47 research outputs found

    Modèles biniveaux pour la réponse de la demande dans les réseaux électriques intelligents

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    This thesis focuses on bilevel optimization, some variants, and an application to optimal price-setting in smart power grids.Bilevel optimization problems are a special subclass of constrained mathematical optimization problems where another problem, the lower level is embedded in the constraints.We consider their application to the optimal pricing of a Time-and-Level-of-Use Demand Response program, allowing an electricity supplier to leverage the flexibility of users through an economic incentive.A generalized form of bilevel optimization is also proposed where the lower level may pick a solution that is not optimal as typically assumed but near-optimal, that is feasible and within a fixed tolerance from an optimal solution.Solving this variant of bilevel optimization requires anticipation of the deviation from optimality and a guarantee that a solution remains feasible even with this deviation.Cette thèse étudie l'optimisation bi-niveau, certaines variantes et une application à la tarification dans les réseaux électriques intelligents.Les problèmes d'optimisation bi-niveaux sont une sous-catégorie de problèmes d'optimisation mathématique contrainte où un deuxième problème ou deuxième niveau estprésent dans les contraintes.Nous étudions leur application à un tariff variable en temps et en niveau de consommation, permettant à un fournisseur d'énergie d'exploiter la flexibilité de consommateurs par des incitations économiques.Une généralisation des problèmes bi-niveaux est également proposée, dans laquelle le deuxième niveau peut sélectionner une solution qui n'est pas optimale contrairement au modèle bi-niveau classique mais quasi-optimale.Résoudre cette variante de problèmes bi-niveaux demande l'anticipation de cette déviation de la solution de deuxième niveau de l'optimalité et garantit qu'une solution au problème bi-niveau sera réalisable malgré cette déviation

    Interaction-Aware Motion Planning for Automated Vehicles

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    Die Bewegungsplanung für automatisierte Fahrzeuge (AVs) in gemischtem Verkehr ist eine herausfordernde Aufgabe. Hierbei bezeichnet gemischter Verkehr, Verkehr bestehend aus von Menschen gefahrenen Fahrzeugen sowie automatisierten Fahrzeugen. Um die Komplexität der Aufgabe zu reduzieren, verwenden state-of-the-art Planungsansätze oft die vereinfachende Annahme, dass das zukünftige Verhalten umliegender Fahrzeuge unabhängig vom Plan des AVs vorhergesagt werden kann. Während die Trennung von Prädiktion und Planung für viele Verkehrssituationen eine hilfreiche Vereinfachung darstellt, werden hierbei Interaktionen zwischen den Verkehrsteilnehmern ignoriert, was besonders in interaktiven Verkehrssituationen zu suboptimalem, übermäßig konservativem Fahrverhalten führen kann. In dieser Arbeit werden zwei interaktionsbewusste Bewegungsplanungsalgorithmen vorgeschlagen, die in der Lage sind übermäßig konservatives Fahrverhalten zu reduzieren. Der Kernaspekt dieser Algorithmen ist, dass Prädiktion und Planung gleichzeitig gelöst werden. Mit diesen Algorithmen können anspruchsvolle Fahrmanöver, wie z. B. das Reißverschlussverfahren in dichtem Verkehr, durchgeführt werden, die mit state-of-the-art Planungsansätzen nicht möglich sind. Der erste Algorithmus basiert auf Methoden der Multi-Agenten-Planung. Interaktionen zwischen Verkehrsteilnehmern werden durch Optimierung gekoppelter Trajektorien mittels einer gemeinsamen Kostenfunktion approximiert. Das Kernstück des Algorithmus ist eine neuartige Multi-Agenten-Trajektorienplanungsformulierung, die auf gemischt-ganzzahliger quadratischer Programmierung (MIQP) basiert. Die Formulierung garantiert global optimale Lösungen und ist somit in der Lage das kombinatorische Problem zu lösen, welches kontinuierliche Methoden auf lokal optimale Lösungen beschränkt. Desweiteren kann durch den vorgestellten Ansatz ein manöverneutrales Verhalten erzeugt werden, das Manöverentscheidungen in ungewissen Situationen aufschieben kann. Der zweite Ansatz formuliert Interaktionen zwischen einem menschlichen Fahrer und einem AV als ein Stackelberg-Spiel. Im Gegensatz zu bestehenden Arbeiten kann der Algorithmus allgemeine nichtlineare Zustands- und Eingabebeschränkungen berücksichtigen. Desweiteren führen wir Mechanismen zur Integration von Kooperation und Rücksichtnahme in die Planung ein. Damit wird übermäßig aggressives Fahrverhalten verhindert, was in der Literatur als ein Problem interaktionsbewusster Planungsmethoden identifiziert wurde. Die Wirksamkeit, Robustheit und Echtzeitfähigkeit des Algorithmus wird durch numerische Experimente gezeigt

