State elimination for mixed-integer optimal control of partial differential equations by semigroup theory

Mixed-integer optimal control problems governed by partial differential equations (MIPDECOs) are powerful modeling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by partial differential equations that have the structure of an abstract ordinary differential equation in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints

We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decom- position approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands

Approximationseigenschaften von Sum-Up Rounding

Optimization problems that involve discrete variables are exposed to the conflict between being a powerful modeling tool and often being hard to solve. Infinite-dimensional processes, as e.g. described by differential equations, underlying the optimization may lead to the need to solve for distributed discrete control variables. This work analyzes approximation arguments that replace the need for solving the optimization problem by the need for first solving a relaxation and second computing appropriate roundings to regain discrete controls. We provide sufficient conditions on rounding algorithms and their grid refinement strategies that allow to prove approximation of the relaxed controls by the discrete controls in weaker topologies, a feature due to the infinite-dimensional vantage point. If the control-to-state mapping of the underlying process exhibits suitable compactness properties, state vector approximation follows in the norm topology as well as, under additional assumptions, optimality principles of the computed discrete controls. The conditions are verified for representatives of the family of Sum-Up Rounding algorithms. We apply the arguments on different classes of mixed-integer optimization problems that are constrained by partial differential equations. Specifically, we consider discrete control inputs, which are distributed in the time domain, for evolution equations that are governed by a differential operator that generates a strongly continuous semigroup, discrete control inputs, which are distributed in multi-dimensional spatial domains, for elliptic boundary value problems and discrete control inputs, which are distributed in space-time cylinders, for evolution equations that are governed by differential operators such that the corresponding Cauchy problem satisfies maximal parabolic regularity. Furthermore, we apply the arguments outside the scope of partial differential equations to a signal reconstruction problem. Computational results illustrate the findings.Optimierungsprobleme mit diskreten Variablen befinden sich im Spannungsfeld zwischen hoher Modellierungsmächtigkeit und oft schwerer Lösbarkeit. Zur Optimierung unendlichdimensionaler Prozesse, z.B. beschrieben mit Hilfe von Differentialgleichungen, kann die Lösung nach verteilten diskreten Kontrollvariablen erforderlich sein. Diese Arbeit untersucht Approximationsargumente, mit deren Hilfe die Notwendigkeit einer Lösung des Optimierungsproblems durch die Notwendigkeit zuerst eine Relaxierung zu lösen und anschließend eine passende Rundung zu berechnen, um wieder diskrete Kontrollvariablen zu erhalten, ersetzt wird. Wir geben hinreichende Bedingungen an Rundungsalgorithmen und ihre Gitterverfeinerungsstrategien an, um eine Approximation der relaxierten Kontrollvariablen mit den diskreten Kontrollvariablen in schwächeren Topologien zu erhalten, was aus der unendlichdimensionalen Betrachtung des Problems folgt. Falls der Steuerungs-Zustands-Operator des zugrundeliegenden Prozesses passende Kompaktheitseigenschaften aufweist, folgen die Approximation der Zustandsvektoren in der Normtopologie und, unter zusätzlichen Bedingungen, Optimalitätsprinzipien für die berechneten diskreten Kontrollvariablen. Die Bedingungen werden für Repräsentanten der Familie von Sum-Up Rounding Algorithmen nachgewiesen. Wir wenden die Argumente auf verschiedene Klassen von gemischt-ganzzahligen Optimierungsproblemen, die von partiellen Differentialgleichungen beschränkt werden, an. Insbesondere betrachten wir diskrete, in der Zeit verteilte, Steuerungen in Evolutionsgleichungen mit Differentialoperatoren, die stark stetige Halbgruppen erzeugen; diskrete, mehrdimensional im Ort verteilte, Steuerungen in elliptischen Randwertproblemen und diskrete, in Ort und Zeit verteilte, Steuerungen in Evolutionsgleichungen mit Differentialoperatoren, deren zugehörige Cauchyprobleme maximale parabolische Regularität aufweisen. Des Weiteren wenden wir die Argumente außerhalb des Kontexts partieller Differentialgleichungen auf ein Signalrekonstruktionsproblem an. Numerische Beispiele illustrieren die gezeigten Resultate

