## Department Mathematik

We consider the gas dynamics on networks modeled by the isothermal Euler equations.
The demands of the customers at the exits of the network
represent a mass flow boundary condition. The true demands are not
exactly known and are modeled by probability distributions.
First, the stationary solutions of the quasilinear isothermal Euler
equations are derived on a single pipe. For this purpose, a real gas model is used.
Important monotonicity properties of the solution, which are of crucial
importance for the extension to networks, are shown. The existence theory
is developed in a more general framework for flow problems on networks.
This makes it possible to treat different coupling conditions at the nodes
and to use various models that are not necessarily restricted to gas, or
active elements along the edges.
For the optimization, the uncertain boundary conditions are represented
by means of chance constraints. These guarantee that the system is kept
within technical pressure limits with a certain probability. The calculation
of the probability is realized by a combination of quasi-Monte Carlo methods
and the spherical radial decomposition of Gaussian random vectors. For
the gradient-based optimization, a result for a gradient representation is
extended to include the case of convex sets of feasible realizations. It is
shown that, for the case of tree networks, the required assumptions are
fulfilled. For the numerical implementation, a multilevel method is proposed
that uses the solution for low sample numbers as a warm start for the more
complex model. This leads to a significant reduction in computational
time.
In the transient case, the linear wave equation is considered on a single
edge. The initial and boundary data at the end of an interval are uncertain
and given by stochastic processes. The other end of the interval is controlled
by a Neumann feedback law. The stochastic processes are approximated
using the Karhunen-Loève theorem. This enables the representation of
the initial-boundary data by a finite number of random variables. As a
stability measure, the probability to stay below a given threshold in the
L∞-norm, is considered. The system can be stabilized by adjusting the
feedback parameter.

In this thesis, we consider model reduction for parameter dependent parabolic PDEs defined on networks with variable composition. For this type of problem, the Reduced Basis Element Method (RBEM), developed by Maday and Rønquist, is a reasonable choice as a solution on the entire domain is not required. The reduction method is based on the idea of constructing a reduced basis for every individual component and coupling the reduced elements using a mortar-like method. However, this decomposition procedure can lead to difficulties, especially for networks consisting of numerous edges. Due to the variable composition of the networks, the solution on the interfaces is extremely difficult to predict. This can lead to unsuitable basis functions and poor approximations of the global solutions.
On the basis of networks consisting of one-dimensional domains, we present an extension of the RBEM which remedies this problem and provides a good basis representation for each individual edge. Essentially this extension makes use of a splinebased boundary parametrization in the local basis construction. To substantiate the approximation properties of the basis representation onto the global solution, we develop an error estimate for local basis construction with Proper Orthogonal Decomposition (POD) or POD-Greedy. Additionally, we provide existence, uniqueness and regularity results for parabolic PDEs on networks with one-dimensional domains, which are essential for the error analysis.
Finally, we illustrate our method with three examples. The first corresponds to the theory presented and shows two different networks of one-dimensional heat equations with varying thermal conductivity. The second and third problem demonstrates the extensibility of the method to component based domains in two dimensions or nonlinear PDEs. These were parts of the research project Life-cycle oriented optimization for a resource and energy efficient infrastructure, funded by the German Federal Ministry of Education and Research.

It is well-known that the full unitary group $\U(\SH)$ over a complex infinite-dimensional separable Hilbert space $\SH$ has Property FH. This means that the first-order cohomology spaces $H^1$ associated to its continuous unitary representations are trivial. This thesis sheds light on several important unitary subgroups of $\U(\SH)$, proving that they do not have Property FH. This is the direct limit group $\U(\infty)$ and its Banach-completions $\U_p(\SH)$, containing $\U(\infty)$ as a dense subgroup. The direct limit group is the countable union of the compact finite-dimensional unitary groups $\U(n)$ (for a positive integer $n$) viewed as increasing subgroups of $\U(\SH)$. A unitary operator $g$ belongs to $\U_p(\SH)$ if and only if the difference $g-\1$ is an operator of $p$th Schatten class (where $p$ is any positive real number greater or equal to $1$). The core of our thesis is the study of unitary highest weight representations of these infinite-dimensional unitary groups and the precise determination of those highest weight representations having a nontrivial first order cohomology space $H^1$. The criteria are simple and depend only on the highest weight of the representation. Thus, we are able to find, for each of the above unitary subgroups, infinitely many nonisomorphic irreducible unitary representations with infinite-dimensional first order 1-cohomology space $H^1$. This is a remarkable observation since it contrasts the well-known classical result that, for each finite-dimensional (connected) semisimple Lie group, there exist at most finitely many (nonisomorphic) continuous irreducible unitary representations with nontrivial 1-cohomology space $H^1$ and, in this case, the space $H^1$ is always finite-dimensional.

