Multistage stochastic optimization:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham [u.a.]
Springer
2014
|
| Series: | Springer series in operations research and financial engineering
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XIV, 301 Seiten Illustrationen |
| ISBN: | 3319088424 9783319088426 |
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MARC
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| 245 | 1 | 0 | |a Multistage stochastic optimization |c Georg Ch. Pflug ; Alois Pichler |
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Record in the Search Index
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| adam_text | Contents
Introduction
.................................................................. 1
1.1
Multistage Decision Models
.......................................... 1
1.2
Fundamentals of Decision Making
................................... 2
1.2.1
Stochastic Problem Formulation
............................ 3
1.2.2
From
Single-Stage
to Multistage Decision Problems
...... 7
1.3
Multistage Stochastic Optimization Versus Dynamic
Optimization
........................................................... 21
1.4
Scenario Trees and Nested Distributions
............................. 23
1.4.1
Nested Distributions
......................................... 29
1.4.2
Equivalence and Minimality
................................. 33
1.4.3
Convex Structures for Scenario Models
.................... 37
The Nested Distance
......................................................... 41
2.1
Distances of Probability Measures
.................................... 42
2.1.1
Semi-Distances Generated by a Class of Test Functions
... 42
2.2
The
Wasserstein
Distance
............................................. 46
2.3
Elementary Properties of the
Wasserstein
Distance
.................. 51
2.3.1
The
Wasserstein
Distance on the Real Line
................ 53
2.4
Alternative Distances as Basis for the
Wasserstein
Distance
........ 55
2.4.1
The Role of the Distance on the Underlying Space
........ 55
2.4.2
Transformation of the Axis, and Fortet-Mourier
Distances
..................................................... 55
2.5
Estimates Involving the
Wasserstein
Distance
....................... 58
2.6
Approximations in the
Wasserstein
Metric
........................... 62
2.7
The
Wasserstein
Distance in a Discrete Framework
................. 63
2.8
Duality for the
Wasserstein
Metric
................................... 65
2.9
Continuity of the Dual Variables,
and the Kantorovich-Rubinstein Theorem
........................... 69
IX
χ
Contents
2.10
Multistage Generalization: The Nested Distance
.................... 71
2.10.1
The Inherited Distance
....................................... 71
2.10.2
The Nested Distance
......................................... 74
2.10.3
The Nested Distance for Trees
.............................. 79
2.11
Dual Representation of the Nested Distance
......................... 88
2.11.1
Martingale Representation of the Nested Distance
......... 91
3
Risk and UtUity Functionals
................................................ 95
3.1
Single-Period Risk and Utility Functionals
.......................... 95
3.2
Examples of Risk and Utility Functionals
............................ 97
3.3
Dual Representation of Risk Functionals
............................. 103
3.3.1
Kusuoka s Representation
................................... 103
3.3.2
The Dual Representation
.................................... 105
3.4
An Alternative Description of Distortion Risk Functionals
......... 110
3.5
The Impact of the Probability Measure on Risk Functionals
........ 114
3.5.1
Compound Concavity and Convex-Concavity
.............. 114
3.5.2
Coştinuity
with Respect to the Probability Measure
....... 117
3.6
Conditional Risk Functionals
......................................... 119
3.6.1
Properties of Conditional Risk Functionals
................. 122
4
From Data to Models
........................................................ 125
4.1
Approximations of Single-Period Probability Distributions
......... 126
4.1.1
Approximation Quality of the Monte Carlo
Generation Method
.......................................... 127
4.1.2
Quasi-Monte
Carlo Approximations
........................ 130
4.1.3
Optimal and Nearly Optimal Single-Period
Discretizations
............................................... 132
4.1.4
The Stochastic Approximation Algorithms
for Multidimensional Quantization
......................... 142
4.1.5
Asymptotic Distribution of Optimal Quantizers
............ 145
4.2
Approximations of Multiperiod Distributions
........................ 149
4.3
Construction of Scenario Trees
....................................... 154
4.3.1
Distance Calculation
......................................... 155
4.3.2
The Construction of Large Trees
............................ 157
4.4
Scenario Tree Reduction
.............................................. 163
4.5
Improvement of Approximating Trees
................................ 166
4.5.1
Improvement of the Probability Measure
................... 167
4.5.2
Improvement of the Paths
................................... 168
4.6
An Alternative View on Approximations
............................. 172
5
Time Consistency
............................................................ 175
5.1
Time Consistency in Stochastic Decision Problems
................. 176
5.2
Time Consistent Risk Functionals
.................................... 179
Contents xi
5.3
Time Consistency and Decomposition
............................... 187
5.3.1
Composition of Risk Functionals
........................... 187
5.3.2
Multistage Decomposition of Risk Functionals:
The Decomposition Theorem
............................... 188
5.4
Martingale Formulations of Time Inconsistent
Stochastic Programs
................................................... 196
5.4.1
Verification Theorems
....................................... 199
5.4.2
An Algorithm for Sequential Improvement
................. 202
5.4.3
Numerical Experiments
...................................... 203
5.5
Dualization of Nonanticipativity Constraints
........................ 205
6
Approximations and Bounds
............................................... 209
6.1
Two-Stage Problems, and Approximation
in the
Wasserstein
Distance
........................................... 209
6.2
Approximation in the Nested Distance Sense
........................ 211
6.3
Bounds
................................................................. 218
6.3.1
Lower Bounds by Changing the Probability Measure
..... 219
6.3.2
Lower Bounds for Replacing the Scenario
Process by Its Expectation
................................... 225
6.3.3
Bounds for Changing the Filtration
......................... 227
6.3.4
Upper Bounds by Inserting (Sub)Solutions
................. 227
6.4
Martingale Properties
.................................................. 228
7
The Problem of Ambiguity in Stochastic Optimization
.................. 229
7.1
Single- or Two-Stage Models: Wasserstein Balls
.................... 234
7.2
Solution Methods for the Single- or Two-Stage Case
................ 237
7.3
The Multistage Case
................................................... 238
7.3.1
A Minimax Theorem
......................................... 241
7.3.2
Ambiguity Sets Defined by Nested
Transportation Kernels
....................................... 245
7.3.3
Algorithmic Solution
........................................ 247
7.4
Example: A Multiperiod Production
/
Inventory Control
Problem
................................................................ 249
7.4.1
Mathematical Modeling Summary
.......................... 251
7.4.2
Computational Results
....................................... 252
8
Examples
...................................................................... 257
8.1
Thermal Electricity Production
....................................... 257
8.2
Hydro Electricity Production
.......................................... 261
8.3
Budget Management for Risk-Prone Countries
...................... 270
A Risk Functionals: Definitions and Notations
.............................. 275
A.