Metaheuristics for Traffic Control and Optimization: Current Challenges and Prospects

Intelligent traffic control at signalized intersections in urban areas is vital for mitigating congestion and ensuring sustainable traffic operations. Poor traffic management at road intersections may lead to numerous issues such as increased fuel consumption, high emissions, low travel speeds, excessive delays, and vehicular stops. The methods employed for traffic signal control play a crucial role in evaluating the quality of traffic operations. Existing literature is abundant, with studies focusing on applying regression and probability-based methods for traffic light control. However, these methods have several shortcomings and can not be relied on for heterogeneous traffic conditions in complex urban networks. With rapid advances in communication and information technologies in recent years, various metaheuristics-based techniques have emerged on the horizon of signal control optimization for real-time intelligent traffic management. This study critically reviews the latest advancements in swarm intelligence and evolutionary techniques applied to traffic control and optimization in urban networks. The surveyed literature is classified according to the nature of the metaheuristic used, considered optimization objectives, and signal control parameters. The pros and cons of each method are also highlighted. The study provides current challenges, prospects, and outlook for future research based on gaps identified through a comprehensive literature review

Optimal speed trajectory and energy management control for connected and automated vehicles

Connected and automated vehicles (CAVs) emerge as a promising solution to improve urban mobility, safety, energy efficiency, and passenger comfort with the development of communication technologies, such as vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I). This thesis proposes several control approaches for CAVs with electric powertrains, including hybrid electric vehicles (HEVs) and battery electric vehicles (BEVs), with the main objective to improve energy efficiency by optimising vehicle speed trajectory and energy management system. By types of vehicle control, these methods can be categorised into three main scenarios, optimal energy management for a single CAV (single-vehicle), energy-optimal strategy for the vehicle following scenario (two-vehicle), and optimal autonomous intersection management for CAVs (multiple-vehicle). The first part of this thesis is devoted to the optimal energy management for a single automated series HEV with consideration of engine start-stop system (SSS) under battery charge sustaining operation. A heuristic hysteresis power threshold strategy (HPTS) is proposed to optimise the fuel economy of an HEV with SSS and extra penalty fuel for engine restarts. By a systematic tuning process, the overall control performance of HPTS can be fully optimised for different vehicle parameters and driving cycles. In the second part, two energy-optimal control strategies via a model predictive control (MPC) framework are proposed for the vehicle following problem. To forecast the behaviour of the preceding vehicle, a neural network predictor is utilised and incorporated into a nonlinear MPC method, of which the fuel and computational efficiencies are verified to be effective through comparisons of numerical examples between a practical adaptive cruise control strategy and an impractical optimal control method. A robust MPC (RMPC) via linear matrix inequality (LMI) is also utilised to deal with the uncertainties existing in V2V communication and modelling errors. By conservative relaxation and approximation, the RMPC problem is formulated as a convex semi-definite program, and the simulation results prove the robustness of the RMPC and the rapid computational efficiency resorting to the convex optimisation. The final part focuses on the centralised and decentralised control frameworks at signal-free intersections, where the energy consumption and the crossing time of a group of CAVs are minimised. Their crossing order and velocity trajectories are optimised by convex second-order cone programs in a hierarchical scheme subject to safety constraints. It is shown that the centralised strategy with consideration of turning manoeuvres is effective and outperforms a benchmark solution invoking the widely used first-in-first-out policy. On the other hand, the decentralised method is proposed to further improve computational efficiency and enhance the system robustness via a tube-based RMPC. The numerical examples of both frameworks highlight the importance of examining the trade-off between energy consumption and travel time, as small compromises in travel time could produce significant energy savings.Open Acces

