Generalized Bounds for Convex Multistage Stochastic Programs (eBook)

Daniel Kuhn (Autor)

eBook Download: PDF

2006
XII, 190 Seiten
Springer Berlin (Verlag)
978-3-540-26901-4 (ISBN)

Infos

This work was completed during my tenure as a scientific assistant and d- 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 c- fronted with complex decision problems under uncertainty. Although usually hard to solve, I quickly learned to appreciate the benefit of stochastic progr- ming models and developed a strong interest in their theoretical properties. Motivated both by practical questions and theoretical concerns, I became p- 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 - 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.

Inhalt

Preface 6
Contents 8
Introduction 11
1.1 Motivation 11
1.2 Previous Research 12
1.3 Objective 14
1.4 Outline 15
Basic Theory of Stochastic Optimization 17
2.1 Modelling Uncertainty 17
2.2 Policies 20
2.3 Constraints 21
2.4 Static and Dynamic Version of a Stochastic Program 24
2.5 Here-and-Now Strategies 35
2.6 Wait-and-See Strategies 35
2.7 Mean-Value Strategies 36
Convex Stochastic Programs 38
3.1 Augmenting the Probability Space 38
3.2 Preliminary Definitions 42
3.3 Regularity Conditions 46
3.4 sup-Projections 49
3.5 Saddle Structure 50
3.6 Subdifferentiability 56
Barycentric Approximation Scheme 59
4.1 Scenario Generation 59
4.2 Approximation of Expectation Functionals 62
4.3 Partitioning 71
4.4 Barycentric Scenario Trees 75
4.5 Bounds on the Optimal Value 82
4.6 Bounding Sets for the Optimal Decisions 85
Extensions 90
5.1 Stochasticity of the Profit Functions 91
5.2 Stochasticity of the Constraint Functions 96
5.3 Synthesis of Results 107
5.4 Linear Stochastic Programs 109
5.5 Bounding Sets for the Optimal Decisions 118
Applications in the Power Industry 120
6.1 The Basic Decision Problem of a Hydropower Producer 122
6.2 Market Power 125
6.3 Lognormal Spot Prices 127
6.4 Lognormal Natural Inflows 128
6.5 Risk Aversion 130
6.6 Numerical Results 133
Conclusions 148
7.1 Summary of Main Results 148
7.2 Future Research 152
A Conjugate Duality 154
B Lagrangian Duality 161
C Penalty-Based Optimization 168
D Parametric Families of Linear Functions 170
E Lipschitz Continuity of sup-Projections 173
References 179
List of Figures 186
List of Tables 187
Index 188

Produkt-Details

• Erscheint lt. Verlag: 30.3.2006
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Zusatzinfo: XII, 190 p. 21 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Allgemeines / Lexika
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management ► Planung / Organisation
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Approximation Technique • Convex Multistage Stochastic Program • Nonconvexities • numerical solution • Regularization • stochastic optimization • Stochastic Processes • Stochastic Programming
• ISBN-10: 3-540-26901-0 / 3540269010
• ISBN-13: 978-3-540-26901-4 / 9783540269014