A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jocobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising

Rate-Independent Gradient Crystal Plasticity Theory -- Robust Algorithmic Formulations based on Incremental Energy Minimization

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Validation of nominations in gas networks and properties of technical capacities

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Optimierung autonom schaltender dynamischer Hybridsysteme

Simulation and optimization based on physical models are indispensable tools in process engineering, since they allow for cost-efficient improvement of design and operation. However, a broad variety of optimization problems are complicated due to the hybrid nature of the dynamic process model, e.g., in chemical processes, power plants and transport vehicles. In these kinds of systems, the continuous dynamics is strongly coupled with discrete phenomena such as transitions between operation modes. This in general leads to nonsmooth or even discontinuous state trajectories which makes the optimization of such systems an extremely challenging task. In this thesis, hybrid dynamic systems with autonomous mode transitions are considered, i.e., systems where the mode transition occurs autonomously when the system state fulfills a certain switching condition.In hybrid dynamic optimization problems, the switching between operation modes needs a specific treatment which eliminates the points of discontinuity or reformulates the model description into a continuous one. In this thesis, two reformulation methods are presented. The key step in both reformulation methods is the introduction of a continuous switching variable which, on the one hand, makes the problem smoother and, on the other hand, retains the hybrid feature in some way. This, generally, leads to a tradeoff of accuracy vs. stability and robustness of the optimization algorithm. The smoothing method proposed in this thesis approximates the instantaneous switch by a fast but smooth transition function. Another approach considered in the present work is smoothing by a penalty approach. In this method, the hybrid dynamic optimization problem is formulated as a bilevel problem. The inner minimization problem will then be replaced by ist optimality conditions. The resulting complementarity constraints enter the objective function of the outer problem as a penalty term. As a third method in this thesis a novel approach is proposed which employs the idea of state event detection and location and extends it to optimization. Once a state event is detected, the system is restarted in a new operation mode. Since the optimization is stopped and restarted at transition points, only problems involving continuous models with continuous state trajectories and gradients need to be solved. Therefore, the direct treatment of hybrid features is avoided and thus the complexity in the solution procedure can be significantly reduced.The functionality and effectivness of all three approaches are illustrated by relatively simple examples. Furthermore, the application to more complex problems with relevance for, e.g. process industries is presented.Die vorliegende Arbeit beschäftigt sich mit der Entwicklung und dem Test von Methoden zur Optimierung autonom schaltender dynamischer Hybridsysteme. Unter “Autonom schaltenden dynamischen Hybridsystemen” werden hier dynamische Systeme verstanden, bei denen gemäß definierten Schaltbedingungen zwischen verschiedenen diskreten Betriebsmodi umgeschaltet wird. Autonom schaltende dynamische Hybridsysteme sind von großer praktischer Relevanz, weil sie bei zahlreichen industriellen Prozessen (z. B. bei Anfahrprozessen oder Verdampfungsprozessen) auftreten. Die Optimierung solcher Prozesse zielt auf hohe Effizienz bei gleichzeitig steigenden Anforderungen an Umweltverträglichkeit und Sicherheit.Bei der Optimierung autonom schaltender dynamischer Hybridsysteme besteht die wissenschaftliche Herausforderung darin, die gemischt diskret-kontinuierlichen Optimierungsprobleme so zu formulieren, dass (i) die Lösungsmethode möglichst allgemein anwendbar ist und (ii) eine Lösung von hinreichender Genauigkeit mit?(iii) vertretbarem Rechenaufwand gefunden werden kann. Die besondere Schwierigkeit ist auf die Nichtglattheit oder sogar Unstetigkeit von Zustandstrajektorien oder deren Gradienten an den Umschaltpunkten zurückzuführen. Die in dieser Arbeit untersuchten Methoden zielen deshalb darauf ab, diese Unstetigkeiten aus den Optimierungsproblemen zu eliminieren. Dies geschieht in der vorgeschlagenen Glättungsmethode mittels der Approximation der das Umschalten beschreibenden Stufenfunktion durch eine Glättungsfunktion. Alternativ wird zur Glättung des Optimierungsproblems ein Strafverfahren verwendet. Hierbei wird zunächst ein Zweiebenenproblem definiert. Das innere Minimierungsproblem wird anschließend durch seine Optimalitätsbedingungen ersetzt und die darin auftretenden komplementären Beschränkungen gehen als Strafterme in die Zielfunktion des äußeren Optimierungsproblems ein.Der dritte methodische Ansatz verwendet die im Rahmen der Simulation dynamischer Hybridsysteme entwickelte Idee zum Anhalten und Neustarten der Integration an den Umschaltpunkten des Systems. Die im Rahmen einer Simulation dem autonomen Umschalten dienenden Schritte werden für Optimierungsprobleme analog realisiert. Dabei unterscheidet sich die konkrete Umsetzung vor allem bezüglich der Bestimmung der Schaltzeiten und des Festhaltens des aktuellen Betriebsmodus.Die Funktionsweise aller untersuchten Methoden wird durch kleine Fallbeispiele im Rahmen der Methodenentwicklung illustriert. Im Anschluss erfolgt jeweils die Anwendung auf komplexere Problemstellungen

