Stochastic Programming

682 Pages·2003·4.571 MB·English

by  A. Ruszczynski, and A. Shapiro (Eds.)

Preface
The area of stochastic programming was created in the middle of the last
century, following fundamental achievements in linear and nonlinear
programming. While it has been quickly realized that the presence of
uncertainty in optimization models creates a need for new problem formul-
ations,manyyearshavepasseduntilthebasicstochasticprogrammingmodels
have been formulated and analyzed. Today, stochastic programming theory
offers a variety of models to address the presence of random data in
optimization problems: chance-constrained models, two- and multi-stage
models, models involving risk measures. New problem formulations appear
almost every year and this variety is one of the strengths of the field.
Stochastic programming can be quite involved, starting with sophisticated
modeling and is based on advanced mathematical tools such as nonsmooth
calculus, abstract optimization, probability theory and statistical techniques.
One of the objectives of this Handbook is to bring these techniques together
and to show how they can be used to analyze and solve stochastic program-
ming models.
Because of the inherent difficulty of stochastic optimization problems, it
took a long time until efficient solution methods have been developed. In the
last two decades a dramatic change in our abilities to solve stochastic
programming problems took place. It is partially due to the progress in large
scale linear and nonlinear programming, in nonsmooth optimization and
integer programming, but mainly it follows the development of techniques
exploiting specific properties of stochastic programming problems. Computa-
tional advances are also due to modern parallel processing technology.
Nowadays we can solve stochastic optimization problems involving tens of
millions of variables and constraints.
Our intention was to bring together leading experts in the most
important sub-fields of stochastic programming to present a rigorous
overview of basic models, methods, and applications of stochastic pro-
gramming. We hope that this Handbook will prove useful to researchers,
students, engineers and economists, who encounter in their work optimiza-
tion problems involving uncertainty. We also hope that our work will
encourage many to undertake research in this exciting and practically impor-
tant field.
v
vi Preface
We want to thank all the Authors involved in this project for their
contributions. We also want to thank Darinka Dentcheva, Shabbir Ahmed,
Tito Homem-de-Mello and Anton Kleywegt, who have helped us to review
and improve several chapters of this Handbook.
Andrzej Ruszczyn´ski and Alexander Shapiro December 2002.
Contents
Preface v
CHAPTER 1
Stochastic Programming Models
A. Ruszczyn´ ski and A. Shapiro 1
1. Introduction 1
2. Two-stage models 11
3. Multistage models 22
4. Robust and min–max approaches to stochastic optimization 48
5. Appendix 55
6. Bibliographic notes 62
References 63
CHAPTER 2
Optimality and Duality in Stochastic Programming
A. Ruszczyn´ ski and A. Shapiro 65
1. Expectation functions 65
2. Two-stage stochastic programming problems 72
3. Multistage models 93
4. Optimality conditions, basic case 97
5. Optimality conditions for multistage models 99
6. Duality, basic case 103
7. Duality for multistage stochastic programs 115
8. Min–max stochastic optimization 122
9. Appendix 126
10. Bibliographic notes 137
References 138
CHAPTER 3
Decomposition Methods
A. Ruszczyn´ ski 141
1. Introduction 141
2. The cutting plane method 144
3. Regularized decomposition 161
4. Trust region methods 175
vii
viii Contents
5. Nested cutting plane methods for multistage problems 180
6. Introduction to dual methods 187
7. The dual cutting plane method 192
8. The augmented Lagrangian method 195
9. Progressive hedging 200
10. Bibliographic notes 207
References 209
