Stochastic programming:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer Acad. Publ.
2010
|
| Series: | Mathematics and its applications
324 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 599 S. |
| ISBN: | 9789048145522 |
Staff View
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| 245 | 1 | 0 | |a Stochastic programming |c by András Prékopa |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer Acad. Publ. |c 2010 | |
| 300 | |a XVIII, 599 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Mathematics and its applications |v 324 | |
| 650 | 0 | 7 | |a Stochastische Optimierung |0 (DE-588)4057625-5 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804143587305193472 |
|---|---|
| adam_text | Contents
Préface
vii
1
General
Theory of Linear Programming
1
1.1
The Simplex and the Lexicographic Simplex Methods
......... 1
1.2
The Duality Theorem
.......................... 10
1.3
Farkas
Theorem on Linear Inequalities
................. 13
1.4 von
Neumann s Theorem on Two-Person Zero-Sum Games
...... 15
1.5
The Dual and Lexicographic Dual Methods
.............. 17
1.6
Discussion of the Tableaux
........................ 20
1.7
Complementary Slackness Theorems
.................. 23
1.8
Exercises and Problems
......................... 27
2
Convex Polyhedra
35
2.1
Definitions
................................. 35
2.2
Parametric Representation of the Solutions of Homogeneous Linear
Inequalities
................................ 42
2.3
Canonical Representation of Convex Polyhedra
............ 48
2.4
Vertices of Convex Polyhedra
...................... 52
2.5
Exercises and Problems
......................... 54
3
Special Problems and Methods
59
3.1
Handling of Problems Containing Free Variables-Revised Methods
. 59
3.2
Individual Upper Bounds
........................ 62
3.3
The Parametric Method
......................... 67
3.4
Cutting Plane Sequences
......................... 71
3.5
Cutting Plane Method for the Solution of the All Integer Variable
Linear Programming Problem
...................... 76
XIV
Contents
3.6
The Dantzig-Wolfe Decomposition Method
.............. 78
3.7
Miscellaneous Remarks
.......................... 81
3.8
Exercises and Problems
......................... 83
Logconcave and Quasi-Concave Measures
87
4.1
Preliminary Notions
........................... 87
4.2
The Basic Theorems of Logconcave Measures
............. 89
4.3
Logconvexity
............................... 95
4.4
Examples of Multivariate Logconcave and Logconvex Probability
Densities
.................................. 97
4.5
Inequalities for Sums and Integrals
................... 99
4.6
Application to Probability Distributions: Generalization of Theorem
4.2.1.................................... 104
4.7
Logconcavity of Discrete Distributions
................. 107
4.8
Theorems on the Binomial and
Poisson
Distributions
.........
Ill
4.9
Exercises and Problems
......................... 121
Moment Problems
125
5.1
Introduction
................................ 125
5.2
Summary of the Chebyshev-Markov Theory and Related Results
. . 128
5.3
Refined Lower and Upper Bounds for the Expectation of a Convex
Function
.................................. 137
5.4
General Moment Problems
........................ 139
5.5
Upper Bounds on the Expectation of a Multivariate Convex
Function
.................................. 146
5.6
Discrete Moment Problems
....................... 152
5.7
The Structure of the Dual Feasible Bases
................ 158
5.8
Generalization and Solutions of Problems
............... 163
5.9
Closed Form Bounds
........................... 168
5.9.1
Lower Bounds,
μι, μ2
are Given
................ 172
5.9.2
Upper Bounds,
μι, μ2
are Given
................ 173
5.9.3
Lower Bounds,
μι, μι, μζ
are Given
.............. 174
5.9.4
Upper Bounds,
μι, μ2, μζ
are Given
.............. 176
5.10
Exercises and Problems
......................... 177
Bounding and Approximation of Probabilities
179
6.1
Introduction
................................ 179
6.2
Sharp Bounds on the Probability that at Least
r
out of
n
Events
Occur, Given 5i,...,5m
.......................... 182
6.2.1
Lower Bounds, Si, S2 Given
................... 186
6.2.2
Upper Bounds, Si, S2 Given
.................. 187
6.2.3
Lower Bounds, Si, S2, S3 Given
................ 187
6.2.4
Upper Bounds, Sx, S2, S3 Given
................ 187
6.2.5
-Upper Bounds, Sb S2, S3, S4, Given
.............. 187
6.3
Hunter s Upper Bound
...................... _ 188
Contents xv
6.4 Application
of Probability Bounds for the Solution of the
Satisfiability Problem
...........................190
6.5
Combined Use of Inclusion-Exclusion and Simulation to Estimate the
Probability of a Composite Event
