Christian Küchler. Stability, Approximation, and Decomposition in Two- and Multistage Stochastic Programming

Transkript

1 Christian Küchler Stability, Approximation, and Decomposition in Two- and Multistage Stochastic Programming

2 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 methods, and practical implementations. The series Stochastic Programming presents original research from this range of topics.

3 Christian Küchler Stability, Approximation, and Decomposition in Two- and Multistage Stochastic Programming VIEWEG+TEUBNER RESEARCH

4 Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über < abrufbar. Dissertation Humboldt-Universität zu Berlin, Auflage 2009 Alle Rechte vorbehalten Vieweg+Teubner GWV Fachverlage GmbH, Wiesbaden 2009 Lektorat: Dorothee Koch Anita Wilke Vieweg+Teubner ist Teil der Fachverlagsgruppe Springer Science+Business Media. Das Werk einschließlich aller seiner Teile ist urheberrechtlich ge schützt. Jede Verwertung außerhalb der engen Grenzen des Ur heber rechts ge - set zes ist ohne Zustimmung des Verlags unzuläs sig und straf bar. Das gilt ins be sondere für Vervielfältigungen, Über setzun gen, Mikro verfil mungen und die Ein speiche rung und Ver ar beitung in elek tro nischen Syste men. Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürften. Umschlaggestaltung: KünkelLopka Medienentwicklung, Heidelberg Druck und buchbinderische Verarbeitung: STRAUSS GMBH, Mörlenbach Gedruckt auf säurefreiem und chlorfrei gebleichtem Papier. Printed in Germany ISBN

5 To my parents

6 Preface The present doctoral thesis has been developed during my employment at the Department of Mathematics at Humboldt-Universität zu Berlin. This employment was enabled and supported by the Bundesministerium für Bildung und Forschung and the Wiener Wissenschafts-, Forschungs- und Technologiefonds. 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 Humboldt-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 decomposition approach in the present form. I would also like to thank him for 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 colleagues from 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 love and support, and, in particular, to Konny for having been so strong and patient during the last years. Christian Küchler

7 Contents 1 Introduction Stochastic Programming Models Approximations, Stability, and Decomposition Contributions Stability of Multistage Stochastic Programs ProblemFormulation ContinuityoftheRecourseFunction Approximations CalmDecisions Stability Recombining Trees for Multistage Stochastic Programs ProblemFormulationandDecomposition NestedBendersDecomposition CutSharing Recombining Scenario Trees AnEnhancedNestedBendersDecomposition Cutting Plane Approximations Dynamic Recombining of Scenarios Upper Bounds ExtendedSolutionAlgorithm Construction of Recombining Trees ATreeGenerationAlgorithm ConsistencyoftheTreeGenerationAlgorithm CaseStudy A Power Scheduling Problem Numerical Results Out-of-Sample Evaluation ProblemFormulation Towards Feasible Solutions... 85

8 x Contents Numerical Examples Scenario Reduction with Respect to Discrepancy Distances DiscrepancyDistances On Stability of Two-Stage and Chance-Constrained Programs ScenarioReduction Bounds and Specific Solutions OrderedSolutionandUpperBound LowerBound TheInnerProblem CriticalIndexSets ReducedCriticalIndexSets Determining the Coefficients Optimal Redistribution Algorithm TheOuterProblem RevisingHeuristics A Glimpse on Low Discrepancy Sequences A Branch and Bound Approach Numerical Experience FurtherResults ANoteonExtendedDiscrepancies MassTransportationandClustering AnEstimationbetweenDiscrepancies Appendix 153 Bibliography 159