CHAPTER 4
Stochastic Integer Programming
F.V. Louveaux and R. Schultz 213
1. Introduction 213
2. Structural properties 215
3. Algorithms 235
References 264
CHAPTER 5
Probabilistic Programming
A. Pre´kopa 267
1. Model constructions 267
2. Convexity theory 272
3. Numerical solution of probabilistic constrained stochastic
programming problems 287
4. Dynamic type stochastic programming problems with
probabilistic constraints 309
5. Bounding, approximation and simulation of probabilities 311
6. Duality and stability 334
7. Selected applications 338
References 345
CHAPTER 6
Monte Carlo Sampling Methods
A. Shapiro 353
1. Introduction 353
2. Statistical properties of SAA estimators 357
3. Exponential rates of convergence 371
4. Validation analysis 382
5. Variance reduction techniques 393
6. Multistage stochastic programming 399
7. Stochastic generalized equations 410
8. Appendix 416
9. Bibliographic notes 421
References 423
Contents ix
CHAPTER 7
Stochastic Optimization and Statistical Inference
G.Ch. Pflug 427
1. Uncertain and ambiguous optimization problems 427
2. The empirical problem 430
3. Properties of statistical estimates 433
4. Risk functionals and Lipschitz properties 445
5. Arithmetic means of of i.i.d. random variables 449
6. Entropic sizes of stochastic programs 458
7. Epiconvergence 461
8. Epipointwise convergence for stochastic programs 467
9. Asymptotic stochastic programs 470
10. Bibliographic remarks 479
References 480
CHAPTER 8
Stability of Stochastic Programming Problems
W. Ro¨ misch 483
1. Introduction 483
2. General stability results 488
3. Stability of two-stage and chance constrained programs 510
4. Approximations of stochastic programs 538
5. Bibliographical notes 547
Acknowledgements 548
References 549
CHAPTER 9
Stochastic Programming in Transportation and Logistics
W.B. Powell and H. Topaloglu 555
1. Introduction 555
2. Applications and issues 557
3. Modeling framework 564
4. A case study: freight car distribution 576
5. The two-stage resource allocation problem 579
6. Multistage resource allocation problems 609
7. Some experimental results 623
8. A list of extensions 627
9. Implementing stochastic programming models in the
real world 629
10. Bibliographic notes 631
References 633
x Contents
CHAPTER 10
Stochastic Programming Models in Energy
S.W. Wallace and S.-E. Fleten 637
1. Introduction 637
2. Electricity in regulated markets 639
3. Electricity in deregulated markets 653
4. Oil 667
5. Gas 670
6. Conclusion 672
Acknowledgements 673
References 673
Subject Index 679
Contents of Previous Volumes 683
A.Ruszczyn´ski and A.Shapiro, Eds., HandbooksinOR &MS, Vol. 10
(cid:1) 2003Elsevier Science B.V.Allrights reserved.
Chapter 1
Stochastic Programming Models
Andrzej Ruszczyn´ski
DepartmentofManagementScienceandInformationSystems,RutgersUniversity,
94RockefellerRd,Piscataway,NJ08854,USA
Alexander Shapiro
SchoolofIndustrialandSystemsEngineering,GeorgiaInstituteofTechnology,Atlanta,
GA30332,USA
Abstract
In this introductory chapter we discuss some basic approaches to modeling of
stochastic optimization problems. We start with motivating examples and then
proceed to formulation of linear, and later nonlinear, two stage stochastic
programming problems. We give a functional description of two stage pro-
grams. After that we proceed to a discussion of multistage stochastic program-
ming and its connections with dynamic programming. We end this chapter by
introducing robust and min–max approaches to stochastic programming.
Finally, in the appendix, we introduce and briefly discuss some relevant
concepts from probability and optimization theories.
Key words: Two stage stochastic programming, expected value solution,
stochasticprogrammingwithrecourse,nonanticipativityconstraints,multistage
stochastic programming, dynamic programming, chance constraints, value at
risk, scenario tree, robust stochastic programming, mean–risk models.
