respect to notation or symbols, but some typos have been eliminated, references updated, and some links and an index was added. The mathematical content is identical. The papers are about the optimization of public transportation systems, i.e., bus networks, railways, and airlines, and its mathematical foundations, i.e., the theory of packing problems. The papers discuss mathematical models, theoretical analyses, algorithmic approaches, and computational aspects of and to problems in this area. Papers 1, 2, and 3 are theoretical. They aim at establishing a theory of packing problems as a general framework that can be used to study traffic optimization problems. Indeed, traffic optimization problems can often be modelled as path packing, partitioning, or covering problems, which lead directly to set packing, partitioning, and covering models. Such models are used in papers 4, 5, and 6 to study a variety of problems concerning the planning of line systems, buses, trains, and crews. The common aim is always to exploit as many degrees of freedom as possible, both at the level of the individual problems by using large-scale integer programming techniques, as well as on a higher level by integrating hitherto separate steps in the planning process.

Description: In solving large polynomial algebraic systems that are too big for standard Gröbner basis techniques one way to make progress is to introduce case distinctions. This divide and conquer technique can be beneficial if the algorithms and computer programs know how to take advantage of inequalities. A further hurdle is the form of the resulting general solutions which often have unnecessarily many branches. In this paper we discuss a procedure to merge solutions by dropping inequalities which are associated with them and, if necessary, by re-parametrizing solutions. In the appendix the usefulness of the procedure is demonstrated in the classification of quadratic Hamiltonians with a Lie-Poisson bracket $e(3)$. This application required the solution of algebraic systems with over 200 unknowns, 450 equations and between 5000 and 9000 terms.

Description: In the paper arguments are given why the concept of static evaluation (SE) has the potential to be a useful extension to Monte Carlo tree search. A new concept of modeling SE through a dynamical system is introduced and strengths and weaknesses are discussed. The general suitability of this approach is demonstrated. A Remark: Among users of the Internet Go server KGS the abbreviation SE is used for 'Score Estimator'. Although different from 'Static Evaluation' a score estimator is easily obtained from static evaluation by adding up probabilities of chains to be alive at the end of the game or points to be owned by White or Black

Description: Linear Poisson brackets on e(3) typical of rigid body dynamics are considered. All quadratic Hamiltonians of Kowalevski type having additional first integral of fourth degree are found. Quantum analogs of these Hamiltonians are listed.

Description: A solution generating technique is developed for $D=5$ minimal supergravity with two commuting Killing vectors based on the $G_2$ U-duality arising in the reduction of the theory to three dimensions. The target space of the corresponding 3-dimensional sigma-model is the coset $G_{2(2)}/(SL(2,R)\times SL(2,R))$. Its isometries constitute the set of solution generating symmetries. These include two electric and two magnetic Harrison transformations with the corresponding two pairs of gauge transformations, three $SL(2,R) \; S$-duality transformations, and the three gravitational scale, gauge and Ehlers transformations (altogether 14). We construct a representation of the coset in terms of $7\times 7$ matrices realizing the automorphisms of split octonions. Generating a new solution amounts to transforming the coset matrices by one-parametric subgroups of $G_{2(2)}$ and subsequently solving the dualization equations. Using this formalism we derive a new charged black ring solution with two independent parameters of rotation.

Description: A classification problem is proposed for supersymmetric %scaling\/-\/in\-va\-ri\-ant evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson\/-\/fermion systems are discovered under the weighting assumption $|f|=|b|=|D_t|=\oh$. The syntax of the \Reduce\ package \SsTools, which was used for intermediate computations, and the applicability of its procedures to the calculus of super\/-\/PDE are described.