This thesis is concerned with modeling, analysis and applications of one-dimensional continua and networks thereof. More precisely, we use the pre-curved and -twisted three-dimensional geometrically exact beam theory to rigorously deduce several well-known models: the pre-curved two-dimensional geometrically exact beam, the pre-curved and -twisted three-dimensional linear Timoshenko beam, as weil as the geometrically nonlinear truss and string.
Based on the abstract theory of first-order quasilinear hyperbolic systems, we show in the second part of this thesis local exact boundary controllability and observability for the second-order system of pre-curved two-dimensional geometrically exact beams. Additionally, we formulate an optimal control problem for this system, derive the adjoint equation and identify conditions, that allow for classical adjoint states.
The one-dimensional models given in this thesis are used in different applications. First, we develop a numerical scheme that solves the optimal control problem for two-dimensional geometrically exact beams. Subsequently, we employ the concept of energetic homogenization to determine effective material properties of a Kirchhoff-Love plate from networks of linear Timoshenko beams and optimize their geometry. With a similar idea, applied at two levels, non-periodic networks of nonlinear strings are homogenized in order to match the behavior of non-woven fiber mats. Finally, the damaging of high-pressure pipes is investigated, which requires a nonlinear path-dependent material law coupled to the three-dimensional geometrically exact beam. In this scenario a creep-damage material law is modeled, numerically implemented and its feasibility to describe piping systems demonstrated.

This thesis is concerned with the efficient and accurate simulation and optimization of linear Timoshenko beam networks subjected to external loads.
For this we develop a solution scheme, which is based on so-called analytic ansatz-functions known to provide analytic solutions
for the unknowns of a single beam subjected to given boundary data. In the present work we prove that this concept can be extended to a network of beams
which enables the analytic calculation of the unique deformation of the whole network.
Moreover, we show that this approach is equivalent to a finite element method where only one finite element with a particular shape function per
beam is required. With this we can provide the analytic solution on each node of the network, from which the state of the whole network can be uniquely determined - no further spatial discretization is needed.
Based on this analytic solution scheme for the simulation problem, we investigate a number of problems from structural optimization, more precisely
topology optimization, material and multi-material optimization, cross section optimization, geometry optimization and simultaneous versions of the before mentioned.
Hereby we provide the problem formulation, analytic formulae for first order derivatives, an implementation and numerical examples for each type of optimization model.
In the next part of this thesis we develop a novel approach for geometry, cross section and material optimization for a system of high-pressure steam pipes in a power plant.
This industrial application is efficiently modeled by including a special parametrization for the geometry of pipes.
Then we formulate
an optimization problem with an objective function considering monetary costs and a large number of constraints.
Moreover, we derive analytic expressions for the first order derivatives of the problem and provide an implementation. We demonstrate the
capabilities of our program with numerical tests for a real-world instance.

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization
(2018)

Natural gas is one of the most important energy sources. Consequently, its transportation through gas networks is an essential task and gives rise to gas transport problems. Such optimization problems involve discrete decisions to switch network elements as valves, control valves, or compressor machines. Moreover, the physical behavior of natural gas is described by differential equations. Thus, when dealing with gas transport optimization, mixed-integer problems constrained by differential equations become relevant. The scientific contribution of this thesis to solve such problems is twofold.
First, three new global algorithms are presented. In general, a typical solution approach transforms the differential equations to linear constraints. This is reasonable as mixed-integer linear programming is the most successful instance of mixed-integer programming. The new global algorithms in this thesis do not rely on this transformation and can work with less information about the underlying differential equation constraints. In an iterative process, mixed-integer linear programs and small nonlinear programs are solved alternately and the correct and finite terminations of the algorithms are proven. An extensive theoretical framework that distinguishes the assumptions on the constraints is set up. The developments allow to solve stationary gas transport optimization problems with ordinary differential equations. In this sense, promising numerical results for the Greek natural gas transport network are shown. Furthermore, the way for more general simulation-based algorithms is paved.
Second, an instantaneous control algorithm for transient gas network optimization with partial differential equations is presented. A new and specific discretization scheme that allows to use mixed-integer linear programs inside of the instantaneous control algorithm is developed for the example of gas. Again, promising numerical results that illustrate the applicability of the approach are shown. These findings pave the way for more research in the field of transient gas network optimization, which, due to its hardness, is often disregarded in the literature.

