Lectures on stochastic programming: modeling and theory
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia, PA
SIAM [u.a.]
2009
|
| Series: | MPS-SIAM series on optimization
9 |
| Subjects: | |
| Online Access: | Table of contents only Publisher description Contributor biographical information Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 436 S. |
| ISBN: | 9780898716870 |
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| adam_text | Titel: Lectures on stochastic programming
Autor: Shapiro, Alexander
Jahr: 2009
Contents
List of Notations xi
Preface xiii
1 Stochastic Programming Models 1
1.1 Introduction............................... 1
1.2 Inventory................................. 1
1.2.1 The News Vendor Problem ................ 1
1.2.2 Chance Constraints .................... 5
1.2.3 Multistage Models..................... 6
1.3 Multiproduct Assembly......................... 9
1.3.1 Two-Stage Model..................... 9
1.3.2 Chance Constrained Model................ 10
1.3.3 Multistage Model..................... 12
1.4 Portfolio Selection............................ 13
1.4.1 Static Model........................ 13
1.4.2 Multistage Portfolio Selection............... 16
1.4.3 Decision Rules....................... 21
1.5 Supply Chain Network Design..................... 22
Exercises..................................... 25
2 Two-Stage Problems 27
2.1 Linear Two-Stage Problems....................... 27
2.1.1 Basic Properties...................... 27
2.1.2 The Expected Recourse Cost for Discrete Distributions . 30
2.1.3 The Expected Recourse Cost for General Distributions .. 32
2.1.4 Optimality Conditions................... 38
2.2 Polyhedral Two-Stage Problems..................... 42
2.2.1 General Properties..................... 42
2.2.2 Expected Recourse Cost.................. 44
2.2.3 Optimality Conditions................... 47
2.3 General Two-Stage Problems...................... 48
2.3.1 Problem Formulation, Interchangeability......... 48
2.3.2 Convex Two-Stage Problems............... 49
2.4 Nonanticipativity ............................ 52
vii
vi ¡i Contents
2.4.1 Scenario Formulation................... 52
2.4.2 Dualization of Nonanticipati vity Constraints....... 54
2.4.3 Nonanticipativity Duality for General Distributions ... 56
2.4.4 Value of Perfect Information . ............... 59
Exercises..................................... 60
3 Multistage Problems 63
3.1 Problem Formulation.......................... 63
3.1.1 The General Setting.................... 63
3.1.2 TheLinearCase...................... 65
3.1.3 Scenario Trees....................... 69
3.1.4 Algebraic Formulation of Nonanticipativity Constraints . 71
3.2 Duality.................................. 76
3.2.1 Convex Multistage Problems............... 76
3.2.2 Optimality Conditions................... 77
3.2.3 Dualization of Feasibility Constraints........... 80
3.2.4 Dualization of Nonanticipativity Constraints....... 82
Exercises..................................... 84
4 Optimization Models with Probabilistic Constraints 87
4.1 Introduction............................... 87
4.2 Convexity in Probabilistic Optimization ................ 94
4.2.1 Generalized Concavity of Functions and Measures .... 94
4.2.2 Convexity of Probabilistically Constrained Sets ..... 106
4.2.3 Connectedness of Probabilistically Constrained Sets ... 113
4.3 Separable Probabilistic Constraints................... 114
4.3.1 Continuity and Differentiability Properties of
Distribution Functions................... 114
4.3.2 p-Efficient Points..................... 115
4.3.3 Optimality Conditions and Duality Theory........ 122
4.4 Optimization Problems with Nonseparable Probabilistic Constraints. . 132
4.4.1 Differentiability of Probability Functions and Optimality
Conditions......................... 133
4.4.2 Approximations of Nonseparable Probabilistic
Constraints......................... 136
4.5 Semi-infinite Probabilistic Problems.................. 144
Exercises..................................... 150
5 Statistical Inference 155
5.1 Statistical Properties of Sample Average Approximation Estimators . . 155
5.1.1 Consistency of SAA Estimators.............. 157
5.1.2 Asymptotics of the SAA Optimal Value.......... 163
5.1.3 Second Order Asymptotics ................ 166
5.1.4 Minimax Stochastic Programs............... 170
5.2 Stochastic Generalized Equations.................... 174
5.2.1 Consistency of Solutions of the SAA Generalized
Equations ......................... 175
Contents ¡x
5.2.2 Asymptotics of SAA Generalized Equations Estimators . 177
5.3 Monte Carlo Sampling Methods.....................180
5.3.1 Exponential Rates of Convergence and Sample Size
Estimates in the Case of a Finite Feasible Set.......181
5.3.2 Sample Size Estimates in the General Case........185
5.3.3 Finite Exponential Convergence..............191
5.4 Quasi-Monte Carlo Methods......................193
5.5 Variance-Reduction Techniques.....................198
5.5.1 Latin Hypercube Sampling ................198
5.5.2 Linear Control Random Variables Method........200
5.5.3 Importance Sampling and Likelihood Ratio Methods . . . 200
5.6 Validation Analysis...........................202
5.6.1 Estimation oftheOptimality Gap.............202
5.6.2 Statistical Testing of Optimality Conditions........207
5.7 Chance Constrained Problems......................210
5.7.1 Monte Carlo Sampling Approach.............210
5.7.2 Validation of an Optimal Solution.............216
5.8 SAA Method Applied to Multistage Stochastic Programming.....220