....................191
6.6
Approximation of Multivariate Normal, Gamma, and Dirichlet
Probability Integrals
...........................194
6.6.1
Multivariate Normal Distribution
................ 195
6.6.2
A Multivariate Gamma Distribution
.............. 197
6.6.3
Dirichlet Distribution
...................... 200
6.6.4
Gradients
............................. 203
6.7
The Probability of a Rectangle in Case of a Multivariate Normal
Distribution
................................ 208
6.8
A Hybrid Method to Compute Multivariate Normal Probabilities
. . 210
6.9
Exercises and Problems
......................... 212
7
Statistical Decisions
219
7.1
Introduction
................................ 219
7.2
The Bernoulli Principle
......................... 221
7.3
Probability Maximization
........................ 223
7.4
Ensuring Safety through Utility
..................... 225
7.5
Choosing Efficient Points
......................... 225
7.6
The Neyman-Pearson Lemma
...................... 226
7.7
Bayesian Decisions
............................ 227
7.8
Decision when the Probability Distribution is Unknown
(The Minimax Principle)
......................... 228
7.9
The Sequential Probability Ratio Test
................. 229
7.10
Two-Stage Methods
........................... 229
7.11
Wald s Theory of Statistical Decision Function
............ 231
8
Static Stochastic Programming Models
233
8.1
Introduction
................................ 233
8.2
Probability Maximization
........................ 235
8.3
Programming under Probabilistic Constraints
............. 235
8.4
Constraints Involving Conditional Expectations and Related
Measures
.................................. 239
8.5
Handling a Random Objective Function
................ 243
8.6
Models where Infeasibility is Penalized
................. 247
8.7
The Newsboy Problem
.......................... 252
8.8
Simultaneous Use of Penalties and Probabilistic Constraint
..... 253
8.9
Utility Functions and Deterministic Equivalents
............ 255
8.10
Stochastic Programming with Multiple Objective Functions
..... 257
8.11
Game Theoretical Formulation
..................... 259
8.12
Exercises and Problems
......................... 264
xvi
Contents
9
Solutions of the Simple Recourse Problem
269
9.1
Introduction
................................ 269
9.2
Primal Method for the Solution of the Simple Recourse Problem
. . 271
9.3
Dual Method for the Solution of the Simple Recourse Problem
. . . 278
9.4
Applications for Deterministic Problems
................ 289
9.5
The Case of the Continuous Distribution
................ 291
9.6
Allocation of Aircraft to Routes under Uncertain Demand
...... 294
9.7
Exercises and Problems
......................... 297
10
Convexity Theory of Probabilistic Constrained Problems
301
10.1
Introduction
................................ 301
10.2
General Convexity Statements
...................... 302
10.3
Some Concavity and Quasi-Concavity Theorems for Probability
Distribution Functions
.......................... 306
10.4
Convexity Statements for Random Linear Constraints
........ 311
10.5
Exercises and Problems
......................... 316
11
Programming under Probabilistic Constraint and Maximizing
Probabilities under Constraints
319
11.1
Introduction
................................ 319
11.2
The Use of the SUMT Interior Point Method with Logarithmic
Barrier Functions
............................. 320
11.3
Application to a Reliability Type Inventory Model
.......... 326
11.4
Application to Serially Linked Reservoir System Design
....... 331
11.5
The Use of a Supporting
Hyperplane
Method
............. 337
11.6
Numerical Examples
........................... 340
11.7
Application of the GRG Method
.................... 346
11.8
Solution by a Primal-Dual Algorithm
.................. 349
11.9
Probabilistic Constraints Involving Discrete Distribution
....... 351
11.10
Applications in Statistics
......................... 357
11.11
A Wafer Design Problem in Semiconductor Manufacturing
...... 363
11.12
The Use of Probability Bounding Techniques in Probabilistic
Constrained Stochastic Programming
.................. 365
11.13
Exercises and Problems
......................... 367
12
Two-Stage Stochastic Programming Problems
373
12.1
Formulation of the Problem
....................... 373
12.2
Mathematical Properties of the Recourse Problem
.......... 377
12.3
Solution of the Recourse Problem by Basis Decomposition Technique
when
ξ
has a Discrete Distribution
................... 380
12.4
Solution of the Recourse Problem by the L-Shaped Method
..... 389
12.5
Solution of the General Recourse Problem by Discretization
..... 395
12.6
Sublinear
