TY - THES A1 - Nowak, Daniel T1 - Nonconvex Nash Games - Solution Concepts and Algorithms N2 - Game theory is a mathematical approach to model competition between several parties, called players. The goal of each player is to choose a strategy, which solves his optimization problem, i.e. minimizes or maximizes his objective function. Due to the competitive setting, this strategy may influence the optimization problems of other players. In the non-cooperative setting each player acts selfish, meaning he does not care about the objective of his opponents. A solution concept for this problem is a Nash equilibrium, which was introduced by John Forbes Nash in his Ph.D. thesis in 1950. Convexity of the optimization problems is a crucial assumption for the existence of Nash equilibria. This work investigates settings, where this convexity assumption fails to hold. The first part of this thesis extends results of Jong-Shi Pang and Gesualdo Scutari from their paper ``Nonconvex Games with Side Constraints'' published in 2011. In this publication, a game with possibly nonconvex objective functions and nonconvex individual and shared inequality constraints was investigated. We extend these results twofold. Firstly, we generalize the individual and shared polyhedral constraints to general convex constraints and, secondly, we introduce convex and nonconvex, individual and shared equality constraints. After a detailed comparison of solution concepts for the generalized Nash game and a related Nash game, we show that so-called quasi-Nash equilibria exist under similar assumptions than in the original work, provided some additional constraint qualification holds. Subsequently, we prove that the existence of Nash equilibria needs additional assumptions on the gradients of the equality constraints. Furthermore, a special case of a multi-leader multi-follower game is investigated. We show the convergence of epsilon-quasi-Nash equilibria to C-stationary points and prove that these are also Clarke-stationary under reasonable assumptions. In the second part of this thesis, an application in computation offloading is investigated. We consider several mobile users that are able to offload parts of a computation task to a connected server. However, the server has limited computation capacities which leads to competition among the mobile users. If a user decides to offload a part of his computation, he needs to wait for the server to finish before he can assemble the results of his computation. This leads to a vanishing constraint in the optimization problem of the mobile users which is a nonconvex and nonsmooth condition. We show the existence of a unique Nash equilibrium for the computation offloading game and provide an efficient algorithm for its computation. Furthermore, we present two extensions to this game, which inherit similar properties and we also show the limitations of these formulations. The third part investigates a hierarchical constrained Cournot game. In the upper level, several firms decide on capacities which act as constraints for the production variables. In the lower level the same firms engage in a Cournot competition, where they choose production variables to maximize profit. The prior chosen capacities are upper bounds on these production variables. This hierarchical setting induces nonconvexity and nonsmoothness in the upper level objective functions. After a detailed sensitivity analysis of the lower level, we give necessary optimality conditions for the upper level, i.e. for the hierarchical Cournot game. Using these conditions, we construct an algorithm which provably finds all Nash equilibria of the game, provided some assumptions are satisfied. This algorithm is numerically tested on several examples which are motivated by the gas market. KW - Game Theory KW - Nash Games KW - Optimization Y1 - 2021 U6 - https://doi.org/10.26083/tuprints-00017637 PB - E-Publishing-Service der TU Darmstadt CY - Darmstadt ER - TY - INPR A1 - Giesselmann, Jan A1 - Egger, Herbert T1 - Stability and asymptotic analysis for instationary gas transport via relative energy estimates N2 - We consider the transport of gas in long pipes and pipeline networks for which the dynamics are dominated by friction at the pipe walls. The governing equations can be formulated as an abstract dissipative Hamiltonian system which allows us to derive perturbation bounds by means of relative energy estimates. As particular consequences, we obtain stability with respect to initial conditions and model parameters and quantitative estimates in the high friction limit. Our results are established in detail for the flow in a single pipe and through the energy-based modelling they naturally generalize also to pipe networks. KW - gas transport on networks KW - asymptotic limits KW - hyperbolic balance laws KW - relative energy estimates KW - singular perturbations Y1 - 2020 ER - TY - INPR A1 - Disser, Yann A1 - Klimm, Max A1 - Weckbecker, David T1 - Fractionally Subadditive Maximization under an Incremental Knapsack Constraint N2 - We consider the problem of maximizing a fractionally subadditive function under a knapsack constraint that