Stochastic Programming

326 Pages·2003·1.851 MB·English

by  Kall P., Wallace S.

Stochastic Programming
Second Edition
Peter Kall
Institutefor Operations Research
and Mathematical Methods of Economics
University of Zurich
CH-8044 Zurich
Stein W. Wallace
Molde UniversityCollege
P.O. Box 2110
N-6402 Molde, Norway
Reference to this text is “Peter Kall and Stein W. Wallace, Stochastic
Programming, John Wiley & Sons, Chichester, 1994”. The text is printed
with permission from the authors. The publisher reverted the rights to the
authorsonFebruary4,2003.Thistextisslightlyupdatedfromthe published
version.
ii STOCHASTICPROGRAMMING
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 A numerical example . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Scenario analysis . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Using the expected value of p . . . . . . . . . . . . . . . 3
1.1.4 Maximizing the expected value of the objective . . . . . 4
1.1.5 The IQ of hindsight . . . . . . . . . . . . . . . . . . . . 5
1.1.6 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Stochastic Programs:General Formulation . . . . . . . . . . . . 21
1.4.1 Measures and Integrals. . . . . . . . . . . . . . . . . . . 21
1.4.2 Deterministic Equivalents . . . . . . . . . . . . . . . . . 31
1.5 Properties of Recourse Problems . . . . . . . . . . . . . . . . . 36
1.6 Properties of Probabilistic Constraints . . . . . . . . . . . . . . 46
1.7 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . 53
1.7.1 The Feasible Set and Solvability . . . . . . . . . . . . . 54
1.7.2 The Simplex Algorithm . . . . . . . . . . . . . . . . . . 64
1.7.3 Duality Statements . . . . . . . . . . . . . . . . . . . . . 70
1.7.4 A Dual Decomposition Method . . . . . . . . . . . . . . 75
1.8 Nonlinear Programming . . . . . . . . . . . . . . . . . . . . . . 80
1.8.1 The Kuhn–Tucker Conditions . . . . . . . . . . . . . . . 83
1.8.2 Solution Techniques . . . . . . . . . . . . . . . . . . . . 89
1.8.2.1 Cutting-plane methods . . . . . . . . . . . . . 90
1.8.2.2 Descent methods . . . . . . . . . . . . . . . . . 93
1.8.2.3 Penalty methods . . . . . . . . . . . . . . . . . 97
1.8.2.4 Lagrangianmethods . . . . . . . . . . . . . . . 98
1.9 Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
iv STOCHASTICPROGRAMMING
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2 Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.1 The Bellman Principle . . . . . . . . . . . . . . . . . . . . . . . 110
2.2 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . 117
2.3 Deterministic Decision Trees. . . . . . . . . . . . . . . . . . . . 121
2.4 Stochastic Decision Trees . . . . . . . . . . . . . . . . . . . . . 124
2.5 Stochastic Dynamic Programming . . . . . . . . . . . . . . . . 130
2.6 Scenario Aggregation . . . . . . . . . . . . . . . . . . . . . . . . 134
2.6.1 Approximate Scenario Solutions . . . . . . . . . . . . . 141
2.7 Financial Models . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.7.1 The Markowitz’ model . . . . . . . . . . . . . . . . . . . 142
2.7.2 Weak aspects of the model . . . . . . . . . . . . . . . . 143
2.7.3 More advanced models . . . . . . . . . . . . . . . . . . . 145
2.7.3.1 A scenario tree . . . . . . . . . . . . . . . . . . 145
2.7.3.2 The individual scenario problems. . . . . . . . 145
2.7.3.3 Practicalconsiderations . . . . . . . . . . . . . 147
2.8 Hydro power production . . . . . . . . . . . . . . . . . . . . . . 147
2.8.1 A small example . . . . . . . . . . . . . . . . . . . . . . 148
2.8.2 Further developments . . . . . . . . . . . . . . . . . . . 150
2.9 The Value of Using a Stochastic Model . . . . . . . . . . . . . . 151
2.9.1 Comparing the Deterministic and Stochastic Objective
Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
2.9.2 Deterministic Solutions in the Event Tree . . . . . . . . 152
2.9.3 Expected Value of Perfect Information . . . . . . . . . . 154
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3 Recourse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3.1 Outline of Structure . . . . . . . . . . . . . . . . . . . . . . . . 159
3.2 The L-shaped Decomposition Method . . . . . . . . . . . . . . 161
3.2.1 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.2.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . 168
3.3 Regularized Decomposition . . . . . . . . . . . . . . . . . . . . 173
3.4 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
3.4.1 The Jensen Lower Bound . . . . . . . . . . . . . . . . . 179
3.4.2 Edmundson–Madansky Upper Bound . . . . . . . . . . 181