Description: We consider the problem of constructing Gardner's deformations for the $N{=}2$ supersymmetric $a{=}4$--\/Korteweg\/--\/de Vries equation; such deformations yield recurrence relations between the super\/-\/Hamiltonians of the hierarchy. We prove the non\/-\/existence %P.~Mathieu's Open problem on constructing for of supersymmetry\/-\/invariant %Gardner's deformations that %solutions, retract to Gardner's formulas for the KdV equation %whenever it is assumed that, under the %respective component reduction. % in the $N{=}2$ super\/-\/field. the solutions . At the same time, we propose a two\/-\/step scheme for the recursive production of the integrals of motion for the $N{=}2$,\ $a{=}4$--\/SKdV. First, we find a new Gardner's deformation of the Kaup\/--\/Boussinesq equation, which is contained in the bosonic limit of the super\/-\/%$N{=}2$,\ $a{=}4$--\/SKdV hierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super\/- /Hamiltonians of the full $N{=}2$, $a{=}4$--\/SKdV hierarchy.

Description: We provide an introduction into the mathematics of and with paths. Not on the shortest, but hopefully on an entertaining path!

Description: Wir stellen in dieser Arbeit ein mathematisches Optimierungsmodell zur Bestimmung eines optimalen Linienplans vor, das sowohl die Fahrzeiten und die Anzahl der Umstiege berücksichtigt als auch die Kosten des Liniennetzes. Dieses Modell deckt wichtige praktische Anforderungen ab, die in einem gemeinsamen Projekt mit den Verkehrsbetrieben in Potsdam (ViP) formuliert wurden. In diesem Projekt wurde der Linienplan 2010 für Potsdam entwickelt. Unsere Berechnungen zeigen, dass die mathematische Optimierung in nichts einer "Handplanung" des Liniennetzes nachsteht. Im Gegenteil, mit Hilfe des Optimierungsprogramms ist es möglich, durch Veränderung der Parameter mehrere verschiedene Szenarien zu berechnen, miteinander zu vergleichen und Aussagen über minimale Kosten und Fahrzeiten zu machen.

Description: The paper gathers evidence showing different dimensions of the game of Go: the continuous and discrete nature of the game and different types of relations between state variables happening on ultra local, local, regional, and global scales. Based on these observations a new continuous local model for describing a board position is introduced. This includes the identification of the basic variables describing a board position and the formulation and solution of a dynamical system for their computation. To be usable as a static evaluation function for a game playing program at least group-wide (regional)aspects will have to be incorporated.

Description: The track allocation problem, also known as train routing problem or train timetabling problem, is to find a conflict-free set of train routes of maximum value in a railway network. Although it can be modeled as a standard path packing problem, instances of sizes relevant for real-world railway applications could not be solved up to now. We propose a rapid branching column generation approach that integrates the solution of the LP relaxation of a path coupling formulation of the problem with a special rounding heuristic. The approach is based on and exploits special properties of the bundle method for the approximate solution of convex piecewise linear functions. Computational results for difficult instances of the benchmark library TTPLIB are reported.

Description: In this paper, we study the neighbourlicity of the polytope $P_{k n}^2$ constituted by the $k$-cliques of the complete graph $K_n$ on $n$ vertices. We prove that this polytope is $3$-, but not $4$-neighbourly. Following a remark of Pierre Duchet, we partially generalize this result to the $k$-clique polytopes of $r$-uniform complete hypergraphs, $P_{kn}^r$. We show that the neighbourlicity of $P_{kn}^r$ is between $r$ and $2^r-1$ whenever $k\geq r+1$ and $n\geq k+r+1$. Computational results indicate that the upper bound is tight.

Description: This paper presents a case study on a railway timetable optimization for the very dense Simplon corridor, a major railway connection in the Alps between Switzerland and Italy. Starting from a detailed microscopic network as it is used in railway simulation, the data is transformed by an automatic procedure to a less detailed macroscopic network, that is sufficient for the purpose of capacity planning and amenable to state-of-the-art integer programming optimization methods. In this way, the macroscopic railway network is saturated with trains. Finally, the corresponding timetable is re-transformed to the microscopic level in such a way that it can be operated without any conflicts among the slots. Using this integer programming based micro-macro aggregation-disaggregation approach, it becomes for the first time possible to generate a profit maximal and conflict free timetable for the complete Simplon corridor over an entire day by a simultaneous optimization of all trains requests. This also allows to to undertake a sensitivity analysis of various problem parameters.