grows over time. An incremental solution to this problem is given by an order in which to include the elements of the ground set, and the competitive ratio of an incremental solution is defined by the worst ratio over all capacities relative to an optimum solution of the corresponding capacity. We present an algorithm that finds an incremental solution of competitive ratio at most $\max\{3.293\sqrt{M},2M\}$, under the assumption that the values of singleton sets are in the range $[1,M]$, and we give a lower bound of $\max\{2.449,M\}$ on the attainable competitive ratio. In addition, we establish that our framework captures potential-based flows between two vertices, and we give a tight bound of 2 for the incremental maximization of classical flows with unit capacities. Y1 - 2021 ER - TY - JOUR A1 - Pfetsch, Marc E. A1 - Schmitt, Andreas T1 - A Generic Optimization Framework for Resilient Systems N2 - This paper addresses the optimal design of resilient systems, in which components can fail. The system can react to failures and its behavior is described by general mixed integer nonlinear programs, which allows for applications to many (technical) systems. This then leads to a three-level optimization problem. The upper level designs the system minimizing a cost function, the middle level represents worst-case failures of components, i.e., interdicts the system, and the lowest level operates the remaining system. We describe new inequalities that characterize the set of resilient solutions and allow to reformulate the problem. The reformulation can then be solved using a nested branch-and-cut approach. We discuss several improvements, for instance, by taking symmetry into account and strengthening cuts. We demonstrate the effectiveness of our implementation on the optimal design of water networks, robust trusses, and gas networks, in comparison to an approach in which the failure scenarios are directly included into the model. Y1 - 2021 ER - TY - INPR A1 - Domschke, Pia A1 - Hiller, Benjamin A1 - Lang, Jens A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview N2 - With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. The idea of a model catalog came to us in the context of the application for the CRC/Transregio 154 ``Mathematical modeling, simulation and optimization using the example of gas networks''. The present English translation is an extension from [P. Domschke, B. Hiller, J. Lang, and C. Tischendorf. Modellierung von Gasnetzwerken: Eine Übersicht. Preprint, TRR 154, 2017]. At this point we would like to thank the DFG for its support. Y1 - 2021 ER - TY - INPR A1 - Grimm, Veronika A1 - Nowak, Daniel A1 - Schewe, Lars A1 - Schmidt, Martin A1 - Schwartz, Alexandra A1 - Zöttl, Gregor T1 - A Tractable Multi-Leader Multi-Follower Peak-Load-Pricing Model with Strategic Interaction N2 - While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems. KW - Game theory KW - Nash-Cournot equilibria KW - Multi-leader multi-follower game KW - Peak-load pricing Y1 - 2020 U6 - https://doi.org/10.1007/s10107-021-01708-0 ER - TY - JOUR A1 - Egger, Herbert A1 - Philippi, Nora T1 - A hybrid discontinuous Galerkin method for transport equations on networks JF - Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples N2 - We discuss the mathematical modeling and numerical discretization of 5 transport problems on one-dimensional networks. Suitable coupling conditions are derived that guarantee conservation of mass across network junctions and dissipation of a mathematical energy which allows us to prove existence of unique solutions. We then consider the space discretization by a hybrid discontinuous Galerkin method which provides a suitable upwind mechanism to handle the transport prob10 lem and allows to incorporate the coupling conditions in a natural manner. In addition, the method inherits mass conservation and stability of the continuous problem. Order optimal convergence rates are established and illustrated by numerical tests. Y1 - 2020 ER - TY - JOUR A1 - Egger, Herbert A1 - Philippi, Nora T1 - On the transport limit of singularly perturbed convection-diffusion problems on networks N2 - We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order O(sqrt(eps)) in the L1(L2)- norm with diffusion eps -> 0. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers. Y1 - 2020 ER - TY - JOUR A1 - Schuster, Michael A1 - Strauch, Elisa A1 - Gugat, Martin A1 - Lang, Jens T1 - Probabilistic Constrained Optimization on Flow Networks N2 - Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability. KW - Probabilistic Constraints KW - Flow Networks KW - Gas Networks KW - Spheric Radial Decomposition KW - Kernel Density Estimator Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1007/s11081-021-09619-x VL - Optimization and Engineering ER - TY - JOUR A1 - Gugat, Martin A1 - Giesselmann, Jan T1 - Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks N2 - The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the $H^1$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon observability inequalities for the $L^2$ and the $H^1$-norm. Y1 - 2020 U6 - https://doi.org/10.1051/cocv/2021061 CY - ESAIM:COCV ER -