Lecture Notes on Optimization

140 Pages·1998·0.744 MB·English

by  Varaiya P.

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Lecture Notes on Optimization
Pravin Varaiya
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Contents
1 INTRODUCTION 1
2 OPTIMIZATIONOVERANOPENSET 7
3 Optimizationwithequalityconstraints 15
4 LinearProgramming 27
5 NonlinearProgramming 49
6 Discrete-time optimalcontrol 75
7 Continuous-timelinearoptimalcontrol 83
8 Coninuous-timeoptimalcontrol 95
9 Dynamicprograming 121
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iv CONTENTS
PREFACE to this edition
Notes on Optimization waspublished in 1971 as part of the Van Nostrand Reinhold Notes on Sys-
tem Sciences, edited by George L. Turin. Our aim was to publish short, accessible treatments of
graduate-level material in inexpensive books (the price of a book in the series was about five dol-
lars). Theeffort wassuccessful for several years. VanNostrand Reinhold wasthen purchased by a
conglomerate whichcancelled NotesonSystemSciencesbecause itwasnotsufficiently profitable.
Books have since become expensive. However, the WorldWide Webhas again madeitpossible to
publishcheaply.
Notes on Optimization has been out of print for 20 years. However, several people have been
using it as a text or as a reference in a course. They have urged me to re-publish it. The idea of
making it freely available over the Web was attractive because it reaffirmed the original aim. The
onlyobstacle wastoretypethemanuscript inLaTex. IthankKateKlohefordoingjustthat.
I would appreciate knowing if you find any mistakes in the book, or if you have suggestions for
(small)changes thatwouldimproveit.
Berkeley, California P.P.Varaiya
September, 1998
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vi CONTENTS
PREFACE
TheseNotesweredevelopedforaten-weekcourseIhavetaughtforthepastthreeyearstofirst-year
graduate students of the University of California at Berkeley. My objective has been to present,
inacompact and unifiedmanner, themainconcepts andtechniques ofmathematical programming
and optimal control to students having diverse technical backgrounds. A reasonable knowledge of
advancedcalculus(uptotheImplicitFunctionTheorem),linearalgebra(linearindependence,basis,
matrixinverse),andlineardifferentialequations(transitionmatrix,adjointsolution)issufficientfor
thereadertofollowtheNotes.
The treatment of the topics presented here is deep. Although the coverage is not encyclopedic,
an understanding of this material should enable the reader to follow much of the recent technical
literatureonnonlinearprogramming,(deterministic) optimalcontrol,andmathematicaleconomics.
TheexamplesandexercisesgiveninthetextformanintegralpartoftheNotesandmostreaderswill
need to attend to them before continuing further. Tofacilitate the useof these Notes asatextbook,
I have incurred the cost of some repetition in order to make almost all chapters self-contained.
However,ChapterVmustbereadbeforeChapterVI,andChapterVIIbeforeChapterVIII.
Theselectionoftopics,aswellastheirpresentation, hasbeeninfluencedbymanyofmystudents
andcolleagues, whohaveread andcriticized earlier drafts. Iwould especially like toacknowledge
thehelpofProfessorsM.Athans,A.Cohen,C.A.Desoer,J-P.Jacob,E.Polak,andMr. M.Ripper. I
alsowanttothankMrs. BillieVrtiakforhermarveloustypinginspiteofstartingfromanotterribly
legible handwritten manuscript. Finally, I want to thank Professor G.L. Turin for his encouraging
andpatienteditorship.
Berkeley, California P.P.Varaiya
November,1971
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Chapter 1
INTRODUCTION
In this chapter, we present our model of the optimal decision-making problem, illustrate decision-
making situations by a few examples, and briefly introduce two more general models which we
cannotdiscussfurther intheseNotes.
1.1 The Optimal Decision Problem