The matching problem is one of the intensely studied combinatorial optimization problems. Nevertheless, many real-world problems cannot be formulated as a pure matching problem. In this thesis we study four variants of the classical matching problem: The resource-constrained
bipartite matching problem, a stochastic matching problem, a recoverable robust matching problem and the quadratic matching problem. The stochastic matching
problem and the recoverable robust matching problem arise from an application to runway scheduling.
This thesis investigates integer as well as mixed-integer reformulations of these four types of matching problems. It consists of two parts. In the first part we study an exact solution approach for the resource-constrained bipartite matching problem and the stochastic matching problem. It reformulates the integer programming formulation (IP) of these problems into a mixed-integer program (MIP) that uses few integer variables. To this end, affine TU decompositions of the constraint matrix
play a major role. We derive several theoretical results arising from the decomposition of the constraint matrices of matching problems with resource constraints. These findings can be extended to the stochastic matching problem as the latter can be modeled as a bipartite matching problem with a special resource constraint. In a computational study we compare different MIP reformulations of resource-constrained and stochastic matching problems with their corresponding IP formulations. We show that in several settings, running times for solving instances to optimality can significantly be reduced, when the new MIP reformulations are used. Furthermore, we examine the complexity status of the recoverable robust matching problem. In a simplified version we can model it as a bipartite matching problem with a quadratic objective in the edge variables.
In the second part we study the quadratic matching problem. It asks for a matching in a graph that optimizes a quadratic objective in the edge variables. In our solution approach, we strengthen the linearized IP formulation by cutting planes that are derived from facets of the corresponding matching problem where only one quadratic term occurs in the objective function. We present different reformulation techniques to strengthen these cutting planes. Based on these methods, we design and implement an exact branch-and-cut approach. We show that root bounds and running times for solving instances to optimality can be improved significantly, when the new approach is applied.

In this thesis we study the structure algebra Z of the stable moment graph (a moment graph that exhibits a certain periodicity property—namely, its edge labels are invariant under translation by an element of the finite coroot lattice) for the case of the affine root system A_1. For a field k and the symmetric algebra S over k associated with the coroot lattice, the structure algebra Z is an S-algebra and in particular, it is an S-module. We study its S-module structure and construct an S-basis. We actually construct two bases: one comes from the perspective of linear algebra, the other comes from the perspective of moment graph theory.
By “setting c equal to zero” in the structure algebra Z, where c denotes a central element of the affine Kac-Moody algebra sl_2, we obtain the S^fin-module Z_c=0, where S^fin denotes the symmetric algebra associated with the finite coroot lattice. This module is “locally finite”, i.e. it can be described in terms of the finite root system A_1 and we show that it is determined by a set of certain divisibility relations. These relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs, and because of this we call them higher-order congruence relations.
Furthermore, the module Z_c=0 inherits the algebra structure from the structure algebra Z. We examine this algebra structure on Z_c=0 by computing the structure constants.
The obtained results apply to an arbitrary affine root system—the studied phenomena occur in all root directions. Within the scope of applications and generalizations of the subgeneric results, the same pattern as in Kato’s theorem appears in our setting of (sheaves on) moment graphs, which enables a categorification of Kato’s theorem.

