Introduction to Stochastic Programming

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Springer Science & Business Media, Feb 2, 2000 - Mathematics - 421 pages
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The aim of stochastic programming is to find optimal decisions in problems which involve uncertain data. This field is currently developing rapidly with contributions from many disciplines including operations research, mathematics, and probability. Conversely, it is being applied in a wide variety of subjects ranging from agriculture to financial planning and from industrial engineering to computer networks. This textbook provides a first course in stochastic programming suitable for students with a basic knowledge of linear programming, elementary analysis, and probability. The authors aim to present a broad overview of the main themes and methods of the subject. Its prime goal is to help students develop an intuition on how to model uncertainty into mathematical problems, what uncertainty changes bring to the decision process, and what techniques help to manage uncertainty in solving the problems. The first chapters introduce some worked examples of stochastic programming and demonstrate how a stochastic model is formally built. Subsequent chapters develop the properties of stochastic programs and the basic solution techniques used to solve them. Three chapters cover approximation and sampling techniques and the final chapter presents a case study in depth. A wide range of students from operations research, industrial engineering, and related disciplines will find this a well-paced and wide-ranging introduction to this subject.
 

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Contents

Introduction and Examples
3
11 A Farming Example and the News Vendor Problem
4
12 Financial Planning and Control
20
13 Capacity Expansion
28
14 Design for Manufacturing Quality
37
15 Other Applications
42
Uncertainty and Modeling Issues
49
22 Deterministic Linear Programs
51
64 Nonlinear Programming in Simple Recourse Problems
225
65 Other Nonlinear ProgrammingBased Methods
231
Multistage Stochastic Programs
233
71 Nested Decomposition Procedures
234
72 Quadratic Nested Decomposition
244
73 Other Approaches to Multiple Stages
251
Stochastic Integer Programs
253
82 Simple Integer Recourse
262

23 Decisions and Stages
52
24 TwoStage Program with Fixed Recourse
54
25 Random Variables and Risk Aversion
61
26 Implicit Representation of the Second Stage
63
27 Probabilistic Programming
64
28 Relationship to Other DecisionMaking Models
67
29 Short Reviews
73
Basic Properties
81
Basic Properties and Theory
83
31 TwoStage Stochastic Linear Programs with Fixed Recourse
84
32 Probabilistic or Chance Constraints
103
33 Stochastic Integer Programs
109
34 TwoStage Stochastic Nonlinear Programs with Recourse
122
35 Multistage Stochastic Programs with Recourse
128
The Value of Information and the Stochastic Solution
137
42 The Value of the Stochastic Solution
139
43 Basic Inequalities
140
44 The Relationship between EVPI and VSS
141
45 Examples
144
46 Bounds on EVPI and VSS
145
Solution Methods
153
TwoStage Linear Recourse Problems
155
51 The LShaped Method
156
52 Feasibility
163
53 The Multicut Version
166
54 Bunching and Other Efficiencies
169
55 Inner Linearization Methods
174
56 Basis Factorization Methods
179
57 Special CasesSimple Recourse and Network Problems
192
Nonlinear Programming Approaches to TwoStage Recourse Problems
199
62 The Piecewise Quadratic Form of the LShaped Method
206
63 Methods Based on the Stochastic Program Lagrangian
215
83 Binary FirstStage Variables
268
84 Other Approaches
276
Approximation and Sampling Methods
283
Evaluating and Approximating Expectations
285
91 Direct Solutions with Multiple Integration
286
92 Discrete Bounding Approximations
288
93 Using Bounds in Algorithms
296
94 Bounds in ChanceConstrained Problems
301
95 Generalized Bounds
305
96 General Convergence Properties
323
Monte Carlo Methods
331
101 General Results for Sampled Problems
332
102 Using Sampling in the LShaped Method
335
103 Stochastic QuasiGradient Methods
343
Uses with Analytical and Empirical Observations
349
Multistage Approximations
353
111 Bounds Based on the Jensen and EdmundsonMadansky Inequalities
354
112 Bounds Based on Aggregation
359
113 Bounds Based on Separable Responses
362
114 Bounds for Specific Problem Structures
366
A Case Study
373
Capacity Expansion
375
121 Model Development
376
122 Demand Distribution Modeling
382
123 Computational Comparisons
383
Sample Distribution Functions
385
A2 Continuous Random Variables
386
References
387
Author Index
411
Subject Index
415
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Page 392 - Decision Problems Under Risk and Chance Constrained Programming: Dilemmas in the Transition'," Management Sci., 29, 6 (June 1983).
Page 409 - Convergence of convex functions, variational inequalities and convex optimization problems" in: RW Cottle, F. Giannessi and J.-L. Lions, Eds., Variational Inequalities and Complementarity Problems (John Wiley, Inc., New York, NY, 1980a) pp. 375-404. [329] RJ-B Wets, "Stochastic multipliers, induced feasibility and nonanticipativity in stochastic programming" in: MAH Dempster, Ed., Stochastic Programming (Academic Press, New York, NY, 1980b).
Page 387 - A combined phase I-phase II projective algorithm for linear programming," Mathematical Programming, 43 (1989), pp.
Page 409 - Large-scale linear programming techniques in stochastic programming" in: Y. Ermoliev and R. Wets, Eds., Numerical Techniques for Stochastic Optimization (Springer- Verlag, Berlin, 1988). [333] RJ-B Wets, "Stochastic programming" in: GL Nemhauser, AHG Rinnooy Kan, and MJ Todd, Eds., Optimization (Handbooks in Operations Research and Management Science; Vol.
Page 409 - Algorithms for frames and lineality spaces of cones," Journal of Research of the National Bureau of Standards — B Mathematics and Mathematical Physics, Vol.
Page 409 - Programming under uncertainty: the equivalent convex program," SIAM Journal on Applied Mathematics 14 (1966) pp. 89-105. [326] RJ-B Wets, "Characterization theorems for stochastic programs," Mathematical Programming 2 (1972) pp.

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