3.4.3 Combinations . . . . . . . . . . . . . . . . . . . . . . . . 184
3.4.4 A Piecewise Linear Upper Bound . . . . . . . . . . . . . 185
3.5 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
3.5.1 Refinementsoftheboundsonthe“Wait-and-See”Solution190
3.5.2 Using the L-shaped Method within Approximation
Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.5.3 What is a Good Partition? . . . . . . . . . . . . . . . . 203
CONTENTS v
3.6 Simple Recourse . . . . . . . . . . . . . . . . . . . . . . . . . . 205
3.7 Integer First Stage . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.7.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . 216
3.7.2 Feasibility Cuts . . . . . . . . . . . . . . . . . . . . . . . 216
3.7.3 Optimality Cuts . . . . . . . . . . . . . . . . . . . . . . 217
3.7.4 Stopping Criteria . . . . . . . . . . . . . . . . . . . . . . 217
3.8 Stochastic Decomposition . . . . . . . . . . . . . . . . . . . . . 217
3.9 Stochastic Quasi-GradientMethods. . . . . . . . . . . . . . . . 225
3.10 Solving Many Similar Linear Programs . . . . . . . . . . . . . . 229
3.10.1 Randomness in the Objective . . . . . . . . . . . . . . . 232
3.11 Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . 233
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
4 Probabilistic Constraints . . . . . . . . . . . . . . . . . . . . . . 243
4.1 Joint Chance Constrained Problems . . . . . . . . . . . . . . . 245
4.2 Separate Chance Constraints . . . . . . . . . . . . . . . . . . . 247
4.3 Bounding Distribution Functions . . . . . . . . . . . . . . . . . 249
4.4 Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . 257
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
5 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
5.1 Problem Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 261
5.1.1 Finding a Frame . . . . . . . . . . . . . . . . . . . . . . 262
5.1.2 Removing Unnecessary Columns . . . . . . . . . . . . . 263
5.1.3 Removing Unnecessary Rows . . . . . . . . . . . . . . . 264
5.2 Feasibility in Linear Programs. . . . . . . . . . . . . . . . . . . 265
5.2.1 A Small Example . . . . . . . . . . . . . . . . . . . . . . 271
5.3 Reducing the Complexity of Feasibility Tests . . . . . . . . . . 273
5.4 Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . 274
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6 Network Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 277
6.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.2 Feasibility in Networks . . . . . . . . . . . . . . . . . . . . . . . 280
6.2.1 The uncapacitated case . . . . . . . . . . . . . . . . . . 286
6.2.2 Comparing the LP and Network Cases . . . . . . . . . . 287
6.3 Generating Relatively Complete Recourse . . . . . . . . . . . . 288
6.4 An Investment Example . . . . . . . . . . . . . . . . . . . . . . 290
6.5 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
6.5.1 Piecewise Linear Upper Bounds . . . . . . . . . . . . . . 295
vi STOCHASTICPROGRAMMING
6.6 Project Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.6.1 PERT as a Decision Problem . . . . . . . . . . . . . . . 303
6.6.2 Introduction of Randomness. . . . . . . . . . . . . . . . 303
6.6.3 Bounds on the Expected Project Duration . . . . . . . . 304
6.6.3.1 Series reductions . . . . . . . . . . . . . . . . . 305
6.6.3.2 Parallelreductions . . . . . . . . . . . . . . . . 305
6.6.3.3 Disregarding path dependences . . . . . . . . . 305
6.6.3.4 Arc duplications . . . . . . . . . . . . . . . . . 306
6.6.3.5 Using Jensen’s inequality . . . . . . . . . . . . 306
6.7 Bibliographical Notes. . . . . . . . . . . . . . . . . . . . . . . . 307
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Preface
Overthe lastfew years,both of the authors,and alsomost othersin the field
of stochastic programming,have said that what we need more than anything
just now is a basic textbook—a textbook that makes the area available not
onlytomathematicians,butalsotostudentsandotherinterestedpartieswho
cannot or will not try to approach the field via the journals. We also felt
the need to provide an appropriate text for instructors who want to include
the subject in their curriculum. It is probably not possible to write such a
book without assuming some knowledge of mathematics, but it has been our
clear goalto avoidwriting a text readable only for mathematicians. We want
the book to be accessible to any quantitatively minded student in business,
economics, computer science and engineering, plus, of course, mathematics.
