Kuhn Generalized Bounds for Convex Multistage Stochastic Programs

1. Auflage 2006
ISBN: 978-3-540-26901-4
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 190 Seiten

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.

Autoren/Hrsg.

Kuhn, Daniel

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Introduction;11
3.1;1.1 Motivation;11
3.2;1.2 Previous Research;12
3.3;1.3 Objective;14
3.4;1.4 Outline;15
4;Basic Theory of Stochastic Optimization;17
4.1;2.1 Modelling Uncertainty;17
4.2;2.2 Policies;20
4.3;2.3 Constraints;21
4.4;2.4 Static and Dynamic Version of a Stochastic Program;24
4.5;2.5 Here-and-Now Strategies;35
4.6;2.6 Wait-and-See Strategies;35
4.7;2.7 Mean-Value Strategies;36
5;Convex Stochastic Programs;38
5.1;3.1 Augmenting the Probability Space;38
5.2;3.2 Preliminary Definitions;42
5.3;3.3 Regularity Conditions;46
5.4;3.4 sup-Projections;49
5.5;3.5 Saddle Structure;50
5.6;3.6 Subdifferentiability;56
6;Barycentric Approximation Scheme;59
6.1;4.1 Scenario Generation;59
6.2;4.2 Approximation of Expectation Functionals;62
6.3;4.3 Partitioning;71
6.4;4.4 Barycentric Scenario Trees;75
6.5;4.5 Bounds on the Optimal Value;82
6.6;4.6 Bounding Sets for the Optimal Decisions;85
7;Extensions;90
7.1;5.1 Stochasticity of the Profit Functions;91
7.2;5.2 Stochasticity of the Constraint Functions;96
7.3;5.3 Synthesis of Results;107
7.4;5.4 Linear Stochastic Programs;109
7.5;5.5 Bounding Sets for the Optimal Decisions;118
8;Applications in the Power Industry;120
8.1;6.1 The Basic Decision Problem of a Hydropower Producer;122
8.2;6.2 Market Power;125
8.3;6.3 Lognormal Spot Prices;127
8.4;6.4 Lognormal Natural Inflows;128
8.5;6.5 Risk Aversion;130
8.6;6.6 Numerical Results;133
9;Conclusions;148
9.1;7.1 Summary of Main Results;148
9.2;7.2 Future Research;152
10;A Conjugate Duality;154
11;B Lagrangian Duality;161
12;C Penalty-Based Optimization;168
13;D Parametric Families of Linear Functions;170
14;E Lipschitz Continuity of sup-Projections;173
15;References;179
16;List of Figures;186
17;List of Tables;187
18;Index;188