Lectures on stochastic programming: modeling and theory
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Philadelphia, PA
SIAM [u.a.]
2014
|
| Edition: | 2. ed. |
| Series: | MOS-SIAM series on optimization
16 |
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVII, 494 S. |
| ISBN: | 9781611973426 |
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| 245 | 1 | 0 | |a Lectures on stochastic programming |b modeling and theory |c Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyński |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Philadelphia, PA |b SIAM [u.a.] |c 2014 | |
| 300 | |a XVII, 494 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a MOS-SIAM series on optimization |v 16 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Stochastic programming | |
| 650 | 0 | 7 | |a Stochastische Optimierung |0 (DE-588)4057625-5 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Dentcheva, Darinka |e Sonstige |0 (DE-588)172986877 |4 oth | |
| 700 | 1 | |a Ruszczyński, Andrzej P. |e Sonstige |0 (DE-588)115267689 |4 oth | |
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Record in the Search Index
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|---|---|
| adam_text | Optimization problems involving stochastic models occur in almost all areas of science
and engineering, such as telecommunications, medicine, and finance. Their existence
compels a need for rigorous ways of formulating, analyzing, and solving such problems
This book focuses on optimization problems involving uncertain parameters and covers
the theoretical foundations and recent advances in areas where stochastic models are
available.
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In Lectures on Stochastic Programming: Modeling and Theory, Second Edition,
the authors introduce new material to reflect recent developments in stochastic
programming:
• an analytical description of the tangent and normal cones of chance constrained
sets;
• analysis of optimality conditions applied to nonconvex problems;
• a discussion of the stochastic dual dynamic programming method;
• an extended discussion of law invariant coherent risk measures and their Kusuoka
representations; and
• in-depth analysis of dynamic risk measures and concepts of time
consistency, including several new results.
This book is intended for researchers working on theory and
applications of optimization. It also is suitable as a text for advanced
. graduate courses in optimization.
Alexander Shapiro is a Professor in the School of industrial and
Systems Engineering at Georgia Institute of Technology. He has
published more than 100 articles in peer-reviewed journals and is the
coauthor of several books.
- Darinka Dentcheva is a Professor of Mathematics at Stevens Institute
of Technology. She works in the areas of decisions under uncertainty,
convex analysis, and stability of optimization problems.
Andrzej Ruszczynski is a Professor of Operations Research at Rutgers
University. His research is devoted to the theory and methods of
---------- optimization under uncertainty and risk.
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For more information about MOS and SIAM books, journals,
conferences, memberships, or activities, contact:
slam..
Mathematical
Optimization Society
Society for industrial
and Applied Mathematics
3600 Market Street, 6th Floor
Philadelphia, PA 19104-2688 USA
+1-215-382-9800 · Fax +1-215-386-7999
siam@siam.org · www.siam.org
Mathematical Optimization Society
3600 Market Street, 6th Floor
Philadelphia, PA 19104-2688 USA
+1-215-382-9800 x319
Fax +1-215-386-7999
service@mathopt.org · www.mathopt.org
M016
Contents p
List of Notations xi
Preface to the Second Edition xiii
Preface to the First Edition xv
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.............................................. 8
1.3.1 Two-Stage Model........................................... 8
1.3.2 Chance Constrained Model................................. 10
1.3.3 Multistage Model......................................... 11
1.4 Portfolio Selection............................................... 12
1.4.1 Static Model............................................. 12
1.4.2 Multistage Portfolio Selection.......................... 16
1.4.3 Decision Rules........................................... 19
1.5 Supply Chain Network Design....................................... 21
Exercises ............................................................... 23
2 Two-Stage Problems 25
2.1 Linear Two-Stage Problems......................................... 25
2.1.1 Basic Properties......................................... 25
2.1.2 The Expected Recourse Cost for Discrete Distributions . . 27
2.1.3 The Expected Recourse Cost for General Distributions .. 30
2.1.4 Optimality Conditions.................................... 35
2.2 Polyhedral Two-Stage Problems................................... 39
2.2.1 General Properties....................................... 39
2.2.2 Expected Recourse Cost................................... 41
2.2.3 Optimality Conditions.................................... 43
2.3 General Two-Stage Problems........................................ 44
2.3.1 Problem Formulation, Interchangeability.................. 44
2.3.2 Convex Two-Stage Problems................................ 46
2.4 Nonanticipativity................................................. 48
2.4.1 Scenario Formulation..................................... 48
