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Introduction to stochastic programming(Second edition).pdf

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    Cover
    Springer Series in Operations Research and Financial Engineering
    Introduction to Stochastic Programming, Second Edition
    ISBN 9781461402367
    Preface
    Preface to the First Edition
    Contents
    Notation
    Part I: Models
    Chapter 1: Introduction and Examples
    1.1 A Farming Example and the News Vendor Problem
    a. The farmer's problem
    b. A scenario representation
    c. General model formulation
    d. Continuous random variables
    e. The news vendor problem
    1.2 Financial Planning and Control
    1.3 Capacity Expansion
    1.4 Design for Manufacturing Quality
    1.5 A Routing Example
    a. Presentation
    b. Wait-and-see solutions
    c. Expected value solution
    d. Recourse solution
    e. Other random variables
    f. Chance-constraints
    1.6 Other Applications
    Chapter 2: Uncertainty and Modeling Issues
    2.1 Probability Spaces and Random Variables
    2.2 Deterministic Linear Programs
    2.3 Decisions and Stages
    2.4 Two-Stage Program with Fixed Recourse
    a. Fixed distribution pattern, fixed demand,ri, vj, tij stochastic
    b. Fixed distribution pattern, uncertain demand
    c. Uncertain demand, variable distribution pattern
    d. Stages versus periods; Two-stage versus multistage
    2.5 Random Variables and Risk Aversion
    2.6 Implicit Representation of the Second Stage
    a. A closed form expression is available for Q(x)
    b. For a given x, Q(x) is computable
    2.7 Probabilistic Programming
    a. Deterministic linear equivalent: a direct case
    b. Deterministic linear equivalent: an indirect case
    c. Deterministic nonlinear equivalent: the case of random constraint coefficients
    2.8 Modeling Exercise
    a. Presentation
    b. Discussion of solutions
    2.9 Alternative Characterizations and Robust Formulations
    2.10 Relationship to Other Decision-Making Models
    a. Statistical decision theory and decision analysis
    b. Dynamic programming and Markov decision processes
    c. Machine learning and online optimization
    d. Optimal stochastic control
    e. Summary
    2.11 Short Reviews
    a. Linear programming
    b. Duality for linear programs
    c. Nonlinear programming and convex analysis
    Part II: Basic Properties
    Chapter 3: Basic Properties and Theory
    3.1 Two-Stage Stochastic Linear Programs with Fixed Recourse
    a. Formulation
    b. Discrete random variables
    c. General cases
    d. Special cases: relatively complete, complete,and simple recourse
    e. Optimality conditions and duality
    f. Stability and nonanticipativity
    3.2 Probabilistic or Chance Constraints
    a. General case
    b. Probabilistic constraints with discrete random variables
    3.3 Stochastic Integer Programs
    a. Recourse problems
    b. Simple integer recourse
    c. Probabilistic constraints
    3.4 Multistage Stochastic Programs with Recourse
    3.5 Stochastic Nonlinear Programs with Recourse
    Chapter 4: The Value of Information and the Stochastic Solution
    4.1 The Expected Value of Perfect Information
    4.2 The Value of the Stochastic Solution
    4.3 Basic Inequalities
    4.4 The Relationship between EVPI and VSS
    a. EVPI = 0 and VSS =0
    b. VSS = 0 and EVPI=0
    4.5 Examples
    4.6 Bounds on EVPI and VSS
    Part III: Solution Methods
    Chapter 5: Two-Stage Recourse Problems
    5.1 The L-Shaped Method
    a. Optimality cuts
    b. Feasibility cuts
    c. Proof of convergence
    d. The multicut version
    5.2 Regularized Decomposition
    5.3 The Piecewise Quadratic Form of the L-shaped Methods
    5.4 Bunching and Other Efficiencies
    a. Full decomposability
    b. Bunching
    5.5 Basis Factorization and Interior Point Methods
    5.6 Inner Linearization Methods and Special Structures
    5.7 Simple and Network Recourse Problems
    5.8 Methods Based on the Stochastic Program Lagrangian
    5.9 Additional Methods and Complexity Results
    Chapter 6: Multistage Stochastic Programs
    6.1 Nested Decomposition Procedures
    6.2 Quadratic Nested Decomposition
    6.3 Block Separability and Special Structure
    6.4 Lagrangian-Based Methods for Multiple Stages
    Chapter 7: Stochastic Integer Programs
    7.1 Stochastic Integer Programs and LP-Relaxation
    7.2 First-stage Binary Variables
    a. Improved optimality cuts
    b. Example with continuous random variables
    7.3 Second-stage Integer Variables
    a. Looking in the space of tenders
    b. Discontinuity points
    c. Algorithm
    7.4 Reformulation
    a. Difficulties of reformulation in stochastic integer programs
    b. Disjunctive cuts
    c. First-stage dependence
    d. An algorithm
    7.5 Simple Integer Recourse
    a. restricted to be integer
    b. The case where S=1, not integral
    7.6 Cuts Based on Branching in the Second Stage
    a. Feasibility cuts
    b. Optimality cuts
    7.7 Extensive Forms and Decomposition
    7.8 Short Reviews
    a. Branch-and-bound
    b. A simple example of valid inequalities
    c. Disjunctive cuts
