Necessary Conditions In Dynamic Optimization

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Principles Of Dynamic Optimization

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Author : Piernicola Bettiol
language : en
Publisher: Springer Nature
Release Date : 2024-06-18

Principles Of Dynamic Optimization written by Piernicola Bettiol and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-06-18 with Mathematics categories.

This monograph explores key principles in the modern theory of dynamic optimization, incorporating important advances in the field to provide a comprehensive, mathematically rigorous reference. Emphasis is placed on nonsmooth analytic techniques, and an in-depth treatment of necessary conditions, minimizer regularity, and global optimality conditions related to the Hamilton-Jacobi equation is given. New, streamlined proofs of fundamental theorems are incorporated throughout the text that eliminate earlier, cumbersome reductions and constructions. The first chapter offers an extended overview of dynamic optimization and its history that details the shortcomings of the elementary theory and demonstrates how a deeper analysis aims to overcome them. Aspects of dynamic programming well-matched to analytical techniques are considered in the final chapter, including characterization of extended-value functions associated with problems having endpoint and state constraints, inverse verification theorems, sensitivity relationships, and links to the maximum principle. This text will be a valuable resource for those seeking an understanding of dynamic optimization. The lucid exposition, insights into the field, and comprehensive coverage will benefit postgraduates, researchers, and professionals in system science, control engineering, optimization, and applied mathematics.

Dynamic Optimization And Differential Games

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Author : Terry L. Friesz
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-08-20

Dynamic Optimization And Differential Games written by Terry L. Friesz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-08-20 with Business & Economics categories.

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.

Dynamic Optimization Of Path Constrained Switched Systems

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Author : Jun Fu
language : en
Publisher: Springer Nature
Release Date : 2023-03-11

Dynamic Optimization Of Path Constrained Switched Systems written by Jun Fu and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-11 with Technology & Engineering categories.

This book provides a series of systematic theoretical results and numerical solution algorithms for dynamic optimization problems of switched systems within infinite-dimensional inequality path constraints. Dynamic optimization of path-constrained switched systems is a challenging task due to the complexity from seeking the best combinatorial optimization among the system input, switch times and switching sequences. Meanwhile, to ensure safety and guarantee product quality, path constraints are required to be rigorously satisfied (i.e., at an infinite number of time points) within a finite number of iterations. Several novel methodologies are presented by using dynamic optimization and semi-infinite programming techniques. The core advantages of our new approaches lie in two folds: i) The system input, switch times and the switching sequence can be optimized simultaneously. ii) The proposed algorithms terminate within finite iterations while coming with a certification of feasibility for the path constraints. In this book, first, we provide brief surveys on dynamic optimization of path-constrained systems and switched systems. For switched systems with a fixed switching sequence, we propose a bi-level algorithm, in which the input is optimized at the inner level, and the switch times are updated at the outer level by using the gradient information of the optimal value function calculated at the optimal input. We then propose an efficient single-level algorithm by optimizing the input and switch times simultaneously, which greatly reduces the number of nonlinear programs and the computational burden. For switched systems with free switching sequences, we propose a solution framework for dynamic optimization of path-constrained switched systems by employing the variant 2 of generalized Benders decomposition technique. In this framework, we adopt two different system formulations in the primal and master problem construction and explicitly characterize the switching sequences by introducing a binary variable. Finally, we propose a multi-objective dynamic optimization algorithm for locating approximated local Pareto solutions and quantitatively analyze the approximation optimality of the obtained solutions. This book provides a unified framework of dynamic optimization of path-constrained switched systems. It can therefore serve as a useful book for researchers and graduate students who are interested in knowing the state of the art of dynamic optimization of switched systems, as well as recent advances in path-constrained optimization problems. It is a useful source of up-to-date optimization methods and algorithms for researchers who study switched systems and graduate students of control theory and control engineering. In addition, it is also a useful source for engineers who work in the control and optimization fields such as robotics, chemical engineering and industrial processes.

Lq Dynamic Optimization And Differential Games

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Author : Jacob Engwerda
language : en
Publisher: John Wiley & Sons
Release Date : 2005-06-17

Lq Dynamic Optimization And Differential Games written by Jacob Engwerda and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-17 with Business & Economics categories.

Game theory is the theory of social situations, and the majority of research into the topic focuses on how groups of people interact by developing formulas and algorithms to identify optimal strategies and to predict the outcome of interactions. Only fifty years old, it has already revolutionized economics and finance, and is spreading rapidly to a wide variety of fields. LQ Dynamic Optimization and Differential Games is an assessment of the state of the art in its field and the first modern book on linear-quadratic game theory, one of the most commonly used tools for modelling and analysing strategic decision making problems in economics and management. Linear quadratic dynamic models have a long tradition in economics, operations research and control engineering; and the author begins by describing the one-decision maker LQ dynamic optimization problem before introducing LQ differential games. Covers cooperative and non-cooperative scenarios, and treats the standard information structures (open-loop and feedback). Includes real-life economic examples to illustrate theoretical concepts and results. Presents problem formulations and sound mathematical problem analysis. Includes exercises and solutions, enabling use for self-study or as a course text. Supported by a website featuring solutions to exercises, further examples and computer code for numerical examples. LQ Dynamic Optimization and Differential Games offers a comprehensive introduction to the theory and practice of this extensively used class of economic models, and will appeal to applied mathematicians and econometricians as well as researchers and senior undergraduate/graduate students in economics, mathematics, engineering and management science.