2.4.2 Dualization of Nonanticipativity Constraints............. 50
vii
viii
Contents
2.4.3 Nonanticipativity Duality for General Distributions .... 52
2.4.4 Value of Perfect Information........................... 55
Exercises .............................................................. 56
3 Multistage Problems 59
3.1 Problem Formulation.............................................. 59
3.1.1 The General Setting...................................... 59
3.1.2 The Linear Case.......................................... 63
3.1.3 Scenario Trees......................................... 67
3.1.4 Filtration Interpretation. .............................. 69
3.1.5 Algebraic Formulation of Nonanticipativity Constraints . 71
3.1.6 Piecewise Affine Policies................................ 75
3.2 Duality............................................................ 78
3.2.1 Convex Multistage Problems ............................ 78
3.2.2 Optimality Conditions ................................. 78
3.2.3 Dualization of Feasibility Constraints................... 82
3.2.4 Dualization of Nonanticipativity Constraints ............ 83
Exercises ................................................................ 85
4 Optimization Models with Probabilistic Constraints 89
4.1 Introduction....................................................... 89
4.2 Convexity in Probabilistic Optimization............................ 95
4.2.1 Generalized Concavity of Functions and Measures........ 95
4.2.2 Convexity of Probabilistically Constrained Sets......... 107
4.2.3 Connectedness of Probabilistically Constrained Sets .... 114
4.3 Separable Probabilistic Constraints............................... 115
4.3.1 Continuity and Differentiability Properties of Distribution
Functions.............................................. 115
4.3.2 p-Efficient Points...................................... 117
4.3.3 The Tangent and Normal Cones of convi^.................. 124
4.3.4 Optimality Conditions and Duality Theory ............... 127
4.4 Optimization Problems with Nonseparable Probabilistic
Constraints..................................................... 142
4.4.1 Differentiability of Probability Functions and Optimality
Conditions............................................... 143
4.4.2 Approximations of Nonseparable Probabilistic
Constraints.............................................. 145
4.5 Semi-Infinite Probabilistic Problems.............................. 153
Exercises ............................................................. 159
5 Statistical Inference 163
5.1 Statistical Properties of Sample Average Approximation Estimators . 163
5.1.1 Consistency of SAA Estimators........................... 165
5.1.2 Asymptotics of the SAA Optimal Value.................... 170
5.1.3 Second Order Asymptotics................................ 173
5.1.4 Minimax Stochastic Programs........................... 177
5.2 Stochastic Generalized Equations.................................. 181
5.2.1 Consistency of Solutions of the SAA Generalized
Equations................................................ 183
5.2.2 Asymptotics of SAA Generalized Equations Estimators . . 184
Contents ix
5.3 Monte Carlo Sampling Methods................................... 187
5.3.1 Exponential Rates of Convergence and Sample Size
Estimates in Case of a Finite Feasible Set............ 188
5.3.2 Sample Size Estimates in the General Case............. 192
5.3.3 Finite Exponential Convergence........................ 198
5.4 Quasi-Monte Carlo Methods...................................... 199
5.5 Variance Reduction Techniques.................................. 205
5.5.1 Latin Hypercube Sampling.............................. 205
5.5.2 Linear Control Random Variables Method................ 206
5.5.3 Importance Sampling and Likelihood Ratio Methods .... 207
5.6 Validation Analysis............................................ 208
5.6.1 Estimation of the Optimality Gap ..................... 209
5.6.2 Statistical Testing of Optimality Conditions.......... 213
5.7 Chance Constrained Problems.................................... 216
5.7.1 Monte Carlo Sampling Approach......................... 216
5.7.2 Validation of an Optimal Solution..................... 222
5.8 SAA Method Applied to Multistage Stochastic Programming....... 226
5.8.1 Statistical Properties of Multistage SAA Estimators.. 226
5.8.2 Complexity Estimates of Multistage Programs........... 231
5.9 Stochastic Approximation Method................................ 234
5.9.1 Classical Approach.................................... 235
5.9.2 Robust SA Approach.................................... 238
5.9.3 Mirror Descent SA Method.............................. 240
5.9.4 Accuracy Certificates for Mirror Descent SA Solutions . . 248
5.10 Stochastic Dual Dynamic Programming Method..................... 252
5.10.1 Approximate Dynamic Programming Approach.............. 253
5.10.2 The Stochastic Dual Dynamic Programming Algorithm . . 256
5.10.3 Convergence Properties of the SDDP Algorithm.......... 259
5.10.4 Risk Averse SDDP Method............................... 260
Exercises ........................................................... 266
6 Risk Averse Optimization 271
6.1 Introduction................................................... 271
6.2 Mean-Risk Models .............................................. 272
6.2.1 Main Ideas of Mean-Risk Analysis...................... 272