This thesis deals with properties of singular orbits in multi-body-systems.
Here, systems are examined, in which the dynamics defining potential solely depends on the relative positions of the particles. Celestial mechanics, which is the motion of any number of particles due to Newton's law of universal gravitation, can be regarded as a standard example. With this in mind, the class of so-called long-ranged and moderated potentials are introduced, which among others also includes the Coulomb field of electrostatics. In these systems, the flow which is induced by the set of differential equations of motion, is generally not complete, and the non-global solutions are called singular. The goal is to show their improbability in due situations, that is to show that the set of initial conditions leading to non-global solution is a null-set in the sense of Lebesgue.
An important distinction of singular orbits are collision and non-collision orbits: the former are defined by the property that all particles have definitive limit points as time approaches the singularity. Previous results in the field of celestial mechanics show the improbability of collisions in general, and the improbability of non-collision singularities in the four-body-problem. Some generalizations to a wider class of potentials is possible with the techniques imposed there, however, certain restrictions like on the homogeneity of the force field cannot be dropped, and even within these restrictions optimal bounds cannot be shown.
Hence, the provision of a new technique seems to be necessary, in order to generalize the results to a wider class of systems, which is done within this work...

We determine some particularly interesting momentum polytopes for multiplicity free quasi-Hamiltonian manifolds by using the methods developed in [Kno16]. This leads to lots of new examples of multiplicity free quasi-Hamiltonian manifolds or equivalently, Hamiltonian loop group actions.
Precisely, we determine all convex quasi-Hamiltonian G-manifolds for G simple and simply connected where the momentum polytope is of rank one for compact Hamiltonian and quasi-Hamiltonian manifolds and we classify all Hamiltonian and q-Hamiltonian manifolds with a surjective moment map.

This thesis aims to investigate the applicability of model order reduction for
gas transport in pipeline-networks. For this purpose, we consider two different
MOR techniques, one for linear systems and the other for nonlinear systems.
A new framework is developed for the linear case, which is composed of lin-
earizing the nonlinear model, performing the spatial discretization based on stag-
gered grids, transforming the linear, time-invariant system in descriptor form into
standard state-space form based on the reduction of the differentiation-index by
applying the singular value decomposition of the system matrices and reducing
the full order model via Gramian projection-based methods, e.g. the balanced
truncation, balancing-free method, singular perturbation approximation and the
optimal Hankel-norm approximation.
The reason for this framework is the development of a stability-preserving
matrix interpolation strategy for parametric reduced order models. In comparison
to the existing matrix interpolation method, new steps include transforming the
reduced order model into modal form, reordering the parametric eigenmodes
according to the correlation and instead of the matrix pencil only interpolating
the eigvenvalues with positive weighting coefficients.
For the nonlinear case, the standard quadratic-bilinearization model order re-
duction method for systems with a single input is extended to systems with multi-
ple inputs by using the properties of the Kronecker product. The method consists
of the quadratic-bilinearization of the nonlinear full order model by employing
new variables for the nonlinear terms, approximating the resulting quadratic-
bilinear system by its homogeneous subsystems of degree k, k ∈ N, and reducing
the subsystems by using moment-matching.
Nonzero initial values may lead to worse error-behavior for the model order
reduction. A method tackling the latter issue is proposed and discussed. Ad-
ditionally, we show that the parametric quadratic-bilinear reduced order models
can be evaluated over a wide parameter range by using matrix interpolation.

We study the connection between a class of tree-valued processes, arising as the evolving
genealogies of population models, and measure-valued processes, that describe the evolution
of the different frequencies of families in the population. It will turn out that these
measure-valued processes describe the evolving genealogies uniquely and this dependence
is continuous, i.e. we will prove that convergence of these measure-valued representations
implies convergence of the corresponding tree-valued processes.
As an example we will show that a collection of measure-valued (neutral) spatial
Fleming-Viot processes can be used to describe the genealogy of the tree-valued spatial
Fleming-Viot process. This allows us to deduce a convergence result of the mean genealogical
distance of the spatial Fleming-Viot process when the size of the geographical space
goes to infinity.
Next, we define a partial order on metric measure spaces. We show that this order
is closed and that, in case of dominance, one can easily calculate the Eurandom distance
of two metric measure-spaces. As an example we will see that the genealogies of two
(neutral) Fleming-Viot processes with different resampling rates dominate each other.
Finally, we will study how to compare a neutral with a non neutral, i.e. selective,
tree-valued Fleming-Viot process.