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development

On the analysis of stochastic optimization and variational inequality problems

Uncertainty has a tremendous impact on decision making. The more connected we get, it seems, the more sources of uncertainty we unfold. For example, uncertainty in the parameters of price and cost functions in power, transportation, communication and financial systems have stemmed from the way these networked systems operate and also how they interact with one another. Uncertainty influences the design, regulation and decisions of participants in several engineered systems like the financial markets, electricity markets, commodity markets, wired and wireless networks, all of which are ubiquitous. This poses many interesting questions in areas of understanding uncertainty (modeling) and dealing with uncertainty (decision making). This dissertation focuses on answering a set of fundamental questions that pertain to dealing with uncertainty arising in three major problem classes: [(1)] Convex Nash games; [(2)] Variational inequality problems and complementarity problems; [(3)] Hierarchical risk management problems in financial networks. Accordingly, this dissertation considers the analysis of a broad class of stochastic optimization and variational inequality problems complicated by uncertainty and nonsmoothness of objective functions. Nash games and variational inequalities have assumed practical relevance in industry and business settings because they are natural models for many real-world applications. Nash games arise naturally in modeling a range of equilibrium problems in power markets, communication networks, market-based allocation of resources etc. where as variational inequality problems allow for modeling frictional contact problems, traffic equilibrium problems etc. Incorporating uncertainty into convex Nash games leads us to stochastic Nash games. Despite the relevance of stochastic generalizations of Nash games and variational inequalities, answering fundamental questions regarding existence of equilibria in stochastic regimes has proved to be a challenge. Amongst other reasons, the main challenge arises from the nonlinearity arising from the presence of the expectation operator. Despite the rich literature in deterministic settings, direct application of deterministic results to stochastic regimes is not straightforward. The first part of this dissertation explores such fundamental questions in stochastic Nash games and variational inequality problems. Instead of directly using the deterministic results, by leveraging Lebesgue convergence theorems we are able to develop a tractable framework for analyzing problems in stochastic regimes over a continuous probability space. The benefit of this approach is that the framework does not rely on evaluation of the expectation operator to provide existence guarantees, thus making it amenable to tractable use. We extend the above framework to incorporate nonsmoothness of payoff functions as well as allow for stochastic constraints in models, all of which are important in practical settings. The second part of this dissertation extends the above framework to generalizations of variational inequality problems and complementarity problems. In particular, we develop a set of almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued and multi-valued mappings. We extend these statements to quasi-variational regimes as well as to stochastic complementarity problems. The applicability of these results is demonstrated in analysis of risk-averse stochastic Nash games used in Nash-Cournot production distribution models in power markets by recasting the problem as a stochastic quasi-variational inequality problem and in Nash-Cournot games with piecewise smooth price functions by modeling this problem as a stochastic complementarity problem. The third part of this dissertation pertains to hierarchical problems in financial risk management. In the financial industry, risk has been traditionally managed by the imposition of value-at-risk or VaR constraints on portfolio risk exposure. Motivated by recent events in the financial industry, we examine the role that risk-seeking traders play in the accumulation of large and possibly infinite risk. We proceed to show that when traders employ a conditional value-at-risk (CVaR) metric, much can be said by studying the interaction between value at risk (VaR) (a non-coherent risk measure) and conditional value at risk CVaR (a coherent risk measure based on VaR). Resolving this question requires characterizing the optimal value of the associated stochastic, and possibly nonconvex, optimization problem, often a challenging problem. Our study makes two sets of contributions. First, under general asset distributions on a compact support, traders accumulate finite risk with magnitude of the order of the upper bound of this support. Second, when the supports are unbounded, under relatively mild assumptions, such traders can take on an unbounded amount of risk despite abiding by this VaR threshold. In short, VaR thresholds may be inadequate in guarding against financial ruin

