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2010 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Dynamic Optimization and Differential Games

verfasst von: Terry L. Friesz

Verlag: Springer US

Buchreihe : International Series in Operations Research & Management Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Dynamic Optimization and Differential Games has been written to address the increasing number of Operations Research and Management Science problems that involve the explicit consideration of time and of gaming among multiple agents. With end-of-chapter exercises throughout, it is a book that can be used both as a reference and as a textbook. It will be useful as a guide to engineers, operations researchers, applied mathematicians and social scientists whose work involves both the theoretical and computational aspects of dynamic optimization and differential games. Included throughout the text are detailed explanations of several original dynamic and game-theoretic mathematical models which are of particular relevance in today’s technologically-driven-global economy: revenue management, oligopoly pricing, production planning, supply chain management, dynamic traffic assignment and dynamic congestion pricing.

The book emphasizes deterministic theory, computational tools and applications associated with the study of dynamic optimization and competition in continuous time. It develops the key results of deterministic, continuous time, optimal control theory from both the classical calculus of variations perspective and the more modern approach of infinite dimensional mathematical programming. These results are then generalized for the analysis of differential variational inequalities arising in dynamic game theory for open loop environments. Algorithms covered include steepest descent in Hilbert space, gradient projection in Hilbert space, fixed point methods, and gap function methods.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
In this book we present the theory of continuous-time dynamic optimization, covering the classical calculus of variations, the modern theory of optimal control, and their linkage to infinite-dimensional mathematical programming. We present an overview of the main classes of practical algorithms for solving dynamic optimization problems and develop some facility with the art of formulating dynamic optimization models. Upon completing our study of dynamic optimization, we turn to dynamic Nash games. Our coverage of dynamic games emphasizes continuous-time variational inequalities and subsumes portions of the classical theory of differential games.
Terry L. Friesz
Chapter 2. Nonlinear Programming and Discrete-Time Optimal Control
Abstract
The primary intent of this chapter is to introduce the reader to the theoretical foundations of nonlinear programming as well as the theoretical foundations of deterministic discrete-time optimal control. In fact, deterministic discrete-time optimal control problems, as we shall see, are actually nonlinear mathematical programs with a very particular type of structure. In a later chapter, we will also discover that deterministic continuous-time optimal control problems are specific instances of mathematical programs in topological vector spaces. Consequently, it is imperative for the student of optimal control to have a command of the foundations of nonlinear programming. Particularly important are the notions of local and global optimality in mathematical programming, the Kuhn-Tucker necessary conditions for optimality in nonlinear programming, and the role played by convexity in making necessary conditions sufficient. Readers already comfortable with finite-dimensional nonlinear programming may wish to go immediately to Section 2.9. We do caution, however, that subsequent chapters of this book assume substantial familiarity with finite-dimensional nonlinear programming, so that an overestimate of one’s nonlinear programming knowledge can be very detrimental to ultimately obtaining a deep understanding of optimal control theory and differential games.
Terry L. Friesz
Chapter 3. Foundations of the Calculus of Variations and Optimal Control
Abstract
In this chapter, we treat time as a continuum and derive optimality conditions for the extremization of certain functionals.We consider both variational calculus problems that are not expressed as optimal control problems and optimal control problems themselves. In this chapter, we relie on the classical notion of the variation of a functional. This classical perspective is the fastest way to obtain useful results that allow simple example problems to be solved that bolster one’s understanding of continuous-time dynamic optimization.
Terry L. Friesz
Chapter 4. Infinite Dimensional Mathematical Programming
Abstract
In this chapter we are concerned with the generalization of finite-dimensional mathematical programming to infinite-dimensional vector spaces. This topic is pertinent to dynamic optimization because dynamic optimization in continuous time de facto occurs in infinite-dimensional spaces since the variable x (t), even if x is a scalar, has an infinity of values for continuous \(t \in \left[t_0, t_f\right] \subseteq \mathfrak{R}^{1}_{+}\) where t f < t 0.
Terry L. Friesz
Chapter 5. Finite Dimensional Variational Inequalities and Nash Equilibria
Abstract