We use the example of hot rolling to develop a comprehensive optimization approach that is based on a mathematical model of the underlying manufacturing process. More precisely, we study an optimal control problem that is designed to minimize cutting scrap while taking industrial specifications and technical limitations into account.
It is well known that the associated control-to-observation map is non-differentiable due to changes of state resulting from elasto-viscoplastic material behavior and frictional contact. However, we still want to apply gradient-based methods to solve the optimal control problem and therefore have to compute derivatives of cost functional and constraints. To resolve this issue, we first regularize all non-differentiabilities before computing sensitivity information via direct differentiation.
We moreover present solution techniques for the regularized problem and discuss numerical results for real-world examples.

The classification of spherical G-varieties X of a connected, reductive, algebraic group G over an algebraically closed field of characteristic 0 can be expressed in a purely combinatorial language. The first goal of this thesis is to define to this combinatorial data an interpretation by means of a reductive group G_X , the dual group of a spherical variety. We show that this group can be embedded into the dual group G^∨ . It appears together with a homomorphism
G_X × SL_2 → G^v ,
where the mapping of the SL_2 -factor is also determined by data coming from the variety. This result should be understood as a contribution to the work of Sakellaridis-Venkatesh. They consider applications of spherical varieties in harmonic analysis, in the context of the Langlands program. For this work the existence of a homomorphism as above is essential.
Moreover, we give the definition of a group G_X^(ass) which is connected to the combinatorics of X and to the group G_X.
A further attempt of this thesis is the investigation of pairs of spherical roots in odd characteristic. We will in particular determine all rank 2 spherical varieties whose spherical roots do not exist in characteristic 0.
The combinatorics over fields of characteristic 2 turn out to be of different nature than over other fields. There we have the important structural property that the valuation cone of a spherical variety is the negative fundamental domain of a finite reflection group. This is in general false in characteristic 2. We will present a family of counterexamples.

Anfang der 1990er Jahre führten Andersen, Jantzen und Soergel ein kombinatorisches Modell ein, um gewisse Darstellungen der folgenden zwei Situationen zu beschreiben:
a) U_p ist die quantisierte einhüllende Algebra einer komplexen Lie-Algebra an einer p-ten Einheitswurzel, wobei p > 1 eine ungerade ganze Zahl ist (und prim zu 3 wenn das zugrundeliegende Wurzelsystem vom Typ G_2 ist),
b) Lie(G_k), wobei G_k eine zusammenhängende, einfach zusammenhängende halbeinfache algebraische Gruppe über einem algebraisch abgeschlossenen Körper k mit char k > h, wobei h die Coxeterzahl des zugrundeliegenden Wurzelsystems ist.
Der Zusammenhang zwischen diesen beiden Situationen ist grundlegend, um die Modulare Lusztig-Vermutung für char p>> 0 zu zeigen. Diese Vermutung stellt eine Charakterformel für Darstellungen zusammenhängender reduktiver algebraischer Gruppen über einem algebraisch abgeschlossenen Körper k mit char k = p > h auf. Mit dem Werk von Andersen, Jantzen und Soergel kann man sie aus ihrem char 0-Analogon herleiten, der Quantum-Vermutung von Lusztig.
Dieses Modell wird durch eine kombinatorische Kategorie K_k und einer vollen Unterkategorie der sogenannten speziellen Objekte M_k realisiert. Die Definition der speziellen Objekte ist sehr technisch. Ein besseres Verständnis oder eine intrinsische Beschreibung durch kategorielle Eigenschaften von M_k könnten beispielsweise zu einer Verbesserung einer oberen Schranke für Ausnahmeprimzahlen der Modularen Lusztig-Vermutung führen.
Diese Arbeit enthält eine detaillierte Untersuchung von M k . Ein Hauptresultat ist die Beschreibung des Verhaltens der Dualität, wobei eine wichtige Klasse der speziellen Objekte aus selbstdualen Objekten besteht. Ein weiteres wichtiges Resultat ist die Beschreibung des Zentrums von M_k.