A Survey on Aerial Swarm Robotics

The use of aerial swarms to solve real-world problems has been increasing steadily, accompanied by falling prices and improving performance of communication, sensing, and processing hardware. The commoditization of hardware has reduced unit costs, thereby lowering the barriers to entry to the field of aerial swarm robotics. A key enabling technology for swarms is the family of algorithms that allow the individual members of the swarm to communicate and allocate tasks amongst themselves, plan their trajectories, and coordinate their flight in such a way that the overall objectives of the swarm are achieved efficiently. These algorithms, often organized in a hierarchical fashion, endow the swarm with autonomy at every level, and the role of a human operator can be reduced, in principle, to interactions at a higher level without direct intervention. This technology depends on the clever and innovative application of theoretical tools from control and estimation. This paper reviews the state of the art of these theoretical tools, specifically focusing on how they have been developed for, and applied to, aerial swarms. Aerial swarms differ from swarms of ground-based vehicles in two respects: they operate in a three-dimensional space and the dynamics of individual vehicles adds an extra layer of complexity. We review dynamic modeling and conditions for stability and controllability that are essential in order to achieve cooperative flight and distributed sensing. The main sections of this paper focus on major results covering trajectory generation, task allocation, adversarial control, distributed sensing, monitoring, and mapping. Wherever possible, we indicate how the physics and subsystem technologies of aerial robots are brought to bear on these individual areas

Application of Metaheuristics in Signal Optimisation of Transportation Networks: A Comprehensive Survey

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.With rapid population growth, there is an urgent need for intelligent traffic control techniques in urban transportation networks to improve the network performance. In an urban transportation network, traffic signals have a significant effect on reducing congestion, improving safety, and improving environmental pollution. In recent years, researchers have been applied metaheuristic techniques for signal timing optimisation as one of the practical solution to enhance the performance of the transportation networks. Current study presents a comprehensive survey of such techniques and tools used in signal optimisation of transportation networks, providing a categorisation of approaches, discussion, and suggestions for future research

Relaxations and Approximations for Mixed-Integer Optimal Control

This thesis treats different aspects of the class of Mixed-Integer Optimal Control Problems (MIOCPs). These are optimization problems that combine the difficulties of underlying dynamic processes with combinatorial decisions. Typically, these combinatorial decisions are realized as switching decisions between the system’s different operations modes. During the last decades, direct methods emerged as the state-of-the-art solvers for MIOCPs. The formulation of a valid, tight and dependable integral relaxation, i.e., the formulation of a model for fractional values, plays an important role for these direct solution methods. We give detailed insight into several relaxation approaches for MIOCPs and compare them with regard to their respective structures. In particular, these are the typical solution’s structures and properties as convexity, problem size and numerical behavior. From these structural properties, we deduce some required specifications of a solver. Additionally, the modeling and subsequent limitation of the switching process directly tackle the class-specific typical issue of chattering solutions. One of the relaxation methods for MIOCPs is the outer convexification, where the binary variables only enter affinely. For the approximation of this relaxation’s solution, we took up on the control approximation problem in integral sense derived by Sager as part of a decomposition approach for MIOCPs with affine binary controls. This problem describes the optimal approximation of fractional controls with binary controls such that the corresponding dynamic process is changed as little as possible. For the multi-dimensional problem, we developed a new heuristic, which for the first time gives a bound that only depends on the control grid and not anymore on the number of the system’s controls. For the generalization of the control approximation problem with additional constraints, we derived a tailored branch-and-bound algorithm, which is based on the properties of the Lagrangian relaxation of the one-dimensional problem. This algorithm beats state-of-the-art commercial solvers for Mixed-Integer Linear Programs (MILPs) for this special approximation problem by several orders of magnitude. Overall, we present several, partially new modeling approaches for MIOCPs together with the accompanying structural properties. On this basis, we develop new theories for the approximation of certain relaxed solutions. We discuss the efficient implementation of the resulting structure exploiting algorithms. This leads to a deeper and better understanding of MIOCPs. We show the practicability of the theoretical observations with the help of four prototypical problems. The presented methods and algorithms allow on their basis the direct development of decision support and analysis tools in practice