Generalized Bounds for Convex Multistage Stochastic Programs

192 Pages·2005·2.35 MB·English

by  Daniel Kuhn (auth.), M. Beckmann, H. P. Künzi, Prof. Dr. G. Fandel, Prof. Dr. W. Trockel, A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten, U. Schittko (eds.)

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
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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