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VenkoluromnP. Applied oplimiZlllion with MATLAB~ Programming I P. Venkalarmlln.

p. com. A Wiley-Intersciencc publicnlion." ISBN 0-471 958.5(c1oth : alk. pnper) 1. MnlhemalicaJ optimization--Dolll procing2.MATLAB. J. Title.

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6

CONTENTS

PREFACE

1 Introductlon

1.1 Optimization Fundamentals I 2 1.1.1 Elements of Problem Formulation I 4 1.1.2 Mathematical Modeling I 10 1.1.3 Nature of Solution I 16 1.1.4 Characteristics of the Search Procedure I 20

1.2 Introduction to MA11..As I 25 1.2.1 Why MATLAB? I 25 1.2.2 MATLAB Installation Issues I 26 1.2.3 Using MATLAB the First Time I 27 1.2.4 Using the Editor / 33 1.2.5 Creating a Code Snippet / 37 1.2.6 Creating a Program I 40

Problems I 44

xiii

1

2 Graphlcal Optlmizatlon 45

2.1 blemDefinition I 45 2.1.1 Example 2.1 I 46 2.1.2 Format for lhe Graphical Display I 47

2.2 Graphical Solution I 48 2.2.1 MATLAB High-Level Graphics Functions I 48 2.2.2 Example 2.I-Graphical Solution I 50 2.2.3 Displaying the Graphics I 53 2.2.4 Customizing the Figure I 54

2.3 Additional Examples I 56 vll

vlll CONTENTS

2.3.1 Example 2.2 I 56 2.3.2 Example 2.3 I 64 2.3.3 Example 2.4 I 73

2.4 Additional MA11..AB Graphics I 79 2.4.1 Handle Graphics I 80 2.4.2 Graphical User Interface I 81 2.4.3 GUI Code I 84

References I 91

Problems I 92

3 Llnear Programmlng

3.1 Problem Definition I 94 3.1.1 Standard Format / 94

3.1.2 Modeling Issues / 98

3.2 Graphical Solution / 107

3.2.1 Example 3.1 / 110

3.2.2 Characteristics of the Solution / II1

3.2.3 Different Solution Types I 114 3.3 Numerical Solution-the Simplex Method I 115

3.3.1 Features of the Simplex Method / 115

3.3.2 Application of Simplex Method / 117 3.3.3 Solution Using MA11..AB / 120 3.3.4 Solution Using MATLAB'S Optimization Toolbox I 123

3.4 Additional Examples / 124

3.4.1 Example 3.2-Transportation Problem / 124 3.4.2 Example 3.3-Equality Constraints and Unrescted

Variables / 130

3.4.3 Example 3.4-A Four-Variable Problem / 134 3.5 Additional Topics in Linear Programming / 138

3.5.1 Primal and Dual Problem / 138

3.5.2 Sensitivity Analysis / 148

Refernces/ 151

Problems / 152

4 Nonlinear Programming

4.1 Problem Definition I 155

93

154

CONTENTS Ix

4.1.1 Problem Formulation-Example 4.1 / 155

4.1.2 Discussion of Consaints/ 157

4.2 Mathematical Concepts / 159

4.2.1 Symbolic Computation Using MATLAB / 159

4.2.2 Ba~ic Mathematical Concepts I 162 4.2.3 Taylor's Theorem/Series / 169

4.3 Graphica1 Solutions / 171 4.3.1 Unconstrained Problem / 171

4.3.2 Equality Constrained Problem / 172 4.3.3 Inequality Constrained Problem / 173 4.3.4 Equality and Inequality Constraints / 174

4.4 Analytica1 Conditions / 175 4.4.1 Unconstrained Problem / 176 4.4.2 Equality Cons inedProblem I 179 4.4.3 Inequality Constrained Optimization I 186 4.4.4 General Optimization Problem / 191

4.5 Exnples/ 194

4.5.1 Example4.2/ 194

4.5.2 Example 4.3 / 196

References / 200

Problems / 201

5 Numerical Technlques-The One-Dlmenslonal Problem 5.1 Problem Definition I 204

5.1.1 Constrained One-Dimensional Problem / 204

5.2 Solution to the Problem / 205

5.2.1 Graphical Solution / 205 5.2.2 Newton-Raphson Technique / 206

5.2.3 Bisection Technique I 209 5.2.4 Polynomial Approximation / 211 5.2.5 Golden Section Method I 214

5.3 Importance of the One-Dimensional Problem / 217

5.4 Additional Examples I 219

203

5.4.1 ample5.2-Il1ustionof Geneml Golden Section Method / 219 5.4.2 Example 5.3 Two-PointBoundary Value Problem I 220 5.4.3 Example 5.4-Root Finding with Golden Section I 223

X CONTENTS

References I 225

Problems I 225

6 Numerlcal Technlques for Unconstralned Optlmlzatlon

6.1 Problem Definition I 227 6.1.1 Example 6.1 I 228 6.1.2 Necessary and Sufficient Conditions I 228 6.1.3 Elements of a Numerica1 Technique I 229