Dualitätskonzepte und sozio-ökonomische Paradigmen mit Ordnungsrelationen zur Mengenoptimierung
(2016)

In der Mengenoptimierung allgemein bedeutet ein minimales Element einer Menge alleine
nicht, dass die gesamte Menge minimal ist. Diese Problematik führt bei Verwaltungsentscheidungen
zu fehlerhaften Informationsgrundlagen. Ziel dieser Arbeit ist,
mit starken und schwachen Ordnungsrelationen, Implikationen für optimale Mengen zu
erhalten. Der Praxisbezug hierbei sind die städtebaulichen Gemengelagen bei Flughafenentwicklungen
mit den soziodemographischen Auswirkungen auf die Stadtentwicklung.
Die gängige Praxis der Deduktion für Wertegrundlagen ist widerlegt, die Darstellung anhand
von Ordnungsrelationen erzielt optimale Lösungen. Dualitätskonzepte mit gewählten
Ordnungsrelationen mit einschränkenden Voraussetzungen, natürlichen Ordnungsstrukturen,
einer speziellen Äquivalenzklasse, der Intervallanalyse und des konjugierten
Störungsansatzes belegen die Charakterisierung volkswirtschaftlicher Gemengelagen neu.

In the simplest geographic setting, an interacting particle system is a $\{0,1\}^{\Z}$ or $\N_{0}^{\Z}$-valued continuous-time stochastic process $(X_{t})_{t\ge0}$, with the interpretation that $X_{t}(\xi)$ denotes the number of particles at the geographic location $\xi\in\Z$ at time $t\ge0$. The term 'particle' is a placeholder whose meaning varies depending on the context. An example is the $\{0,1\}^{\Z}$-valued contact process, which models the spread of an infection in a linearly arranged population: the positions of the 1s in the state $X_{t}$ indicate where the particles (=’infections’) are located, i.e. which individuals are infected at time t. In the course of time, the number and the locations of the infections change according to certain stochastic rules after random times.
The goal of this thesis is to examine the overall shape of a given interacting particle system that has been started with a single particle at the origin, at a given finite time t. The starting point is a result of L. Gray (1991) and L. Gray and E. Andjel (2014): For the $\{0,1\}^{\Z}$-valued contact process with nearest neighbour infections, and for fixed t, the map
$\Z \to [0,1]$, $\xi \mapsto E [ X_{t}(\xi) ]$
is 'bell-shaped': it is symmetric around the origin and is, when restricted to $\N_{0}$, monotonically decreasing. Gray calls this map a 'population profile'.
This thesis proves similar shape results for a class of $\N_{0}^{\Z}$-valued systems that we call branching random walks with interactions. The base process is called branching random walks process, which is a simple spatial population model in which the particles can be interpreted as ‘individuals’: Particles perform independent continuous-time random walks on $\Z$, and additionally, after certain random times, a given particle either dies or branches. In the latter case, the particle is replaced by two identical copies that behave independently henceforth. From this base process, we derive multilevel branching random walks (in which particles are grouped into clans which themselves undergo certain dynamics), coupled branching random walks (in which additionally -- at certain spatial locations and certain times -- local catastrophes diminish the local population), and coalescing branching random walks (in which the death rate of a given particle is higher if many other particles are around, which can be interpreted in terms of a competition for scarce resources).
Broadly speaking, we prove that suitable generalizations of Gray's population profile are 'bell-shaped' for our class of particle systems at any given finite time, given that the respective system is initialized with one particle at the origin, and every interaction kernel is itself 'bell-shaped' in a suitable sense. As a corollary, we obtain a criterion whose verification would allow to remove the nearest-neighbour assumption that Gray needed to impose on the contact process.
Regarding the proof techniques, the focus on finite time horizons comes with the challenge to find novel combinatorial arguments. Further proof concepts are to distinguish particles via labels, which leads to the notion of tree-indexed random walks; to exploit that the 'bell-shape' is preserved by elementary operations such as convolutions, and by less elementary operations such as so-called tree-indexed convolutions; and to invoke a function-valued duality.
We briefly indicate how to transfer our results to other geographical spaces rather than $\Z$, most notably to the hierarchical group $\Omega_{N}$ and to the hypercube $H_{N}$ (N=2,3,...). Finally, also a side-project concerning local survival of particle systems on Cayley graphs, initiated by J. Swart, is discussed.

