Stability, Approximation, and Decomposition in Two- and Multistage Stochastic Programming

178 Pages·2009·1.77 MB·English

by  Christian Kuchler

Christian Küchler
Stability, Approximation, and Decomposition
in Two- and Multistage Stochastic Programming
VIEWEG+TEUBNER RESEARCH
Stochastic Programming
Editor:
Prof. Dr. Rüdiger Schultz
Uncertainty is a prevailing issue in a growing number of optimization prob-
lems in science, engineering, and economics. Stochastic programming
offers a flexible methodology for mathematical optimization problems
involving uncertain parameters for which probabilistic information is avail -
able. This covers model formulation, model analysis, numerical solution
m ethods, and practical implementations. The series ”Stochastic Program-
ming“ presents original research from this range of topics.
Christian Küchler
Stability, Approximation,
and Decomposition
in Two- and Multistage
Stochastic Programming
VIEWEG+TEUBNER RESEARCH
Bibliografische Information der Deutschen Nationalbibliothek
Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der
Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über
<http://dnb.d-nb.de> abrufbar.
Dissertation Humboldt-Universität zu Berlin, 2009
1. Auflage 2009
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To my parents
Preface
Thepresentdoctoralthesishasbeendevelopedduringmyemploymentatthe
Department of Mathematics at Humboldt-Universität zu Berlin. This em-
ploymentwasenabledandsupportedbytheBundesministerium für Bildung
und Forschung and the Wiener Wissenschafts-, Forschungs- und Technolo-
giefonds.
First of all, it is a great pleasure to express my gratitude to my advisor
Prof.Werner Römisch who enabled for me a most pleasant and inspiring
time at Humboldt-Universität. Prof.Römisch supported me with numerous
discussions and suggestions, incessantly encouraged me to follow my own
ideas, and made me attend various workshops and conferences that gave me
many opportunities to present and discuss my work.
Very special thanks also go to my colleague Stefan Vigerske from Hum-
boldt-Universität for the intensive collaboration whose results are also part
of this thesis. In particular, it was his profound knowledge in optimization
and programming which had allowed to implement the recombining tree de-
compositionapproachinthepresentform. Iwouldalsoliketothankhimfor
never tiring to discuss the many facets of computer manipulation as well as
mathematics.
Special thanks also go to Prof.René Henrion from Weierstraß-Institut in
Berlin for the very pleasant and fruitful collaboration on scenario reduction,
resulting in parts of this thesis. I also want to thank him for numerous
motivating and inspiring discussions.
I would also like to thank my colleaguesfrom the BMBF network for the
successful cooperation, especially Alexa Epe from Ruhr-Universität Bochum
and Oliver Woll from Universität Duisburg-Essen. Furthermore, I want to
thank all of my friends and colleagues from Humboldt-Universität for their
support, many helpful discussions, and the enjoyable common coffee breaks.
In particular, I thank Dr.Andreas Eichhorn, Thomas Surowiec, and, once
again, Stefan Vigerske for proofreading parts of this thesis. I also want to
thank Dr.Holger Heitsch for providing me his scenred software.
Finally, I want to express my deepest gratitude to my family for all their
loveandsupport,and,inparticular,toKonnyforhavingbeensostrongand
patient during the last years.
Christian Küchler
Contents
1 Introduction 1
1.1 Stochastic Programming Models . . . . . . . . . . . . . . . . . 1
1.2 Approximations, Stability, and Decomposition . . . . . . . . . 4
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Stability of Multistage Stochastic Programs 9
2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Continuity of the Recourse Function . . . . . . . . . . . . . . 13
2.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Calm Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Recombining Trees for Multistage Stochastic Programs 33
