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Decomposition Techniques in Mathematical Programming.pdf

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    Decomposition Techniques in Mathematical Programming
    Antonio J. Conejo Enrique Castillo Roberto Mínguez Raquel García-Bertrand Decomposition Techniques in Mathematical Programming Engineering and Science Applications ABC
    Professor Antonio J. Conejo Universidad de Castilla – La Mancha E.T.S. Ingenieros Industriales Avda. Camilo José Cela s/n 13071 Ciudad Real Spain E-mail: antonio.conejo@uclm.es Professor Enrique Castillo Escuela de Ingenieros de Caminos Universidad de Cantabria Avda. de los Castros s/n 39005 Santander Spain E-mail: castie@unican.es Dr. Roberto Mínguez Universidad de Castilla – La Mancha E.T.S. Ingenieros de Caminos Avda. Camilo José Cela s/n 13071 Ciudad Real Spain E-mail: roberto.minguez@uclm.es Dr. Raquel García-Bertrand Universidad de Castilla – La Mancha E.T.S. Ingenieros Industriales Avda. Camilo José Cela s/n 13071 Ciudad Real Spain E-mail: raquel.garcia@uclm.es Library of Congress Control Number: 2005934995 ISBN-10 3-540-27685-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-27685-2 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11511946 89/TechBooks 5 4 3 2 1 0
    To N´uria, Mireia Zhen and Olaia Xiao To my family To my parents, my sister, my grandma, and my aunt To Jos´e Agust´ın, to my brothers Javier and Jorge Luis, and especially to my parents
    Preface Optimization plainly dominates the design, planning, operation, and con- trol of engineering systems. This is a book on optimization that considers particular cases of optimization problems, those with a decomposable struc- ture that can be advantageously exploited. Those decomposable optimization problems are ubiquitous in engineering and science applications. The book considers problems with both complicating constraints and complicating vari- ables, and analyzes linear and nonlinear problems, with and without inte- ger variables. The decomposition techniques analyzed include Dantzig-Wolfe, Benders, Lagrangian relaxation, Augmented Lagrangian decomposition, and others. Heuristic techniques are also considered. Additionally, a comprehensive sensitivity analysis for characterizing the solution of optimization problems is carried out. This material is particularly novel and of high practical interest. This book is built based on many clarifying, illustrative, and computa- tional examples, which facilitate the learning procedure. For the sake of clar- ity, theoretical concepts and computational algorithms are assembled based on these examples. The results are simplicity, clarity, and easy-learning. We feel that this book is needed by the engineering community that has to tackle complex optimization problems, particularly by practitioners and researchers in Engineering, Operations Research, and Applied Economics. The descriptions of most decomposition techniques are available only in complex and specialized mathematical journals, difficult to understand by engineers. A book describing a wide range of decomposition techniques, emphasizing problem-solving, and appropriately blending theory and application, was not previously available. The book is organized in five parts. Part I, which includes Chapter 1, pro- vides motivating examples and illustrates how optimization problems with de- composable structure are ubiquitous. Part II describes decomposition theory, algorithms, and procedures. Particularly, Chapter 2 and 3 address solution procedures for linear programming problems with complicating constraints and complicating variables, respectively. Chapter 4 reviews and summarizes
    VIII Preface duality theory. Chapter 5 describes decomposition techniques appropriate for continuous nonlinear programming problems. Chapter 6 presents decompo- sition procedures relevant for mixed-integer linear and nonlinear problems. Chapter 7 considers specific decomposition techniques not analyzed in the previous chapters. Part III, which includes Chapter 8, provides a comprehen- sive treatment of sensitivity analysis. Part IV provides in Chapter 9 some case studies of clear interest for the engineering profession. Part V contains some of the codes in GAMS used throughout the book. Finally, Part VI contains the solutions of the even exercises proposed throughout the book. Relevant features of this book are 1. It provides an appropriate blend of theoretical background and practical applications in engineering and science. 2. Many examples, clarifying, illustrative, and computational, are provided. 3. Applications encompass electrical, mechanical, energy, and civil engineer- ing as well as applied mathematics and applied economics. 4. The theoretical background of the book is deep enough to be of interest 5. Practical applications are developed up to working algorithms that can to applied mathematicians. be readily used. 6. The book includes end-of-chapter exercises and the solutions of the even numbered exercises are given in a Part VI. This makes the book very practical as a textbook for graduate and postgraduate courses. 7. The book addresses decomposition in linear programming, mixed-integer linear programming, nonlinear programming, and mixed-integer nonlinear programming. It provides rigorous decomposition algorithms as well as heuristic ones. Required background to fully understand this book is moderate and in- cludes elementary algebra and calculus, and basic knowledge of linear and nonlinear programming. Over the last two decades, the two senior authors of this book have been involved in research projects that required the solution of large-scale complex optimization problems. We have received advice and relevant observations from many colleagues. We would like to express our appreciation to Prof. Gerald B. Shebl´e from Iowa State University, Prof. Mohammad Shahideh- pour from Illinois Institute of Technology, Prof. Francisco D. Galiana from McGill University, Prof. V´ıctor H. Quintana from University of Waterloo, Prof. Francisco J. Prieto from Universidad Carlos III of Madrid, and Prof. Benjamin F. Hobbs from Johns Hopkins University. We are also thankful to quite a few colleagues and former students for suggestions and insightful observations that have improved our book. Partic- ularly, we would like to thank Prof. Steven A. Gabriel from the University of Maryland, and Prof. Bruce F. Wollenberg from the University of Minnesota.
    We are deeply grateful to the University of Castilla-La Mancha, Spain, for providing us with an outstanding research environment. Preface IX Ciudad Real and Santander, Spain June 2005 A.J. Conejo E. Castillo R. M´ınguez R. Garc´ıa-Bertrand
    Contents Part I Motivation and Introduction 1.1 1.2 1.3 3 1 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Linear Programming: Complicating Constraints . . . . . . . . . . . . Transnational Soda Company . . . . . . . . . . . . . . . . . . . . . 1.3.1 8 Stochastic Hydro Scheduling . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 River Basin Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.3 1.3.4 Energy Production Model . . . . . . . . . . . . . . . . . . . . . . . . 23 Linear Programming: Complicating Variables. . . . . . . . . . . . . . . 28 Two-Year Coal and Gas Procurement . . . . . . . . . . . . . . 28 1.4.1 Capacity Expansion Planning . . . . . . . . . . . . . . . . . . . . . 32 1.4.2 1.4.3 The Water Supply System . . . . . . . . . . . . . . . . . . . . . . . . 36 Nonlinear Programming: Complicating Constraints . . . . . . . . . . 39 1.5.1 Production Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.5.2 Operation of a Multiarea Electricity Network. . . . . . . . 42 The Wall Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.5.3 1.5.4 Reliability-based Optimization of a Rubblemound Breakwater . . . . . . . . . . . . . . . . . . . . 48 Nonlinear Programming: Complicating Variables . . . . . . . . . . . . 53 1.6.1 Capacity Expansion Planning: Revisited . . . . . . . . . . . . 53 1.7 Mixed-Integer Programming: Complicating Constraints . . . . . . 55 Unit Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.8 Mixed-Integer Programming: Complicating Variables . . . . . . . . 57 Capacity Expansion Planning: Revisited 2 . . . . . . . . . . 57 1.8.1 The Water Supply System: Revisited . . . . . . . . . . . . . . . 60 1.8.2 1.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.7.1 1.4 1.5 1.6
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