    Part IV: Approximation and Sampling Methods
    Chapter 8: Evaluating and Approximating Expectations
    8.1 Direct Solutions with Multiple Integration
    8.2 Discrete Bounding Approximations
    8.3 Using Bounds in Algorithms
    8.4 Bounds in Chance-Constrained Problems
    8.5 Generalized Bounds
    a. Extensions of basic bounds
    b. Bounds based on separable functions
    c. General-moment bounds
    8.6 General Convergence Properties
    Chapter 9: Monte Carlo Methods
    9.1 Sample Average Approximation and Importance Samplingin the L-Shaped Method
    9.2 Stochastic Decomposition
    9.3 Stochastic Quasi-Gradient Methods
    9.4 Sampling Methods for Probabilistic Constraints and Quantiles
    9.5 General Results for Sample Average Approximation and Sequential Sampling
    Chapter 10: Multistage Approximations
    10.1 Extensions of the Jensen and Edmundson-Madansky Inequalities
    10.2 Bounds Based on Aggregation
    10.3 Scenario Generation and Distribution Fitting
    10.4 Multistage Sampling and Decomposition Methods
    10.5 Approximate Dynamic Programming and Special Cases
    a. Network revenue management
    b. Vehicle allocation problems
    c. Piecewise-linear separable bounds
    d. Nonlinear bounds and a production planning example
    e. Extensions
    Appendix A Sample Distribution Functions
    A.1 Discrete Random Variables
    A.2 Continuous Random Variables
    References
    Author Index
    Subject Index
    Springer Series in Operations Research and Financial Engineering Series Editors: Thomas V. Mikosch University of Copenhagen Laboratory of Actuarial Mathematics DK-1017 Copenhagen Denmark mikosh@act.ku.dk Sidney I. Resnick Cornell University School of Operations Research and Industrial Engineering Ithaca, NY 14853 U.S.A. sirl@cornell.edu Stephen M. Robinson University of Wisconsin-Madison Department of Industrial Engineering Madison, WI 53706 U.S.A. smrobins@wisc.edu For further volumes: http://www.springer.com/series/3182
    John R. Birge • Franc¸ois Louveaux Introduction to Stochastic Programming Second Edition 123
    John R. Birge Booth School of Business University of Chicago 5807 South Woodlawn Avenue Chicago, Illinois 60637 USA john.birge@chicagobooth.edu Franc¸ois Louveaux Department of Business Administration University of Namur Rempart de la Vierge 8 B-5000, Namur Belgium francois.louveaux@fundp.ac.be ISSN 1431-8598 ISBN 978-1-4614-0236-7 DOI 10.1007/978-1-4614-0237-4 Springer New York Dordrecht Heidelberg London e-ISBN 978-1-4614-0237-4 Library of Congress Control Number: 2011929942 Mathematics Subject Classification (2010): 37N40, 46N10, 49L20, 49Mxx (all), 49N30, 49N15, 90-01, 90B50, 90C05, 90C06, 90C15, 90C39 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
    To Richard and Joelle, Sebastien, J´erˆome, Quentin, and G´eraldine.
    Preface Since the publication of the first edition of this book, we have been encouraged by the growing interest in stochastic programming and its application in a variety of areas, including routine use in many industries from transportation and logistics to finance and energy. We have also been heartened by the many new methodological and theoretical advances within the field. In this second edition, we have attempted to capture aspects of both recent applications and models as well as new practically relevant methods and theory. As in the first edition, our primary goal is to provide students and other readers with an appreciation of how to build uncertainty into an optimization model, what differences in decisions might result from recognizing the presence of uncertainty, and how and what kinds of models are amenable to solution. We have focused the second edition on satisfying these main objectives while also uncovering basic research questions to give beginning researchers a foundation upon which to build more in-depth knowledge. To help make the relevant issues in modeling, solving, and analyzing stochas- tic programs more evident, we have incorporated more examples than in the first edition so that the each of the main modeling, solution, and analysis processes are illustrated with a detailed example. We have also added many exercises whose so- lutions provide additional insights into stochastic programming concepts and tools. Many of these exercises assume the availability of software to solve basic linear and nonlinear optimization models and to construct algorithmic procedures involv- ing matrix operations. Since we view completing these exercises as a key part of understanding the material, instructors should ensure that students have adequate programming skills to implement the methods described in the book. Besides additional examples and exercises throughout the book, we have re- organized the material to improve the logical flow and to eliminate unnecessary or complicating issues before explaining the most practically relevant material. Spe- cific changes in the second edition include the following: • a new section (Section 1.5) and routing example in Chapter 1; • a worked-out modeling exercise (Section 2.8) and a section on risk modeling and robust formulation (Section 2.9 in Chapter 2; vii
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