3.1 Problem Formulation and Decomposition . . . . . . . . . . . . 35
3.1.1 Nested Benders Decomposition . . . . . . . . . . . . . 37
3.1.2 Cut Sharing . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.3 Recombining Scenario Trees . . . . . . . . . . . . . . . 39
3.2 An Enhanced Nested Benders Decomposition . . . . . . . . . 40
3.2.1 Cutting Plane Approximations. . . . . . . . . . . . . . 41
3.2.2 Dynamic Recombining of Scenarios . . . . . . . . . . . 45
3.2.3 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.4 Extended Solution Algorithm . . . . . . . . . . . . . . 53
3.3 Construction of Recombining Trees . . . . . . . . . . . . . . . 55
3.3.1 A Tree Generation Algorithm . . . . . . . . . . . . . . 57
3.3.2 Consistency of the Tree Generation Algorithm . . . . . 60
3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.4.1 A Power Scheduling Problem . . . . . . . . . . . . . . 77
3.4.2 Numerical Results . . . . . . . . . . . . . . . . . . . . 79
3.5 Out-of-Sample Evaluation . . . . . . . . . . . . . . . . . . . . 83
3.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . 84
3.5.2 Towards Feasible Solutions . . . . . . . . . . . . . . . . 85
x Contents
3.5.3 Numerical Examples . . . . . . . . . . . . . . . . . . . 89
4 Scenario Reduction with Respect to Discrepancy Distances 97
4.1 Discrepancy Distances . . . . . . . . . . . . . . . . . . . . . . 98
4.2 On Stability of Two-Stage and Chance-Constrained Programs 100
4.3 Scenario Reduction . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 Bounds and Specific Solutions . . . . . . . . . . . . . . . . . . 106
4.4.1 Ordered Solution and Upper Bound . . . . . . . . . . . 106
4.4.2 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . 110
4.5 The Inner Problem . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1 Critical Index Sets . . . . . . . . . . . . . . . . . . . . 114
4.5.2 Reduced Critical Index Sets . . . . . . . . . . . . . . . 115
4.5.3 Determining the Coefficients . . . . . . . . . . . . . . . 116
4.5.4 Optimal Redistribution Algorithm. . . . . . . . . . . . 121
4.6 The Outer Problem . . . . . . . . . . . . . . . . . . . . . . . . 124
4.6.1 Revising Heuristics . . . . . . . . . . . . . . . . . . . . 124
4.6.2 A Glimpse on Low Discrepancy Sequences . . . . . . . 127
4.6.3 A Branch and Bound Approach . . . . . . . . . . . . . 127
4.7 Numerical Experience. . . . . . . . . . . . . . . . . . . . . . . 131
4.8 Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.8.1 A Note on Extended Discrepancies . . . . . . . . . . . 138
4.8.2 Mass Transportation and Clustering . . . . . . . . . . 140
4.8.3 An Estimation between Discrepancies . . . . . . . . . . 144
Appendix 153
Bibliography 159
List of Figures
3.1 Recombining and non-recombining scenario trees. . . . . . . . 40
3.2 Electricity demand and wind turbine power curve . . . . . . . 78
3.3 Out-of-sample values for the power scheduling model . . . . . 91
3.4 Out-of-sample values for the swing option model . . . . . . . . 95
4.1 Recourse function of Example 4.2.1 . . . . . . . . . . . . . . . 104
4.2 Supporting and non-supporting polyhedra . . . . . . . . . . . 117
4.3 Rectangular discrepancy for different reduction techniques . . 125
4.4 Cumulative distribution functions of Example 4.6.1 . . . . . . 126
4.5 Cell discrepancy and running time for the Forward Selection. . 134
4.6 Low discrepancy points with adjusted probabilities . . . . . . 135
4.7 Perturbation of the optimal value of Example 4.7.1 . . . . . . 136
4.8 Recourse function of Example 4.7.2 . . . . . . . . . . . . . . . 137
4.9 Perturbation of the optimal value of Example 4.7.2 . . . . . . 138
4.10 Probabilities adjusted with respect to αBrect and ζ2 . . . . . . 142
4.11 αBrect and ζ2 for different weighting factors . . . . . . . . . . . 143
4.12 Optimal mass transportation with respect to αBrect and ζ2 . . 144
4.13 A polyhedron and the corresponding set M∦ . . . . . . . . . . 147
4.14 Enlargements of a polyhedron . . . . . . . . . . . . . . . . . . 149
4.15 Polyhedral singularity of multivariate normal distributions . . 151