1 Introduction
1.1 Motivation
Uncertainty is the key ingredient in many decision problems. Financial
planning, airline scheduling, unit commitment in power systems are just few
examplesofareasinwhichignoringuncertaintymayleadtoinferiororsimply
wrongdecisions.Oftenthereisavarietyofwaysinwhichtheuncertaintycanbe
1
2 A. Ruszczyn´ski and A. Shapiro
formalized and over the years various approaches to optimization under
uncertainty were developed. We discuss a particular approach based on
probabilistic models of uncertainty. By averaging possible outcomes or
consideringprobabilitiesofeventsofinterest wecandefinethe objectivesand
the constraints of the corresponding mathematical programming model.
To formulate a problem in a consistent way, a number of fundamental
assumptionsneedtobemadeaboutthenatureofuncertainty,ourknowledge
of it, and the relations of decisions to the observations made. In order to
motivate the main concepts let us start by discussing the following classical
example.
Example 1 (Newsvendor Problem). A newsvendor has to decide about the
quantity x of newspapers which he purchases from a distributor at the begin-
ningofadayatthecostofcperunit.Hecansellanewspaperatthepricesper
unitandunsoldnewspaperscanbereturnedtothevendoratthepriceofrper
unit. It is assumed that 0(cid:1)r<c<s. If the demand D, i.e., the quantity of
newspaperswhichheisabletosellataparticularday,turnsouttobegreater
thanorequaltotheorderquantityx,thenhemakestheprofitsx(cid:2)cx¼(s(cid:2)c)x,
while if D is less than x, his profit is sDþr(x(cid:2)D)(cid:2)cx¼(r(cid:2)c)xþ(s(cid:2)r)D.
ThustheprofitisafunctionofxandDandisgivenby
(cid:1)
ðs(cid:2)cÞx, if x(cid:1)D,
Fðx,DÞ¼ ð1:1Þ
ðr(cid:2)cÞxþðs(cid:2)rÞD, if x>D:
The objective of the newsvendor is to maximize his profit. We assume that
thenewsvendorisveryintelligent(hehasPh.D.degreeinmathematicsfroma
prestigious university and sells newspapers now), so he knows what he is
doing. The function F((cid:3),D) is a continuous piecewise linear function with
positive slope s(cid:2)c for x<D and negative slope r(cid:2)c for x>D. Therefore, if
thedemandDisknown,thenthebestdecisionistochoosetheorderquantity
x*¼D. However, in reality D is not known at the time the order decision
has to be made, and consequently the problem becomes more involved.
Sincethenewsvendorhasthisjobforawhilehecollecteddataandhasquitea
good idea about the probability distribution of the demand D. That is, the
demand D is viewed now as a random variable with a known, or at least well
estimated,probabilitydistributionmeasuredbythecorrespondingcumulative
distribution function (cdf) G(w):¼P(D(cid:1)w). Note that since the demand
cannotbenegative,itfollowsthatG(w)¼0foranyw<0.BytheLawofLarge
Numberstheaverageprofitoveralongperiodoftimetendstotheexpectedvalue
Z 1
E½Fðx,DÞ(cid:4)¼ Fðx,wÞdGðwÞ:
0
Ch. 1. Stochastic Programming Models 3
Therefore, from the statistical point of view it makes sense to optimize the
objective function on average, i.e., to maximize the expected profit E[F(x,D)].
This leads to the following stochastic programming problem1
Max(cid:2)fðxÞ:¼E½Fðx,DÞ(cid:4)(cid:3): ð1:2Þ
x(cid:5)0
Notethatwetreatherexasacontinuousratherthanintegervariable.This
makes sense if the quantity of newspapers x is reasonably large.
Inthepresentcaseitisnotdifficulttosolvetheaboveoptimizationproblem
in a closed form. Let us observe that for any D(cid:5)0, the function F((cid:3),D)
isconcave(andpiecewiselinear).Therefore,theexpectedvaluefunctionf((cid:3))is
also concave. Suppose for a moment that G((cid:3)) is continuous at a point x>0.