Description: Mixed integer programming (MIP) has become one of the most important techniques in Operations Research and Discrete Optimization. SCIP (Solving Constraint Integer Programs) is currently one of the fastest non-commercial MIP solvers. It is based on the branch-and-bound procedure in which the problem is recursively split into smaller subproblems, thereby creating a so-called branching tree. We present ParaSCIP, an extension of SCIP, which realizes a parallelization on a distributed memory computing environment. ParaSCIP uses SCIP solvers as independently running processes to solve subproblems (nodes of the branching tree) locally. This makes the parallelization development independent of the SCIP development. Thus, ParaSCIP directly profits from any algorithmic progress in future versions of SCIP. Using a first implementation of ParaSCIP, we were able to solve two previously unsolved instances from MIPLIB2003, a standard test set library for MIP solvers. For these computations, we used up to 2048 cores of the HLRN~II supercomputer.

Description: The thesis presents the analysis of a reduced model for modulation of internal gravity waves by deep convective clouds. The starting point for the derivation are conservation laws for mass, momentum and energy coupled with a bulk micro-physics model describing the evolution of mixing ratios of water vapor, cloud water and rain water. A reduced model for the identified scales of the regime is derived, using multi-scale asymptotics. The closure of the model employs conditional averaging over the horizontal scale of the convective clouds. The resulting reduced model is an extension of the anelastic equations, linearized around a constant background state, which are well-known from meteorology. The closure of the model is achieved purely by analytical means and involves no additional physically motivated assumptions. The essential new parameter arising from the coupling to a micro-physics model is the area fraction of saturated regions on the horizontal scale of the convective clouds. It turns out that this parameter is constant on the employed short timescale. Hence the clouds constitute a constant background, modulating the characteristics of propagation of internal waves. The model is then investigated by analytical as well as numerical means. Important results are, among others, that in the model moisture (i) inhibits propagation of internal waves by reducing the modulus of the group velocity, (ii) reduces the angle between the propagation direction of a wave-packet and the horizontal, (iii) causes critical layers and (iv) introduces a maximum horizontal wavelength beyond which waves are no longer propagating but become evanescent. The investigated examples of orographically generated gravity waves also feature a significant reduction of vertical momentum flux by moisture. The model is extended by assuming systematically small under-saturation, that is saturation at leading order. The closure is similar to the original case but requires additional assumptions. The saturated area fraction in the obtained model is no longer constant but now depends nonlinearly on vertical displacement and thus on vertical velocity.

Description: Vehicle rotation planning for long distance passenger railways is a fundamental problem in rail transport. It deals with the allocation of vehicles to trips in a cyclic weekly schedule. Its result is an assignment of each trip to a follow-on trip which will be serviced by the same vehicle. To take so-called regularity, which is an important requirement, into account, vehicle rotation planning can be modeled as a hyperassignment problem. This is a generalization of the assignment problem to directed hypergraphs we propose. We prove that the hyperassignment problem is NP-hard for the practically relevant cases and that the canonical integer linear programming (ILP) formulation results in large integrality gaps and arbitrarily high basis matrix determinants. Our main contribution is an extended ILP formulation, which provably implies important classes of inequalities, e. g., all clique inequalities. Clique inequalities are of great importance, because as calculations with practical data show they highly reduce the LP-IP gap. The extended formulation can be solved by column generation. We propose fast combinatorial algorithms for the pricing subproblem.

Description: In optimal control problems with nonlinear time-dependent 3D PDEs, the computation of the reduced gradient by adjoint methods requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. Since the state enters into the adjoint equation, the storage of a 4D discretization is necessary. We propose a lossy compression algorithm using a cheap predictor for the state data, with additional entropy coding of prediction errors. Analytical and numerical results indicate that compression factors around 30 can be obtained without exceeding the FE discretization error.

Description: We consider an optimal control problem from hyperthermia treatment planning and its barrier regularization. We derive basic results, which lay the groundwork for the computation of optimal solutions via an interior point path-following method. Further, we report on a numerical implementation of such a method and its performance at an example problem.