    An algorithm for the global resolution of linear stochastic bilevel programs

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    The aim of this thesis is to find a technique that allows for the use of decomposition methods known from stochastic programming in the framework of linear stochastic bilevel problems. The uncertainty is modeled as a discrete, finite distribution on some probability space. Two approaches are made, one using the optimal value function of the lower level, whereas the second technique uses the Karush-Kuhn-Tucker conditions of the lower level. Using the latter approach, an integer-programming based algorithm for the global resolution of these problems is presented and evaluated

    Robust optimization, game theory, and variational inequalities

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (p. 193-109).We propose a robust optimization approach to analyzing three distinct classes of problems related to the notion of equilibrium: the nominal variational inequality (VI) problem over a polyhedron, the finite game under payoff uncertainty, and the network design problem under demand uncertainty. In the first part of the thesis, we demonstrate that the nominal VI problem is in fact a special instance of a robust constraint. Using this insight and duality-based proof techniques from robust optimization, we reformulate the VI problem over a polyhedron as a single- level (and many-times continuously differentiable) optimization problem. This reformulation applies even if the associated cost function has an asymmetric Jacobian matrix. We give sufficient conditions for the convexity of this reformulation and thereby identify a class of VIs, of which monotone affine (and possibly asymmetric) VIs are a special case, which may be solved using widely-available and commercial-grade convex optimization software. In the second part of the thesis, we propose a distribution-free model of incomplete- information games, in which the players use a robust optimization approach to contend with payoff uncertainty.(cont.) Our "robust game" model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative, distribution-free equilibrium concept, for which, in contrast to ex post equilibria, existence is guaranteed. We show that computation of "robust-optimization equilibria" is analogous to that of Nash equilibria of complete- information games. Our results cover incomplete-information games either involving or not involving private information. In the third part of the thesis, we consider uncertainty on the part of a mechanism designer. Specifically, we present a novel, robust optimization model of the network design problem (NDP) under demand uncertainty and congestion effects, and under either system- optimal or user-optimal routing. We propose a corresponding branch and bound algorithm which comprises the first constructive use of the price of anarchy concept. In addition, we characterize conditions under which the robust NDP reduces to a less computationally demanding problem, either a nominal counterpart or a single-level quadratic optimization problem. Finally, we present a novel traffic "paradox," illustrating counterintuitive behavior of changes in cost relative to changes in demand.by Michele Leslie Aghassi.Ph.D

    Optimization with mixed-integer, complementarity and bilevel constraints with applications to energy and food markets

