Preface	7
	Contents	8
	List of Figures	10
	List of Tables	11
	Index of Notation	12
	Chapter 1 Introduction	14
	1.1 Stochastic Programming Models	14
	1.2 Approximations, Stability, and Decomposition	17
	Approximations	17
	Stability	18
	Decomposition	19
	1.3 Contributions	20
	Chapter 2 Stability of Multistage Stochastic Programs	22
	2.1 Problem Formulation	23
	2.2 Continuity of the Recourse Function	26
	2.3 Approximations	34
	2.4 Calm Decisions	38
	2.5 Stability	41
	Chapter 3 Recombining Trees for Multistage Stochastic Programs	45
	3.1 Problem Formulation and Decomposition	47
	3.1.1 Nested Benders Decomposition	49
	3.1.2 Cut Sharing	50
	3.1.3 Recombining Scenario Trees	51
	3.2 An Enhanced Nested Benders Decomposition	52
	3.2.1 Cutting Plane Approximations	53
	Optimality and Feasibility Cuts	54
	Nested Benders Decomposition Algorithm	56
	3.2.2 Dynamic Recombining of Scenarios	57
	3.2.3 Upper Bounds	63
	3.2.4 Extended Solution Algorithm	65
	3.3 Construction of Recombining Trees	67
	3.3.1 A Tree Generation Algorithm	69
	Transition Probabilities	70
	3.3.2 Consistency of the Tree Generation Algorithm	72
	3.4 Case Study	89
	3.4.1 A Power Scheduling Problem	89
	3.4.2 Numerical Results	91
	Cut Sharing	91
	Aggregation of Decision Points	93
	Dynamic Aggregation	94
	3.5 Out-of-Sample Evaluation	95
	3.5.1 Problem Formulation	96
	3.5.2 Towards Feasible Solutions	97
	Feasibility Restoration	98
	3.5.3 Numerical Examples	101
	Power Scheduling	102
	Swing Option Exercising	104
	Chapter 4 Scenario Reduction with Respect to Discrepancy Distances	109
	4.1 Discrepancy Distances	110
	4.2 On Stability of Two-Stage and ChanceConstrained Programs	112
	Mixed-Integer Two-Stage Programs	113
	Chance-Constrained Programs	116
	4.3 Scenario Reduction	117
	4.4 Bounds and Specific Solutions	118
	4.4.1 Ordered Solution and Upper Bound	118
	4.4.2 Lower Bound	122
	4.5 The Inner Problem	125
	4.5.1 Critical Index Sets	126
	4.5.2 Reduced Critical Index Sets	127
	A Sufficient Optimality Condition	127
	4.5.3 Determining the Coefficients	128
	4.5.4 Optimal Redistribution Algorithm	133
	4.6 The Outer Problem	136
	4.6.1 Revising Heuristics	136
	4.6.2 A Glimpse on Low Discrepancy Sequences	139
	4.6.3 A Branch and Bound Approach	139
	Breadth-First Search Heuristics	141
	4.7 Numerical Experience	143
	Optimal Redistribution	144
	Support Selection	145
	Back to Stochastic Programming	147
	4.8 Further Results	150
	4.8.1 A Note on Extended Discrepancies	150
	4.8.2 Mass Transportation and Clustering	152
	Duality and Reduced Costs	152
	Further Numerical Experience	155
	4.8.3 An Estimation between Discrepancies	156
	Upper Bound	158
	Lower Bound	161
	Determining the Polyhedral Singularity	162
	Appendix	164
	Bibliography	7