6.2 Numerical Techniques-Nongradient Methods I 230 6.2.1 Random Wa1k I 230 6.2.2 Pattem Search / 234 6.2.3 PoweJ1's Method I 238

6.3 Numerical Thniques-Gradient-BasedMethods I 241 '6.3.1 Steepest Descent Method I 241 6.3.2 Conjugate Gradient (F1etcher-Reeves) Method I 244 6.3.3 Davidon-FJetcher-PowelI Method I 246 6.3.4 Broydon-Fletcher-GoldfartShannoMethod I 249

6.4 Numerical Techniques-Second Order I 251 6.5 Additional Examples I 253

6.5.1 Example 6.2-Rosenbrock Problem / 253

227

6.5.2 Example 6.3-Three-Dimensional Aow nraRo ngDiskI 255 6.5.3 ExampJe 6.4-Fitting Bezier Parametric Curvcs I 258

References I 262

Problems I 263

7 Numerlcal Techniques for Constralned Optlmlzatlon 265

7.1 Problem Definition I 266 7.1.1 Problem Formulation-Example 7.1 I 266 7.1.2 Necessary Conditions / 267 7.1.3 Elements of a Numerical Technique I 269

7.2 Indirect Methods for Constrained Optimization I 270 7.2.1 Exterior Penalty Function (EPF) MeodI 271 7.2.2 Augmented Lagrange Multiplier (ALM) Method I 276

7.3 Direct Methods for ConsainedOptimization I 281 7.3.1 Sequentia1 Linear Programming (SLP) I 284 7.3.2 Sequential Quadratic Programming (SQP) I 289

CONTENTS xl

7.3.3 Genera1ized Reduced Gradient (GRG) Method I 297 7.3.4 SuentialGradient Restoration AJgorithm (SGRA) I 302

7.4 Additiona1 ExampJes I 307 7.4.1 ExampJe 7.2-FJagpoJe Problem I 307 7.4.2 ExampJe 7.3-Beam Design I 310 7.4.3 ExampJe 7.4 Optima1Control I 313

References I 316

Problems I 316

8 Dlscrete Optlmlzation

8.1 Concepts in Discrete Programming / 320 8.1.1 Problem Relaxation I 321 8.1.2 Discrete Optimal Solution I 322

8.2 Discrete Optimization Techniques I 324 8.2.1 Exhaustive Enumeration I 326 8.2.2 Branch and Bound I 329 8.2.3 Dynamic Programming / 336

8.3 Additional Examples I 341 8.3.1 Exnmple 8.4--1 Beum Design I 341 8.3.2 Zero-One Integer Programming I 343

References I 348

Problems I 348

9 Global Optlmlzation

9.1 Problem Definition I 351 9.1.1 Global Minimum / 351 9.1.2 Nature of the Solution I 354 9.1.3 Elements of a Numerica1 Technique I 356

9.2 Numerical Thniqusand Additional Examples I 357 9.2.1 Simulated Annealing (SA) I 358 9.2.2 Genetic Algorithm (GA) I 366

References I 377 Problems I 378

10 Optlmization Toolbox from MATlAB

10.1 The Optimization Toolbox I 380

318

350

379

xll CONTENTS

10.1.1 Programs I 380 10.1.2 Using Programs I 382 10.1.3 Setting Optimization Parameters I 384

10.2 Examples I 385 10.2.1 Linear Programming I 385 10.2.2 Quadratic Programming I 386 10.2.3 Unconstrained Optimization I 388 10.2.4 Constrained Optimization I 389

Reference I 391

Index 393

PREFACE

lesu~ject of optimization is receiving serious attentionomengineers scientists managers and most everybody else. This is driven by competion quality assurance cost of production and finally the success of the business entee.lgnoringepractice of optimization is not an option during current times. Optimization is practiced through softwar programsand requires significant

computer resources. The techniques of optimization have not changed significantly in centyears but tlle areas of applications have mushroomed at a significant rate. Successfully embedding the use of optimization in professional practice requires at least three Pquisites.beyinclude mathematical modeling of the design problem knowledge of computer programming and knowledge of optimization techniques. Many special-purpose optimization software packages relax requiredknowledge of programming are available today. To use them efficiently the remaining two areas still have to be addressed. lereare several excellent books on the subject of optimization a few of them

released recently. Most of them covrthe subject in depth. which is necessary because thmathematicalmodels are nonlinear and require special techniqusthat are usually notpaofany core curriculum. All ofebooks assumeereader is familiar wilh a programming languageaditionallyFORTRAN and recently C. Another assumplion hquentlymade is theoptimizationchniqueswi1l be implemented in a mainframe computing environmenl. Such a combination is exemelydifficu1t for the self-Iearner even with enormous motivation. An explosion of inexpensive desktop computing resources aids engineering design

