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Generalized Bounds for Convex Multistage Stochastic Programs PDF

192 Pages·2005·2.35 MB·English
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Lecture Notes in Economics and Mathematical Systems 548 Founding Editors: M. Beckmann H. P. Ktinzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversitat Hagen Feithstr. 140/AVZ H, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut fiir Mathematische Wirtschaftsforschung (IMW) Universitat Bielefeld Universitatsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kiirsten, U. Schittko Daniel Kuhn Generalized Bounds for Convex Multistage Stochastic Programs <£J Springer Author Daniel Kuhn Universitat St. Gallen Institut fur Unternehmensforschung (HSG) BodanstraBe 6 9000 St. Gallen Switzerland Library of Congress Control Number: 2004109705 ISSN 0075-8442 ISBN 3-540-22540-4 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany 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: Camera ready by author Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper 42/3130Di 5 4 3 2 10 To Barbara Preface This work was completed during my tenure as a scientific assistant and doc- toral student at the Institute for Operations Research at the University of St. Gallen. During that time, I was involved in several industry projects in the field of power management, on the occasion of which I was repeatedly con- fronted with complex decision problems under uncertainty. Although usually hard to solve, I quickly learned to appreciate the benefit of stochastic program- ming models and developed a strong interest in their theoretical properties. Motivated both by practical questions and theoretical concerns, I became par- ticularly interested in the art of finding tight bounds on the optimal value of a given model. The present work attempts to make a contribution to this important branch of stochastic optimization theory. In particular, it aims at extending some classical bounding methods to broader problem classes of practical relevance. This book was accepted as a doctoral thesis by the University of St. Gallen in June 2004.1 am particularly indebted to Prof. Dr. Karl Frauendorfer for su- pervising my work. I am grateful for his kind support in many respects and the generous freedom I received to pursue my own ideas in research. My gratitude also goes to Prof. Dr. Georg Pflug, who agreed to co-chair the dissertation committee. With pleasure I express my appreciation for his encouragement and continuing interest in my work. Of course, this book would not have achieved its present form without the support of my fellow colleagues. I enjoyed the numerous stimulating dis- cussions with Karsten Linowsky about topics of scientific or secular interest. Moreover, I benefited a lot from the experience and expertise of Jens Giissow and Georg Ostermaier in thef ieldo f power management and applied stochas- tic optimization. A special thanks goes to Olivier Schmid and Patrick Wirth who shared with me their insights into the subtleties of the barycentric ap- proximation scheme. I am also truly thankful to Manfred Grollmann for com- puter support and transportation services. Furthermore, I am indebted to VIII Preface Michael Schiirle who had always an open ear for my questions and cease- lessly regaled me with anecdotes from the past. A particular debt of thanks is owed to Gido Haarbriicker and Dominik Boos for continuously appealing to my mathematical conscience and to Ulrich Jacobi for competent assistance with computer problems. I also acknowledge the collegiality and manifold help of Massimo Cutaia and Jerome Roller and the fruitful cooperation with Daniel Hofstetter during the doctoral seminars. St. Gallen, June 2004 Daniel Kuhn Contents 1 Introduction 1 1.1 Motivation 1 1.2 Previous Research 2 1.3 Objective 4 1.4 Outline 5 2 Basic Theory of Stochastic Optimization 7 2.1 Modeling Uncertainty 7 2.2 Policies 10 2.3 Constraints 1 2.4 Static and Dynamic Stochastic Programs 14 2.5 Here-and-Now Strategies 25 2.6 Wait-and-Se Strategies 25 2.7 Mean-Value Strategies 26 3 Convex Stochastic Programs 29 3.1 Augmenting the Probability Space 29 3.2 Preliminary Definitions 3 3.2.1 Slater's Constraint Qualification 3 3.2.2 Convex Functions 35 3.2.3 Block-Diagonal Autoregresive Proceses 36 3.3 Regularity Conditions 37 3.4 sup-Projections 40 3.5 Sadle Structure 41 3.6 Subdiferentiability 47 4 Barycentric Approximation Scheme 51 4.1 Scenario Generation 51 4.2 Approximation of Expectation Functionals 54 4.2.1 Jensen's Inequality 5 4.2.2 Edmundson-Madansky Inequality 57 X Contents 4.2.3 Lower Barycentric Approximation 59 4.2.4 Upper Barycentric Approximation 62 4.3 Partitioning 63 4.4 Barycentric Scenario Tres 67 4.5 Bounds on the Optimal Value 74 4.6 Bounding Sets for the Optimal Decisions 7 5 Extensions 83 5.1 Stochasticity of the Profit Functions 84 5.2 Stochasticity of the Constraint Functions 89 5.3 Synthesis of Results 10 5.4 Linear Stochastic Programs 102 5.4.1 D.c. Functions 105 5.4.2 Generalized Bounds for Linear Stochastic Programs. . . . 106 5.5 Bounding Sets for the Optimal Decisions I l l 6 Applications in the Power Industry 113 6.1 The Basic Decision Problem of a Hydropower Producer 115 6.2 Market Power 18 6.3 Lognormal Spot Prices 120 6.4 Lognormal Natural Inflows 121 6.5 Risk Aversion 123 6.6 Numerical Results 126 6.6.1 Model Parameterization 126 6.6.2 Discretization of the Probability Space 128 6.6.3 Results of the Reference Problem 129 6.6.4 Hydro Scheduling Problem with Market Power 130 6.6.5 Lognormal Prices 131 6.6.6 Hydro Scheduling Problem with Lognormal Inflows . . . . 133 6.6.7 Hydro Scheduling Problem with Risk-Aversion 137 7 Conclusions 141 7.1 Summary of Main Results 141 7.2 Future Research 145 A Conjugate Duality 147 B Lagrangian Duality 15 C Penalty-Based Optimization 163 D Parametric Families of Linear Functions 165 E Lipschitz Continuity of sup-Projections 169 References 175 Contents XI List of Figures 183 List of Tables 185 Index 187 Introduction 1.1 Motivation Almost any technical or economic decision problem includes some degree of uncertainty about the values to assign to some problem-specific parameters. The best decision strategies with respect to some objective criterion must be found on the basis of the a priori information about these uncertainties. If it is possible to assign a probability distribution to the random parameters, the determination of an optimal decision strategy gives rise to a stochastic optimization model, also referred to as a stochastic program. However, the solution of stochastic programs poses severe difficulties, especially in the mul- tistage case. If the underlying probability space is continuous, the stochas- tic program represents an optimization problem over an infinite-dimensional function space. Then, analytical solutions are hardly available, and nontrivial problems of practical relevance must always be solved numerically. However, numerical solution requires discretization of the continuous probability space. One should select a discrete probability measure with finite support and solve the stochastic program with respect to this discrete auxiliary measure, instead of the continuous original measure. In doing so, one effectively approximates the original stochastic program by an optimization problem over a finite- dimensional Euclidean space, which is numerically tractable. The selection of an appropriate discrete probability measure is referred to as scenario generation and represents a primary challenge in the field of stochastic programming. It is indispensable that the solution of the approx- imate problem can be related in some way to the solution of the original problem, i.e. the exact solution of the auxiliary problem should provide an approximate solution of the original stochastic program. Ideally, one can find a discrete probability measure such that the optimal value and the optimizer set of the associated auxiliary stochastic program are, in a quantitative sense, close to the optimal value and the optimizer set of the original optimization

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