5.8.1 Statistical Properties of Multistage SAA Estimators ...221
5.8.2 Complexity Estimates of Multistage Programs......226
5.9 Stochastic Approximation Method ...................230
5.9.1 Classical Approach.....................230
5.9.2 Robust S A Approach....................233
5.9.3 Mirror Descent SA Method................236
5.9.4 Accuracy Certificates for Mirror Descent SA Solutions . . 244
Exercises.....................................249
6 Risk Averse Optimization 253
6.1 Introduction...............................253
6.2 Mean-Risk Models............................254
6.2.1 Main Ideas of Mean-Risk Analysis............254
6.2.2 Semideviations.......................255
6.2.3 Weighted Mean Deviations from Quantités........256
6.2.4 Average Value-at-Risk...................257
6.3 Coherent Risk Measures.........................261
6.3.1 Differentiability Properties of Risk Measures.......265
6.3.2 Examples of Risk Measures................269
6.3.3 Law Invariant Risk Measures and Stochastic Orders . . . 279
6.3.4 Relation to Ambiguous Chance Constraints........285
6.4 Optimization of Risk Measures.....................288
6.4.1 Dualization of Nonanticipativity Constraints.......291
6.4.2 Examples..........................295
6.5 Statistical Properties of Risk Measures.................300
6.5.1 Average Value-at-Risk...................300
6.5.2 Absolute Semideviation Risk Measure..........301
6.5.3 Von Mises Statistical Functionals.............304
6.6 The Problem of Moments........................306
Contents
6.7 Multistage Risk Averse Optimization..................308
6.7.1 Scenario Tree Formulation.................308
6.7.2 Conditional Risk Mappings................315
6.7.3 Risk Averse Multistage Stochastic Programming.....318
Exercises.....................................328
Background Material
333
7.2
7.1 Optimization and Convex Analysis...................334
7.1.1 Directional Differentiability................334
7.1.2 Elemente of Convex Analysis...............336
7.1.3 Optimization and Duality.................339
7.1.4 Optimality Conditions...................346
7.1.5 Perturbation Analysis...................351
7.1.6 Epiconvergence......................357
Probability................................359
7.2.1 Probability Spaces and Random Variables ........359
7.2.2 Conditional Probability and Conditional Expectation ... 363
7.2.3 Measurable Multifunctions and Random Functions .... 365
7.2.4 Expectation Functions...................368
7.2.5 Uniform Laws of Large Numbers.............374
7.2.6 Law of Large Numbers for Random Sets and
Subdifferentials......................379
7.2.7 Delta Method .......................382
7.2.8 Exponential Bounds of the Large Deviations Theory . . . 387
7.2.9 Uniform Exponential Bounds...............393
7.3
Elements of Functional Analysis ....................399
7.3.1 Conjugate Duality and Differentiability..........401
7.3.2 Lattice Structure......................403
Exercises.....................................405
8 Bibliographical Remarks
Bibliography
Index
407
415
431
|
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| author | Shapiro, Alexander 1949- Dentcheva, Darinka Ruszczyński, Andrzej P. |
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| ctrlnum | (OCoLC)699297003 (DE-599)BVBBV035818126 |
| dewey-full | 519.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| language | English |
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| spelling | Shapiro, Alexander 1949- Verfasser (DE-588)130425850 aut Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyski Philadelphia, PA SIAM [u.a.] 2009 XV, 436 S. txt rdacontent n rdamedia nc rdacarrier MPS-SIAM series on optimization 9 Includes bibliographical references and index Stochastic programming Stochastische Optimierung (DE-588)4057625-5 gnd rswk-swf Stochastische Optimierung (DE-588)4057625-5 s DE-604 Dentcheva, Darinka Verfasser (DE-588)172986877 aut Ruszczyński, Andrzej P. Verfasser (DE-588)115267689 aut MPS-SIAM series on optimization 9 (DE-604)BV023088428 9 http://www.loc.gov/catdir/enhancements/fy0916/2009022942-t.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0916/2009022942-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0916/2009022942-b.html Contributor biographical information HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018676941&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shapiro, Alexander 1949- Dentcheva, Darinka Ruszczyński, Andrzej P. Lectures on stochastic programming modeling and theory MPS-SIAM series on optimization Stochastic programming Stochastische Optimierung (DE-588)4057625-5 gnd |
| subject_GND | (DE-588)4057625-5 |
| title | Lectures on stochastic programming modeling and theory |
| title_auth | Lectures on stochastic programming modeling and theory |
| title_exact_search | Lectures on stochastic programming modeling and theory |
| title_full | Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyski |
| title_fullStr | Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyski |
| title_full_unstemmed | Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyski |
| title_short | Lectures on stochastic programming |
| title_sort | lectures on stochastic programming modeling and theory |
| title_sub | modeling and theory |
| topic | Stochastic programming Stochastische Optimierung (DE-588)4057625-5 gnd |
| topic_facet | Stochastic programming Stochastische Optimierung |
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