9 List of Figures 3.1 Recombining and non-recombining scenario trees Electricity demand and wind turbine power curve Out-of-sample values for the power scheduling model Out-of-sample values for the swing option model Recourse function of Example Supporting and non-supporting polyhedra Rectangular discrepancy for different reduction techniques Cumulative distribution functions of Example Cell discrepancy and running time for the Forward Selection Low discrepancy points with adjusted probabilities Perturbation of the optimal value of Example Recourse function of Example Perturbation of the optimal value of Example Probabilities adjusted with respect to α Brect and ζ α Brect and ζ 2 fordifferentweightingfactors Optimal mass transportation with respect to α Brect and ζ A polyhedron and the corresponding set M Enlargementsofapolyhedron Polyhedral singularity of multivariate normal distributions.. 151

10 List of Tables 3.1 Parameters of the power scheduling model RunningtimesofBendersDecomposition Running times with and without using upper bounds Runningtimesfordifferentaggregationparameters Out-of-sample values for different aggregation parameters Out-of-sample values for different aggregation parameters Number of supporting polyhedra and critical index sets RunningtimesofAlgorithm Runningtimesofvarioussupportselectionalgorithms...134

11 Index of Notation 1 B ( ) the indicator function of the set B, the standard scalar product in R m I the cardinality of the finite set I a, ā a closed polyhedron in R s induced by a, ā R k, see p.116 α B the discrepancy distance w.r.t. the system B, seep.98 B a system of Borel subsets of R s,seep.98 B cell the system of closed cells in R s,seep.98 B cl the system of closed subsets of R s,seep.98 B conv thesystemofclosed,convexsubsetsofr s,seep.98 B ph,k the system of polyhedra in R s having at most k vertices, see p.98 B ph,w the system of polyhedra in R s each of whose facets parallels a facet of [0, 1] s or pos W,seep.98 B rect the system of closed, s-dimensional rectangles in R s C feas a set of feasibility cuts, see p.43 C opt a set of optimality cuts, see p.42 δ ξ the Dirac measure in ξ B the topological boundary of the Borel set B d the Pompeiu-Hausdorff distance, see p.12 D F a probability metric with ζ-structure, see p.100 Δ B the optimal value of the scenario reduction problem, see p.107 dist(a, B) the distance between the point a R n and the set B R n, see p.12 E[ξ] the expectation of the random variable ξ F a class of Borel measurable mappings, see p.100 F p a class of locally Lipschitz continuous mappings, see p.101 F p a class of uniformly bounded, piecewise locally Lipschitz continuous mappings, see p.101 F p,b a class of uniformly bounded, locally Lipschitz continuous mappings, see p.101 I B the system of critical index sets, see p.114 IB the system of reduced critical index sets, see p.115

12 Index of Notation xvi int S the topological interior of the set S Λ Rj the set of representative nodes at time R j,seep.40 M m [1,T ] a set of Borel measurable mappings, see p.11 pos W the positive cone of a (d m)-matrix W, i.e., pos W = {Wy : y R m +} Q t (, ) the recourse function at time t, seep.13 P, Q Borel probability measures P t the probability distribution of the random variable ξ t under the measure P, seep.11 P [t] the probability distribution of the random variable ξ [t] under the measure P, seep.11 P the set of supporting polyhedra, see p.117 P p (Ξ) the set of all Borel probability measures on Ξ R s with finite absolute moments of order p 1, see p.101 R the set of real numbers R + the set of non-negative real numbers, i.e., R + =[0, ) R the set of extended real numbers, R = R {, + } ξ an R s -valued random variable or stochastic process ξ [t] the random vector (ξ 1,...,ξ t ) ξ [s,t] the random vector (ξ s,...,ξ t ) for s, t N with s t S(ξ) the set of decisions that are feasible w.r.t. the process ξ, see p.12 S n the standard simplex in R n, see p.105 v(ξ),v(p) the optimal value of a stochastic program, see p.12 Ξ t the support of the measure P t Ξ [t] the support of the measure P [t] ζ p the p-th order Fortet-Mourier metric, see p.101 ζ p,bph,k an extended polyhedral discrepancy, see p.101 ζ p,bph,w an extended polyhedral discrepancy, see p.101