The planning of a whole power plant involves diversivied specializations in engineering and can take up to several years.
Within the planning procedure the routing of the high pressure pipes transporting the hot steam from the main boiler to the steam turbines exerts a strong influence on efficiency of the power plant.
The design of pipe routing is usually done by a specially trained and experienced engineer with the aid of CAE software.
The starting point of the problem is given by a rough outline of the design space and specified points where steam is produced and expended.
In addition to the routing, placement of hangers has to be specified to hold the pipe into place and meet technical restrictions.
Even with the help of engineering software the planning procedure often involves trial and error, which makes the design phase cumbersome - especially if unforeseen obstacles have to be incorporated in a late stage of the project.
This thesis tries to lay a first foundation of an algorithmic engineering aid to help speed up the routing of the pipe and suggest a placing for the hangers.
After discretizing the design space, the problem brings together two different main ingredients.
On the one hand a very well-known combinatorial problem of finding a shortest path or a Steiner tree in a graph has to be considered as we want to obtain a solution that resembles the shape of a pipe.
On the other hand we have to deal with the mechanics of the pipe that has to withstand its self-weight and forces due to the heated steam.
Bringing together all necessary prerequisites we start by deriving a mixed-integer nonlinear model (MINLP) that captures all combinatorial and nonlinear aspects of our problem.
To solve real-world instances we test several approaches: solving the non-convex MINLP, linearizing the problem, reformulating to a MISOCP, and decomposing the problem.
The last technique yields the fastest results and can be further accelerated by employing an infeasibility check.
The test can be further extended to specially tailored cutting planes that help to provide a better bound for the subproblem.
Moreover, we prove hardness results for some occurring problems and models of our application.
We test the proposed algorithms and formulations on real-world instances and explain some interesting insights into the solutions of the procedures.

In dieser Arbeit wird eine neue Strategie zur Lösung von Optimierungsproblemen mit lokal Lipschitz-stetiger Zielfunktion vorgestellt. Wir werden insbesondere nicht voraussetzen, dass die Zielfunktion semismooth ist.
Die Grundlage dieser Strategie bilden die sogenannten stetigen äußeren Subdifferentiale, die den mangelnden Informationsgehalt eines Subdifferentials beheben. Es wird ein Abstiegsverfahren basierend auf diesen äußeren Subdifferentialen entwickelt und dessen Konvergenz bewiesen. In diesem Zusammenhang wird untersucht, welche Eigenschaften eines Subdifferentials essentiell zur Optimierung nichtdifferenzierbarer Funktionen sind, und damit wird die Frage nach einer geeigneten Wahl eines Subdifferentials beantwortet.
Ein weiterer Abschnitt der Arbeit widmet sich der Konstruktion stetiger äußerer Subdifferentiale. Dabei nimmt die Klasse der Optimalwertfunktionen einen besonderen Stellenwert ein. Die bei der Konstruktion zum Vorschein kommenden Schwierigkeiten werden anhand akademischer Beispiele diskutiert.
Im letzten Abschnitt der Arbeit wird ein neues Verfahren zur Minimierung lokal Lipschitz-stetiger Optimalwertfunktionen vorgestellt (BTO). Die Grundlage bilden Bundle-Trust-Region-Ideen der nichtglatten, konvexen Optimierung und die Armijo-Regel zur Bestimmung einer geeigneten Schrittweite in der glatten Optimierung. Von Bundle- Trust-Region-Verfahren wird die Idee zur Bildung einer Modellfunktion adaptiert, die auf den in dieser Arbeit zuvor eingeführten approximativen stetigen äußeren Subdifferentialen basiert.
Wir verallgemeinern weiterhin die Strategie zur Anpassung des Trust-Region-Radius und präsentieren ein neues Verfahren zur sukzessiven Verbesserung der Modellfunktion. Abschließend wird die globale Konvergenz des BTO-Verfahrens bewiesen.

Let G and T be topological groups, α : T → Aut(G) a homomorphism defining a continuous action of T on G and G ♯ := G ⋊α T the corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯, ) of G ♯ whose restriction to G remains irreducible. First, we prove that, for T = , this is the case for any irreducible positive energy representation of G ♯, i.e. for which the one-parameter group Ut := π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations of G to representations of G ♯ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers for W*-dynamical systems, we also derive a characterization of the continuous positive definite functions on G that extend to G ♯.