So what do we mean by a quantitatively minded student? We assume that
the reader of this book has had a basic course in calculus, linear algebra
and probability. Although most readers will have a background in linear
programming(whichreplacesthe needfora specific coursein linearalgebra),
we provide an outline of all the theory we need from linear and nonlinear
programming. We have chosen to put this material into Chapter 1, so that
the reader who is familiar with the theory can drop it, and the reader who
knows the material, but wonders about the exact definition of some term, or
who is slightly unfamiliar with our terminology, can easily check how we see
things. We hope that instructors will find enough material in Chapter 1 to
cover specific topics that may have been omitted in the standard book on
optimization used in their institution. By putting this material directly into
the running text, we have made the book more readable for those with the
minimalbackground.But,atthesametime,wehavefounditbesttoseparate
what is new in this book—stochastic programming—from more standard
material of linear and nonlinear programming.
Despite this clear goal concerning the level of mathematics, we must
admit that when treating some of the subjects, like probabilistic constraints
(Section 1.6 and Chapter 4), or particular solution methods for stochastic
programs, like stochastic decomposition (Section 3.8) or quasi-gradient
viii STOCHASTICPROGRAMMING
methods (Section3.9),we havehadto use a slightly moreadvancedlanguage
in probability. Although the actual information found in those parts of the
book is made simple, some terminology may here and there not belong to
the basic probability terminology.Hence, for these parts, the instructor must
either provide some basic backgroundin terminology,or the reader should at
leastconsultcarefullySection1.4.1,wherewehavetriedtoputtogetherthose
terms and concepts from probability theory used later in this text.
Within the mathematical programming community, it is common to split
the field into topics such as linear programming, nonlinear programming,
networkflows,integerandcombinatorialoptimization,and,finally,stochastic
programming. Convenient as that may be, it is conceptually inappropriate.
It puts forwardthe idea that stochastic programming is distinct from integer
programmingthesamewaythatlinearprogrammingisdistinctfromnonlinear
programming. The counterpart of stochastic programming is, of course,
deterministic programming. We have stochastic and deterministic linear
programming,deterministicandstochasticnetworkflowproblems,andsoon.
Althoughthisbookmostlycoversstochasticlinearprogramming(sincethatis
the best developed topic), we also discuss stochastic nonlinear programming,
integer programming and network flows.
Since we have let subject areas guide the organization of the book, the
chapters are of rather different lengths. Chapter 1 starts out with a simple
examplethatintroducesmanyoftheconceptstobeusedlateron.Temptingas
it may be, we strongly discourage skipping these introductory parts. If these
parts are skipped, stochastic programming will come forward as merely an
algorithmicandmathematicalsubject,whichwillservetolimittheusefulness
of the field. In addition to the algorithmic and mathematical facets of the
field, stochastic programming also involves model creation and specification
of solution characteristics. All instructors know that modelling is harder to
teach than are methods. We are sorry to admit that this difficulty persists
in this text as well. That is, we do not provide an in-depth discussion of
modellingstochasticprograms.Thetextisnotfreefromdiscussionsofmodels
andmodelling,however,anditisourstrongbeliefthatacoursebasedonthis
text is better (and also easier to teach and motivate) when modelling issues
are included in the course.
Chapter 1 contains a formal approach to stochastic programming, with a
discussion of different problem classes and their characteristics. The chapter
ends with linear and nonlinear programming theory that weighs heavily in
stochastic programming. The reader will probably get the feeling that the
parts concerned with chance-constrained programming are mathematically
more complicated than some parts discussing recourse models. There is a
good reason for that: whereas recourse models transform the randomness
containedinastochasticprogramintoonespecialparameterofsomerandom
vector’s distribution, namely its expectation, chance constrainedmodels deal
PREFACE ix
more explicitly with the distribution itself. Hence the latter models may
be more difficult, but at the same time they also exhaust more of the
informationcontainedintheprobabilitydistribution.However,withrespectto
applications,thereisnogenerallyvalidjustificationtostatethatanyoneofthe
two basic model types is “better” or “more relevant”.As a matter of fact, we
know of applications for which the recourse model is very appropriate and of
othersforwhichchanceconstraintshavetobemodelled,andevenapplications
are known for which recourse terms for one part of the stochastic constraints
andchanceconstraintsforanotherpartweredesigned.Hence,inafirstreading
or an introductory course, one or the other proof appearing too complicated
cancertainlybe skippedwithout harm.However,to geta validpicture about
stochasticprogramming,thestatementsaboutbasicpropertiesofbothmodel
typesaswellastheideasunderlyingthevarioussolutionapproachesshouldbe
noticed.Althoughthebasiclinearandnonlinearprogrammingisputtogether
in one specific part of the book, the instructor or the reader should pick up
the subjects as they are needed for the understanding of the other chapters.