Then
Z x Z 1
fðxÞ¼ ½ðr(cid:2)cÞxþðs(cid:2)rÞw(cid:4)dGðwÞþ ðs(cid:2)cÞxdGðwÞ:
0 x
Using integration by parts it is possible to calculate then that
Z x
fðxÞ¼ðs(cid:2)cÞx(cid:2)ðs(cid:2)rÞ GðwÞdw: ð1:3Þ
0
The function f((cid:3)) is concave, and hence continuous, and therefore formula
(1.3) holds even if G((cid:3)) is discontinuous at x. It follows that f((cid:3)) is
differentiable at x iff (that is, if and only if) G((cid:3)) is continuous at x, in which
case
f0ðxÞ¼s(cid:2)c(cid:2)ðs(cid:2)rÞGðxÞ: ð1:4Þ
Consider the inverse G(cid:2)1((cid:1)):¼min{x:G(x)(cid:5)(cid:1)} function2 of the cdf G,
which is defined for (cid:1)2(0,1). Since f((cid:3)) is concave, a necessary and sufficient
condition for x*>0 to be an optimal solution of problem (1.2) is that
f0(x*)¼0, provided that f((cid:3)) is differentiable at x*. Note that because
r<c<s, it follows that 0<(s(cid:2)c)/(s(cid:2)r)<1. Consequently, an optimal
solution of (1.2) is given by
(cid:4)s(cid:2)c(cid:5)
x* ¼G(cid:2)1 : ð1:5Þ
s(cid:2)r
This holds even if G((cid:3)) is discontinuous at x*. It is interesting to note
that G(0) is equal to the probability that the demand D is zero, and
1 Thenotation‘‘:¼’’meansequalbydefinition.
2 RecallthatG(cid:2)1((cid:1))iscalledthe(cid:1)-quantileofthecdfG.
4 A. Ruszczyn´ski and A. Shapiro
hence if this probability is positive and (s(cid:2)c)/(s(cid:2)r)(cid:1)G(0), then the optimal
solution x*¼0.
Clearly the above approach explicitly depends on the knowledge of the
probability distribution of the demand D. In practice the corresponding cdf
G((cid:3))isneverknownexactlyandcouldbeapproximated(estimated)atbest.In
thepresentcasetheoptimalsolutionisgiveninaclosedformandthereforeits
dependenceon G((cid:3)) can be easily evaluated. Itis well known that (cid:1)-quantiles
are robust (stable) with respect to small perturbations of the corresponding
cdf G((cid:3)), providedthat (cid:1) is nottoo close to0 or 1. In general, it is important
toinvestigatesensitivityofaconsideredstochasticprogrammingproblemwith
respect to the assumed probability distributions.
The following deterministic optimization approach is also often used for
decision making under uncertainty. The random variable D is replaced by its
mean (cid:2)¼E[D], and then the following deterministic optimization problem is
solved:
Max Fðx,(cid:2)Þ: ð1:6Þ
x(cid:5)0
A resulting optimal solution x is sometimes called the expected value
solution. In the present example, the optimal solution of this deterministic
optimization problem is x¼(cid:2). Note that the mean solution x can be very
differentfromthesolutionx* givenin(1.5).Itiswellknownthatthequantiles
are much more stable to variations of the cdf G than the corresponding
mean value. Therefore, the optimal solution x* of the stochastic optimization
problem is more robust with respect to variations of the probability
distributionsthananoptimalsolutionxofthecorrespondingdeterministicopti-
mization problem. This should be not surprising since the deterministic
problem (1.6) can be formulated in the framework of the stochastic
programming problem (1.2) by considering the trivial distribution of D being
identically equal to (cid:2).
For any x, F(x,D) is concave in D. Therefore the following Jensen’s
inequality holds:
Fðx, (cid:2)Þ(cid:5)E½Fðx, DÞ(cid:4):
Hence
max Fðx,(cid:2)Þ(cid:5)max E½Fðx,DÞ(cid:4):
x(cid:5)0 x(cid:5)0
Thus the optimal value of the deterministic optimization problem is
biased upward relative to the optimal value of the stochastic optimization
problem. This should be also not surprising since the optimization problem