Dynamic Optimization Second Edition

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Author : Morton I. Kamien
language : en
Publisher: Courier Corporation
Release Date : 2013-04-17

Dynamic Optimization Second Edition written by Morton I. Kamien and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

Since its initial publication, this text has defined courses in dynamic optimization taught to economics and management science students. The two-part treatment covers the calculus of variations and optimal control. 1998 edition.

Elements Of Dynamic Optimization

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Author : Alpha C. Chiang
language : en
Publisher:
Release Date : 2000

Elements Of Dynamic Optimization written by Alpha C. Chiang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Economics, Mathematical categories.

INTRODUCTION 1.

Necessary Conditions In Dynamic Optimization

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Author : Francis Clarke
language : en
Publisher: American Mathematical Soc.
Release Date : 2005

Necessary Conditions In Dynamic Optimization written by Francis Clarke and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.

This monograph derives necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion. These conditions constitute a new state of the art, subsuming, unifying, and substantially extending the results in the literature. The Euler, Weierstrass and transversality conditions are expressed in their sharpest known forms. No assumptions of boundedness or convexity are made, no constraint qualifications imposed, and only weak pseudo-Lipschitz behavior is postulated on the underlying multifunction. The conditions also incorporate a 'stratified' feature of a novel nature, in which both the hypotheses and the conclusion are formulated relative to a given radius function.When specialized to the calculus of variations, the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard. They also apply to parametrized control systems, giving rise to new and stronger maximum principles of Pontryagin type. The final chapter is devoted to a different issue, that of the Hamiltonian necessary condition. It is obtained here, for the first time, in the case of nonconvex values and in the absence of any constraint qualification; this has been a longstanding open question in the subject. Apart from the final chapter, the treatment is self-contained, and calls upon only standard results in functional and nonsmooth analysis.

Dynamic Programming And The Calculus Of Variations

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Author : Dreyfus
language : en
Publisher: Academic Press
Release Date : 1965-01-01

Dynamic Programming And The Calculus Of Variations written by Dreyfus and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1965-01-01 with Computers categories.

Dynamic Programming and the Calculus of Variations

Foundations Of Dynamic Economic Analysis

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Author : Michael R. Caputo
language : en
Publisher: Cambridge University Press
Release Date : 2005-01-10

Foundations Of Dynamic Economic Analysis written by Michael R. Caputo and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-01-10 with Business & Economics categories.

Foundations of Dynamic Economic Analysis presents a modern and thorough exposition of the fundamental mathematical formalism used to study optimal control theory, i.e., continuous time dynamic economic processes, and to interpret dynamic economic behavior. The style of presentation, with its continual emphasis on the economic interpretation of mathematics and models, distinguishes it from several other excellent texts on the subject. This approach is aided dramatically by introducing the dynamic envelope theorem and the method of comparative dynamics early in the exposition. Accordingly, motivated and economically revealing proofs of the transversality conditions come about by use of the dynamic envelope theorem. Furthermore, such sequencing of the material naturally leads to the development of the primal-dual method of comparative dynamics and dynamic duality theory, two modern approaches used to tease out the empirical content of optimal control models. The stylistic approach ultimately draws attention to the empirical richness of optimal control theory, a feature missing in virtually all other textbooks of this type.

Dynamic Programming

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Author : John O.S. Kennedy
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Dynamic Programming written by John O.S. Kennedy and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Social Science categories.

Humans interact with and are part of the mysterious processes of nature. Inevitably they have to discover how to manage the environment for their long-term survival and benefit. To do this successfully means learning something about the dynamics of natural processes, and then using the knowledge to work with the forces of nature for some desired outcome. These are intriguing and challenging tasks. This book describes a technique which has much to offer in attempting to achieve the latter task. A knowledge of dynamic programming is useful for anyone interested in the optimal management of agricultural and natural resources for two reasons. First, resource management problems are often problems of dynamic optimization. The dynamic programming approach offers insights into the economics of dynamic optimization which can be explained much more simply than can other approaches. Conditions for the optimal management of a resource can be derived using the logic of dynamic programming, taking as a starting point the usual economic definition of the value of a resource which is optimally managed through time. This is set out in Chapter I for a general resource problem with the minimum of mathematics. The results are related to the discrete maximum principle of control theory. In subsequent chapters dynamic programming arguments are used to derive optimality conditions for particular resources.