That way, it will be easier to pick out exactly those parts of the theory that
the students or readers do not know already.
Chapter 2 starts out with a discussion of the Bellman principle for
solving dynamic problems, and then discusses decision trees and dynamic
programminginbothdeterministicandstochasticsettings.Therethenfollows
a discussionofthe rathernew approachofscenarioaggregation.We conclude
the chapter with a discussion of the value of using stochastic models.
Chapter 3 covers recourse problems. We first discuss some topics from
Chapter 1 in more detail. Then we consider decomposition procedures
especially designed for stochastic programs with recourse. We next turn to
the questions of bounds and approximations, outlining some major ideas
and indicating the direction for other approaches. The special case of simple
recourseis then explained,before we show how decompositionprocedures for
stochastic programs fit into the framework of branch-and-cut procedures for
integerprograms.Thismakesitpossibletodevelopanapproachforstochastic
integer programs. We conclude the chapter with a discussion of Monte-Carlo
based methods, in particular stochastic decomposition and quasi-gradient
methods.
Chapter 4 is devoted to probabilistic constraints. Based on convexity
statementsprovidedinSection1.6,oneparticularsolutionmethodisdescribed
forthecaseofjointchanceconstraintswithamultivariatenormaldistribution
of the right-hand side. For separate probabilistic constraints with a joint
normal distribution of the coefficients, we show how the problem can be
transformed into a deterministic convex nonlinear program. Finally, we
address a problem very relevant in dealing with chance constraints: the
problem of how to construct efficiently lower and upper bounds for a
multivariate distribution function, and give a first sketch of the ideas used
x STOCHASTICPROGRAMMING
in this area.
Preprocessing is the subject of Chapter 5. “Preprocessing” is any analysis
thatiscarriedoutbeforetheactualsolutionprocedureiscalled.Preprocessing
canbeusefulforsimplifyingcalculations,butthemainpurposeistofacilitate
a tool for model evaluation.
We conclude the book with a closer look at networks (Chapter 6). Since
these arenothing else thanspecially structuredlinear programs,we candraw
freelyfromthetopicsinChapter3.However,the addedstructureofnetworks
allowsmanysimplifications.We discussfeasibility,preprocessingandbounds.
We conclude the chapter with a closer look at PERT networks.
Each chapter ends with a short discussion of where more literature can be
found, some exercises, and, finally, a list of references.
Writingthisbookhasbeenbothinterestinganddifficult.Sinceitisthefirst
basic textbook totally devoted to stochastic programming, we both enjoyed
and suffered from the fact that there is, so far, no experience to suggest how
such a book should be constructed. Are the chapters in the correct order?
Is the level of difficulty even throughout the book? Have we really captured
the basics of the field? In all cases the answer is probably NO. Therefore,
dear reader, we appreciate all comments you may have, be they regarding
misprints,plainerrors,orsimply goodideasabouthowthis shouldhavebeen
done. And also, if you produce suitable exercises, we shall be very happy to
receive them, and if this book ever gets revised, we shall certainly add them,
and allude to the contributor.
About 50% of this text served as a basis for a course in stochastic
programmingatTheNorwegianInstituteofTechnologyinthefallof1992.We
wish to thank the students for putting up with a very preliminary text, and
for finding such an astonishing number of errors and misprints. Last but not
least, we owe sincere thanks to Julia Higle (University of Arizona, Tucson),
DiethardKlatte(UniverityofZurich),JanosMayer(UniversityofZurich)and
Pavel Popela (Technical University of Brno) who have read the manuscript1
verycarefullyandfixednotonlylinguisticbugsbutpreventedusfromquitea
numberofcrucialmistakes.Finallywehighlyappreciatethegoodcooperation
and very helpful comments provided by our publisher. The remaining errors
are obviously the sole responsibility of the authors.
Zurich and Trondheim, February 1994 P. K. and S.W.W.
1 WritteninLATEX