1
Multiperiod Risk Functionals
......................................... 279
A.2 Information
Monotonicity
............................................. 280
Xli
Contents
В
Minimax Theorems.......................................................... 283
С
Comparison of Weighted Norms
........................................... 287
D
The Canonical Construction for Nested Distributions
................... 289
Bibliography
...................................................................... 291
Index
299
Georg
Ch.
Pflug ·
Alois Pichler
Multistage Stochastic Optimization
Multistage stochastic optimization problems appear in many ways in finance, insurance,
energy production and trading, logistics and transportation, among other areas. They
describe decision situations under uncertainty and with a longer planning horizon.
This book contains a comprehensive treatment of today
s
state of the art in multistage
stochastic optimization. It covers the mathematical backgrounds of approximation
theory as well as numerous practical algorithms and examples for the generation and
handling of scenario trees. A special emphasis is put on estimation and bounding
of the modeling error using novel distance concepts, on time consistency and the
role of model ambiguity in the decision process. An extensive treatment of examples
from electricity production, asset liability management and inventory control concludes
the book.
|
| any_adam_object | 1 |
| author | Pflug, Georg 1951- Pichler, Alois 1966- |
| author_GND | (DE-588)115267670 (DE-588)1020884851 |
| author_facet | Pflug, Georg 1951- Pichler, Alois 1966- |
| author_role | aut aut |
| author_sort | Pflug, Georg 1951- |
| author_variant | g p gp a p ap |
| building | Verbundindex |
| bvnumber | BV042198715 |
| classification_rvk | QH 424 SK 870 SK 890 |
| ctrlnum | (OCoLC)898427623 (DE-599)BVBBV042198715 |
| dewey-full | 519.23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.23 |
| dewey-search | 519.23 |
| dewey-sort | 3519.23 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV042198715 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:15:06Z |
| institution | BVB |
| isbn | 3319088424 9783319088426 |
| language | English |
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| physical | XIV, 301 Seiten Illustrationen |
| publishDate | 2014 |
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| publisher | Springer |
| record_format | marc |
| series2 | Springer series in operations research and financial engineering |
| spelling | Pflug, Georg 1951- Verfasser (DE-588)115267670 aut Multistage stochastic optimization Georg Ch. Pflug ; Alois Pichler Cham [u.a.] Springer 2014 XIV, 301 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Springer series in operations research and financial engineering Stochastische Optimierung (DE-588)4057625-5 gnd rswk-swf Statistische Entscheidungstheorie (DE-588)4077850-2 gnd rswk-swf Statistische Entscheidungstheorie (DE-588)4077850-2 s Stochastische Optimierung (DE-588)4057625-5 s DE-604 Pichler, Alois 1966- Verfasser (DE-588)1020884851 aut Erscheint auch als Online-Ausgabe 978-3-319-08843-3 Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027637651&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Augsburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027637651&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Pflug, Georg 1951- Pichler, Alois 1966- Multistage stochastic optimization Stochastische Optimierung (DE-588)4057625-5 gnd Statistische Entscheidungstheorie (DE-588)4077850-2 gnd |
| subject_GND | (DE-588)4057625-5 (DE-588)4077850-2 |
| title | Multistage stochastic optimization |
| title_auth | Multistage stochastic optimization |
| title_exact_search | Multistage stochastic optimization |
| title_full | Multistage stochastic optimization Georg Ch. Pflug ; Alois Pichler |
| title_fullStr | Multistage stochastic optimization Georg Ch. Pflug ; Alois Pichler |
| title_full_unstemmed | Multistage stochastic optimization Georg Ch. Pflug ; Alois Pichler |
| title_short | Multistage stochastic optimization |
| title_sort | multistage stochastic optimization |
| topic | Stochastische Optimierung (DE-588)4057625-5 gnd Statistische Entscheidungstheorie (DE-588)4077850-2 gnd |
| topic_facet | Stochastische Optimierung Statistische Entscheidungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027637651&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027637651&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT pfluggeorg multistagestochasticoptimization AT pichleralois multistagestochasticoptimization |