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    In this dissertation, we discuss three classes of nonconvex optimization problems, namely, mixed-integer programming, nonlinear complementarity problems, and mixed-integer bilevel programming. For mixed-integer programming, we identify a class of cutting planes, namely the class of cutting planes derived from lattice-free cross-polytopes, which are proven to provide good approximations to the problem while being efficient to compute. We show that the closure of these cuts gives an approximation that depends only on the ambient dimension and that the cuts can be computed efficiently by explicitly providing an algorithm to compute the cut coefficients in O(n2n)O(n2^n) time, as opposed to solving a nearest lattice-vector problem, which could be much harder. For complementarity problems, we develop a first-order approximation algorithm to efficiently approximate the covariance of the decision in a stochastic complementarity problem. The method can be used to approximate the covariance for large-scale problems by solving a system of linear equations. We also provide bounds to the error incurred in this technique. We then use the technique to analyze policies related to the North American natural gas market. Further, we use this branch of nonconvex problems in the Ethiopian food market to analyze the regional effects of exogenous shocks on the market. We develop a detailed model of the food production, transportation, trade, storage, and consumption in Ethiopia, and test it against exogenous shocks. These shocks are motivated by the prediction that teff, a food grain whose export is banned now, could become a super grain. We present the regional effects of different government policies in response to this shock. For mixed-integer bilevel programming, we develop algorithms that run in polynomial time, provided a subset of the input parameters are fixed. Besides the Σ2p\Sigma^p_2-hardness of the general version of the problem, we show polynomial solvability and NPNP-completeness of certain restricted versions of this problem. Finally, we completely characterize the feasible regions represented by each of these different types of nonconvex optimization problems. We show that the representability of linear complementarity problems, continuous bilevel programs, and polyhedral reverse-convex programs are the same, and they coincide with that of mixed-integer programs if the feasible region is bounded. We also show that the feasible region of any mixed-integer bilevel program is a union of the feasible regions of finitely many mixed-integer programs up to projections and closures

    International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

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    The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

    Robust Design in Game Theory: Bayesian Optimization Approach to Minimax Problems with Equilibrium Constraints

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    Modern engineering systems have become increasingly complex due to the integration of human actors and advanced artificial intelligence, both of which can be interpreted as intelligent agents. Game theory is a mathematical framework that provides an explanatory model for systems constituted of those intelligent agents. It postulates that the apparent behavior of a system is an equilibrium resulting from each agent within the system individually optimizing their own objectives. Thus, designing an intelligent system is to identify a configuration such that its equilibrium is desirable with respect to some external criteria. However, equilibria are often not unique and form sets that lack topological properties on which optimization heavily relies on, e.g., convexity, connectedness, or even compactness in some cases. The unsureness nature, i.e., uncertainty, of equilibria also appeals for another common design criterion: robustness. In this context, a robust design should reach worst-case optimality to avoid sensitivity to the eventual outcome among all possible equilibria. In this dissertation, I incorporate both the game theoretical aspect and the robustness requirement of system design using the formulation of minimax problems with equilibrium constraints. The complexity of the problem structure and the non-uniqueness of potential equilibria require a new solution strategy different from traditional gradient based methods. I propose a Bayesian approach which infers the probabilistic belief of the optimality of a design given sampled objective function values. Due to the anisotropic natural of systems of independent agents, I then revisit the original Kushner’s Wiener process prior instead of radial basis kernel prior despite their popularity for other global optimization applications. I also derive theoretical results on sample maxima and their locations, develop an effective method to decompose the search space into independent regions, and design necessary adaptations to take into account the equilibrium constraints and minimax objective. Finally, I discuss a few applications of the proposed design framework

    DECENTRALIZED ALGORITHMS FOR NASH EQUILIBRIUM PROBLEMS – APPLICATIONS TO MULTI-AGENT NETWORK INTERDICTION GAMES AND BEYOND

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    Nash equilibrium problems (NEPs) have gained popularity in recent years in the engineering community due to their ready applicability to a wide variety of practical problems ranging from communication network design to power market analysis. There are strong links between the tools used to analyze NEPs and the classical techniques of nonlinear and combinatorial optimization. However, there remain significant challenges in both the theoretical and algorithmic analysis of NEPs. This dissertation studies certain special classes of NEPs, with the overall purpose of analyzing theoretical properties such as existence and uniqueness, while at the same time proposing decentralized algorithms that provably converge to solutions. The subclasses are motivated by relevant application examples
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