6.2.2 Semideviations........................................ 273
6.2.3 Weighted Mean Deviations from Quantiles............... 274
6.2.4 Average Value-at-Risk................................. 275
6.3 Coherent Risk Measures......................................... 279
6.3.1 Differentiability Properties of Risk Measures......... 284
6.3.2 Examples of Risk Measures............................. 289
6.3.3 Law Invariant Risk Measures .......................... 298
6.3.4 Spectral Risk Measures ............................... 302
6.3.5 Kusuoka Representations............................... 306
6.3.6 Probability Spaces with Atoms ........................ 313
6.3.7 Stochastic Orders..................................... 316
6.4 Ambiguous Chance Constraints................................... 318
6.5 Optimization of Risk Measures.................................. 325
6.5.1 Dualization of Nonanticipativity Constraints ......... 328
6.5.2 Interchangeability Principle for Risk Measures........ 329
x Contents
6.5.3 Examples............................................... 331
6.6 Statistical Properties of Risk Measures......................... 336
6.6.1 Average Value-at-Risk.................................. 336
6.6.2 Absolute Semideviation Risk Measure ................... 341
6.6.3 Von Mises Statistical Functionals...................... 343
6.7 The Problem of Moments.......................................... 346
6.8 Multistage Risk Averse Optimization ............................ 351
6.8.1 Scenario Tree Formulation ............................ 351
6.8.2 Conditional Risk Mappings.............................. 357
6.8.3 Dynamic Risk Measures.................................. 362
6.8.4 Risk Averse Multistage Stochastic Programming.......... 366
6.8.5 Time Consistency of Multiperiod Problems............... 369
6.8.6 Minimax Approach to Risk Averse Multistage
Programming............................................ 375
6.8.7 Portfolio Selection and Inventory Model Examples...... 378
Exercises ............................................................. 381
7 Background Material 387
7.1 Optimization and Convex Analysis................................ 388
7.1.1 Directional Differentiability........................ 388
7.1.2 Elements of Convex Analysis............................ 390
7.1.3 Optimization and Duality............................... 393
7.1.4 Optimality Conditions.................................. 400
7.1.5 Perturbation Analysis.................................. 405
7.1.6 Epiconvergence......................................... 412
7.2 Probability................................................... 413
7.2.1 Probability Spaces and Random Variables................ 413
7.2.2 Conditional Probability and Conditional Expectation . . . 418
7.2.3 Measurable Multifunctions and Random Functions ..... 419
7.2.4 Expectation Functions.................................. 422
7.2.5 Uniform Laws of Large Numbers ......................... 428
7.2.6 Law of Large Numbers for Risk Measures................. 432
7.2.7 Law of Large Numbers for Random Sets and
Subdifferentials....................................... 437
7.2.8 Delta Method .......................................... 440
7.2.9 Exponential Bounds of the Large Deviations Theory .... 445
7.2.10 Uniform Exponential Bounds............................. 450
7.3 Elements of Functional Analysis................................. 456
7.3.1 Conjugate Duality and Differentiability................ 460
7.3.2 Paired Locally Convex Topological Vector Spaces........ 462
7.3.3 Lattice Structure...................................... 463
7.3.4 Interchangeability Principle........................... 464
Exercises ............................................................. 465
8 Bibliographical Remarks 467
Bibliography 475
Index 491
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| id | DE-604.BV042142940 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:13:45Z |
| institution | BVB |
| isbn | 9781611973426 |
| language | English |
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| series | MOS-SIAM series on optimization |
| series2 | MOS-SIAM series on optimization |
| spelling | Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyński 2. ed. Philadelphia, PA SIAM [u.a.] 2014 XVII, 494 S. txt rdacontent n rdamedia nc rdacarrier MOS-SIAM series on optimization 16 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 Shapiro, Alexander 1949- Sonstige (DE-588)130425850 oth Dentcheva, Darinka Sonstige (DE-588)172986877 oth Ruszczyński, Andrzej P. Sonstige (DE-588)115267689 oth MOS-SIAM series on optimization 16 (DE-604)BV039542569 16 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027582882&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027582882&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lectures on stochastic programming modeling and theory MOS-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 Ruszczyński |
| title_fullStr | Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyński |
| title_full_unstemmed | Lectures on stochastic programming modeling and theory Alexander Shapiro ; Darinka Dentcheva ; Andrzej Ruszczyński |
| 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 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027582882&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=027582882&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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