Fuel Optimal Control Algorithms for Connected and Automated Plug-In Hybrid Vehicles

Improving the fuel economy of light-duty vehicles (LDV) is a compelling solution to stabilizing Greenhouse Gas (GHG) emissions and decreasing the reliance on fossil fuels. Over the years, there has been a considerable shift in the market of LDVs toward powertrain electrification, and plug-in hybrid electric vehicles (PHEVs) are the most cost-effective in avoiding GHG emissions. Meanwhile, connected and automated vehicle (CAV) technologies permit energy-efficient driving with access to accurate trip information that integrates traffic and charging infrastructure. This thesis aims at developing optimization-based algorithms for controlling powertrain and vehicle longitudinal dynamics to fully exploit the potential for reducing fuel consumption of individual PHEVs by utilizing CAV technologies. A predictive equivalent minimization strategy (P-ECMS) is proposed for a human-driven PHEV to adjust the co-state based on the difference between the future battery state-of-charge (SOC) obtained from short-horizon prediction and a future reference SOC from SOC node planning. The SOC node planning, which generates battery SOC reference waypoints, is performed using a simplified speed profile constructed from segmented traffic information, typically available from mobile mapping applications. The PHEV powertrain, consisting of engine and electric motors, is mathematically modeled as a hybrid system as the state is defined by the values of the continuous variable, SOC, and discrete modes, hybrid vehicle (HV), and electric vehicle (EV) modes with the engine on/off. As a hybrid system, the optimal control of PHEVs necessitates a numerical approach to solving a mixed-integer optimization problem. It is of interest to have a unified numerical algorithm for solving such mixed-integer optimal control problems with many states and control inputs. Based on a discrete maximum principle (DMP), a discrete mixed-integer shooting (DMIS) algorithm is proposed. The DMIS is demonstrated in successfully addressing the cranking fuel optimization in the energy management of a PHEV. It also serves as the foundation of the co-optimization problem considered in the remaining part of the thesis. This thesis further investigates different control designs with an increased vehicle automation level combining vehicle dynamics and powertrain of PHEVs in within-a-lane traffic flow. This thesis starts with a sequential (or decentralized) optimization and then advances to direct fuel minimization by simultaneously optimizing the two subsystems in a centralized manner. When shifting toward online implementation, the unique challenge lies in the conflict between the long control horizon required for global optimality and the computational power limit. A receding horizon strategy is proposed to resolve the conflict between the horizon length and the computation complexity, with co-states approximating the future cost. In particular, the co-state is updated using a nominal trajectory and the temporal-difference (TD) error based on the co-state dynamics. The remaining work aims to develop a unified model predictive control (MPC) framework from the powertrain (PT) control of a human-driven to the combined vehicle dynamics (VD) and PT control of an automated PHEV. In the unified framework, the cost-to-go (the fuel consumption as the economic cost) is represented by the co-state associated with the battery SOC dynamics. In its application to automated PHEVs, a control barrier function (CBF) is augmented as an add-on block to modify the vehicle-level control input for guaranteed safety. This unified MPC framework allows for systematically evaluating the fuel economy and drivability performance of different levels and structures of optimization strategies.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169876/1/dichencd_1.pd