Upper Bounding Technique for the Recourse Function
... 400
12.7
Regularized Decomposition Method for Minimizing a Sum of
Polyhedral Functions
........................... 403
Contents
xvii
12.8
Stochastic Decomposition and Conditional Stochastic
Decomposition
.............................. 406
12.9
Stochastic
Quasigradient
Methods
................... 414
12.10
Two-Stage Stochastic Programming Formulations Using Probabilis¬
tic Constraint
............................... 417
12.11
Two-Stage Stochastic Integer Programming
.............. 420
12.12
Exercises and Problems
......................... 421
13
Multi-Stage Stochastic Programming Problems
425
13.1
Formulation of the Problem
.......................425
13.2
Probabilistic Constrained Formulation
.................431
13.3
Basis Decomposition Technique Applied to a Multi-Stage Stochastic
Programming Problem
..........................434
13.4
L-Shaped Technique Applied to a Multi-Stage Stochastic Program¬
ming Problem
...............................439
13.5
The Method of Scenario Aggregation
..................444
14
Special Cases and Selected Applications
447
14.1
A Network Recourse Problem
...................... 447
14.2
Electric Power Generation Capacity Expansion under Uncertainty
. 448
14.3
Models Including the Transmission System
............... 452
14.4
Computing Power System Reliability
.................. 458
14.5
Optimal Scheduling of
a
Hydrothermal
Generating System
...... 466
14.6
Optimal Control of a Storage Level
................... 468
14.7
An Example for Optimal Control of Reservoirs
............ 479
14.8
Two-Sector Multi-Stage Economic Planning
.............. 485
14.9
A PERT Optimization Problem
..................... 486
14.10
Finance Problems
............................. 492
14.11
Diet and Animal Feed Problems
..................... 499
15
Distribution Problems
501
15.1
Formulation of the Problem
....................... 501
15.2
The Random Linear Programming Problem
.............. 502
15.3
The Continuity of the Optimum Value of a Linear Programming
Problem
.................................. 507
15.4
Computation of Characteristics of the Random Optimum Value
. . . 509
15.5
Asymptotic Distribution of the Optimum in Case of a Highly Stable
Basis
.................................... 513
15.6
Laws of Large Numbers for Random Linear Programs
........ 519
15.7
Laws of Large Numbers for Random Knapsack Problems
....... 526
15.8
The Beardwood-Halton-Hammersley Theorem for the Random
Traveling Salesman Problem
....................... 533
15.9
Some Inequalities
............................. 533
15.10
Exercises and Problems
......................... 537
xvüi
Contents
Appendix. The Multivariate
Normal Distribution
541
Bibliography
551
Author Index
589
Subject Index
595
|
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| author_facet | Prékopa, András 1929- |
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| building | Verbundindex |
| bvnumber | BV036879389 |
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| ctrlnum | (OCoLC)706091396 (DE-599)BVBBV036879389 |
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| id | DE-604.BV036879389 |
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| indexdate | 2024-07-09T22:50:02Z |
| institution | BVB |
| isbn | 9789048145522 |
| language | English |
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| spelling | Prékopa, András 1929- Verfasser (DE-588)121144909 aut Stochastic programming by András Prékopa Dordrecht [u.a.] Kluwer Acad. Publ. 2010 XVIII, 599 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 324 Stochastische Optimierung (DE-588)4057625-5 gnd rswk-swf Stochastische Optimierung (DE-588)4057625-5 s DE-604 Mathematics and its applications 324 (DE-604)BV008163334 324 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020794814&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Prékopa, András 1929- Stochastic programming Mathematics and its applications Stochastische Optimierung (DE-588)4057625-5 gnd |
| subject_GND | (DE-588)4057625-5 |
| title | Stochastic programming |
| title_auth | Stochastic programming |
| title_exact_search | Stochastic programming |
| title_full | Stochastic programming by András Prékopa |
| title_fullStr | Stochastic programming by András Prékopa |
| title_full_unstemmed | Stochastic programming by András Prékopa |
| title_short | Stochastic programming |
| title_sort | stochastic programming |
| topic | Stochastische Optimierung (DE-588)4057625-5 gnd |
| topic_facet | Stochastische Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020794814&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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