In this chapter, we lay the foundation for turning our focus from dynamic optimization, which has been the subject of preceding chapters, to the notion of a dynamic game. To fully appreciate the material presented in subsequent chapters, we must in the present chapter review some of the essential features of the theory of finite-dimensional variational inequalities and static noncooperative mathematical games. Today many economists and engineers are exposed to the notion of a game-theoretic equilibrium that we study in this chapter, namely Nash equilibrium. Yet, the relationship of such equilibria to certain nonextremal problems known as fixed-point problems, variational inequalities and nonlinear complementarity problems is not widely understood. It is the fact that, as we shall see, Nash and Nash-like equilibria are related to and frequently equivalent to nonextremal problems that makes the computation and qualitative investigation of such equilibria so tractable. Although the static games discussed in this chapter are really steady states of dynamic games, we are, for the most part, indifferent in this chapter to any underlying dynamics. We also comment that readers familiar with finite-dimensional variational inequalities and static Nash games may wish to skip this chapter.
Terry L. Friesz
Chapter 6. Differential Variational Inequalities and Differential Nash Games
Abstract
In this chapter we focus on extending the notion of a noncooperative Nash equilibrium to a dynamic, continuous-time setting. The dominant mathematical perspective we will employ is that of a differential variational inequality. In fact we shall see that many of the results obtained in the previous chapter for finite-dimensional variational inequalities and static games carry over with some slight modifications to the dynamic, continuous-time setting we now address.
Terry L. Friesz
Chapter 7. Optimal Economic Growth
Abstract
The theory of optimal economic growth is a branch of economic theory that makes direct and sophisticated use of the theory of optimal control. As such, the models of optimal economic growth that have been devised and reported in the economics literature are relatively easy for a person who has mastered the material of Chapters 3 and 4 of this book to comprehend. Among other things, this chapter shows how aspatial optimal economic growth theory may be extended to study optimal growth of interdependent regions in a national economy. Moreover, working through the analyses presented in this chapter provides a means of assessing and improving one’s mastery of the key mathematical concepts from the theory of optimal control that were introduced in previous chapters, especially the analysis and interpretation of optimality conditions and singular controls.
Terry L. Friesz
Chapter 8. Production Planning, Oligopoly and Supply Chains
Abstract
In this chapter we develop models that describe how prices, production rates and distribution activities evolve over time and influence one another for three output market structures: 1. perfect competition 2. monopoly, and 3. oligopoly. In particular, we apply the material from previous chapters to the modeling and computation of production, distribution, and supply chain decisions made by firms operating within the three competitive environments mentioned above. Throughout this chapter our perspective is deterministic, and the dynamic games considered are open loop in nature with perfect initial information.We begin with aspatial models and move to models with explicit network path flows. We shall deal exclusively with finite terminal times and see that policies near the terminal time are of great importance to the lifetime profitability of firms. One of our goals will be to study how policies on inventory remaining at the terminal time as will as the value of such residual inventories when liquidated can influence operations throughout a firm’s history.
Terry L. Friesz
Chapter 9. Dynamic User Equilibrium
Abstract
Dynamic traffic assignment (DTA) is the positive (descriptive) modeling of time-varying flows of automobiles on road networks consistent with established traffic flow theory and travel demand theory. Dynamic user equilibrium (DUE) is one type of DTA wherein the effective unit travel delay, including early and late arrival penalties, of travel for the same purpose is identical for all utilized path and departure time pairs. In the context of planning, DUE is usually modelled for the within-day time scale based on demands established on a day-to-day time scale.
Terry L. Friesz
Chapter 10. Dynamic Pricing and Revenue Management
Abstract
An active and rapidly growing applied operations research discipline is the field known as revenue management (RM). The principal intent of revenue management is to extract all unused willingness to pay from consumers of differentiated services and products. Talluri and van Ryzin (2004) provide a comprehensive introduction to most aspects of the theory and practice of revenue management. For this chapter, our goal is to illustrate and solve some differentialNash games that occur in network revenue management and that provide critical information about pricing, resource allocation, and demand management to retailers and service providers.
Terry L. Friesz
Backmatter
Metadaten
Titel
Dynamic Optimization and Differential Games
verfasst von
Terry L. Friesz
Copyright-Jahr
2010
Verlag
Springer US
Electronic ISBN
978-0-387-72778-3
Print ISBN
978-0-387-72777-6
DOI
https://doi.org/10.1007/978-0-387-72778-3

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