TheseNotesshowhowtoarriveatanoptimaldecisionassumingthatcompleteinformationisgiven.
Thephrasecompleteinformation isgivenmeansthatthefollowingrequirements aremet:
1. Thesetofallpermissible decisions isknown,and
2. Thecostofeachdecision isknown.
When these conditions are satisfied, the decisions can be ranked according to whether they incur
greater or lesser cost. An optimal decision is then any decision which incurs the least cost among
thesetofpermissible decisions.
Inordertomodeladecision-makingsituationinmathematicalterms,certainfurtherrequirements
mustbesatisfied, namely,
1. Thesetofalldecisions canbeadequately represented asasubsetofavectorspacewitheach
vectorrepresenting adecision, and
2. Thecostcorresponding tothesedecisions isgivenbyareal-valued function.
Someillustrations willhelp.
Example 1: The Pot Company (Potco) manufacturers a smoking blend called Acapulco Gold.
The blend is made up of tobacco and mary-john leaves. For legal reasons the fraction (cid:11) of mary-
johninthemixturemustsatisfy 0 < (cid:11) < 1. Fromextensivemarketresearch Potcohasdetermined
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their expected volume ofsales asafunction of(cid:11)andtheselling price p. Furthermore, tobacco can
bepurchased atafixedprice,whereasthecostofmary-john isafunction oftheamountpurchased.
If Potco wants to maximize its profits, how much mary-john and tobacco should it purchase, and
whatpricepshould itset?
Example 2: Tough University provides “quality” education to undergraduate and graduate stu-
dents. In an agreement signed with Tough’s undergraduates and graduates (TUGs), “quality” is
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2 CHAPTER1. INTRODUCTION
defined as follows: every year, each u (undergraduate) must take eight courses, one of which is a
seminarandtherestofwhicharelecturecourses,whereaseachg(graduate)musttaketwoseminars
and fivelecture courses. Aseminar cannot havemore than20 students and alecture course cannot
havemorethan40students. TheUniversityhasafacultyof1000. TheWearyOldRadicals(WORs)
haveacontractwiththeUniversitywhichstipulatesthateveryjuniorfacultymember(thereare750
of these) shall be required to teach six lecture courses and two seminars each year, whereas every
senior faculty member (there are 250 of these) shall teach three lecture courses and three seminars
each year. TheRegents ofTouch rate Tough’s President at(cid:11) points per uand (cid:12) points per g “pro-
cessed” bytheUniversity. Subjecttotheagreements withtheTUGsandWORshowmanyu’sand
g’sshouldthePresidentadmittomaximizehisrating?
Example 3: (See Figure 1.1.) An engineer is asked to construct a road (broken line) connection
pointatopointb. Thecurrentprofileofthegroundisgivenbythesolidline. Theonlyrequirement
isthatthefinalroadshouldnothaveaslopeexceeding0.001. Ifitcosts$cpercubicfoottoexcavate
orfilltheground, howshouldhedesigntheroadtomeetthespecifications atminimumcost?
Example 4: Mr. Shell is the manager of an economy which produces one output, wine. There
aretwofactors ofproduction, capital andlabor. IfK(t)andL(t)respectively are thecapital stock
used andthe labor employed attimet,then therate ofoutput ofwine W(t)attimeisgiven bythe
production function
W(t) = F(K(t);L(t))
AsManager,Mr. ShellallocatessomeoftheoutputrateW(t)totheconsumption rateC(t),and
theremainderI(t)toinvestment incapitalgoods. (Obviously, W,C,I,andK arebeingmeasured
in a common currency.) Thus, W(t) = C(t)+I(t) = (1−s(t))W(t) where s(t) = I(t)=W(t)
.
b

.
a
Figure1.1: Admissablesetofexample.
2 [0;1] isthefraction ofoutput which issaved andinvested. Suppose thatthecapital stock decays
exponentially with time at a rate (cid:14) > 0, so that the net rate of growth of capital is given by the
followingequation:
K_(t) = d K(t) (1.1)
dt
= −(cid:14)K(t)+s(t)W(t)
= −(cid:14)K(t)+s(t)F(K(t);L(t)):
Thelaborforceisgrowingataconstant birthrateof(cid:12) > 0. Hence,