practi today.Paralleling this development iseavailability of extensivesouon the Internet both for learning and deployment. This is joined by the availability of sofsystemsthat provide opportunityfor handling mathematics graphics. and programming in a consistent manner. These software systems are significant1y easy to master compar tothe higher-level programming languages of the pviousyeS.lisbook primarily seeks to harness thisiangulationof srvicesto provide a practical approach to the study of design optimization.lebook uses MATI.AB@ to i11ustrate and implement the various techniques of optimization. MATLAB is a product from MathWorks. Inc. This book attempts 10 accomplish two important obj tives.The fir

xlv PREFACE

of symbolic. numerical. and graphical fl turesofMATLABltintegratesispowerful combination duringnslationof many algorithms into app1ied numerical techniques for design optimization. There is a constant enhancement of the programming skills throughout thbook.The second 0ectiveis thprimarytask of book-tocommunicate and demonstrate various lIumerical techniques that are

currenlusedin tlte area oloptimal des.AH of thc numerical techniquesMEsupported by MMLAB code available

computer files. These files are available on a companion web site www.wilev.comlvenkat.ltis necessary to visit the web site to download thc files to fo l l o w a l l of t h e E X tl I n p i es -Th e b oo k w i l l b e e ntiallyincomplet.ewitho0utthcsc files. At the web site you will also find uscful links 10 other resources a web course on MATLAB programmingand updated information about Applied Optintimuon with MATl.AB@ Programming. The decision to locate the files on a web site pmvidEs an avenue to shadow the

dynamic changes in MATLAs sof1ware itsclf which is bcing transformed in a major way almost every ycar.lt provides a mechanism 10 correct bugs iecode in a timely manner. It estabishes an opportunity to improvecodethrough reader suggcstions and provides a way foreauthor 10 keep thc rcader engaged with new dcvclopments. It docut down the sizc of the book enormously by locating all of the pag ofcodeoutside the book. An unusual feature of the book iscinclusion of discrete optimization and global

optimization. Tradilional continuous design optimization techniques do take up a significant portion of the book. Continuous problems generally allow the algorithms to be mathematically developed in a convincing manncr andeideas could becore of original a1gorithms dcvc10pcd by the readers to address their own special necds. Every chaptcr includes additional nontrivial cxamp1esatoftcn prescnt a novel use or extcnsion of thc optimization techniques in the chapter. The mathcmatical algorithms and cxamples ar accompaniedby MA' ABcode available at the companion web sue-Thenumerical techniques usually include strong graphical support to il1ustratecopcration of thc techniques. Discrete andglobal optimization techniques are basedon verydifferentalgorithms

and.chof thcm can easily justify full-fledged independent courscs detcdexclusively 10 thcir S1udy. In a comprchensivc book such

PREFACE XV

classroom setting withou1 computational cxperience is also feasib1e. Iniscase the a1gorithms can be presented and the nurnerical results illustrated. The senior/gradua1e s1udents in various discipines. especially engincering are the targc1 audience. Optimization techniqucs are an importanl 1001 to solve dcsign problems in all professional areas of study. Many illustrations areomthe area of mechanical engineering reflectingexperienceofeauor.Independent learnel's particuIarly professionals who need to understand the subject

of optimization should also find the book very useful. A reader who has access to MATLAB software c us thebook to its best advantage. The book is largely self-contained and develops all ncccssary mathematical concepts when ncedcd. Abstract mathematical idcas of optimi tion introduccdgraphically and through iIlustrative examples in thc book. In many instances the operation of thc numcrical technique isatcdto show how the design is changing with iterations. This has been possible because MATLAB graphics are rich in fcatures and simple to incooratc.making Ulcm cffcctive for convcyingeidcas. It is for this reasonereader is cxposed to graphical programming early on incIuding a chapter on graphical optimization. To balance thc development of programming skills and thpresentationof optimization concepts programming issues are incrementally explored and implemented. Investment of time in dvelopingprogramming skills isemost effective way to imbibe them. Thc author is nOl aware of shortcuts. Studcnts arc expected to program independently. make mistakes debug errors. and incorporate improvements as ptof their learning experience delivered throughisbook. The book proceeds at a brisk pace to provide experiencc in MATLA8 programming

and communicate ideas in optimization in a reasonable number of pages. This could not be possible without taking advantage of Ihe accompanying web site 10 locate much of the code for algorithms and examples. Merely running the code will n01 significantly ecethe reader's programming skills. It is important for the student to understand the code also. To sistinisprocess the code carries Iiberal comments often corresponding to the algorithm that is being implemcnted. The auorexpects ereader to have readecomments. The accompanying code allows the instructor to assign both programming assignments as well as so

xvl PREFACE

approaches the instructor can choose to include only a subset of continuous optimization algorithms to allow the inclusion ofesepopular techniques. Ifestudent has access to the Optimization Toolbox from MATlABenit can be integrated intoecourse for handling the final design project. The last chapter provides an overview on the use of the toolbox. This book started out using MATI.AB Version 5. About halfway through the book

etransition to MATlAB Version 5.3 was complete. At the completion of the m uscriptMATLAB Version 6 wshipping for some time. As of this tirne Version 6.1 is available. Most institutional MATlAB licenses are usually based on subscription sotheywill havelatestversion ofesoftware. In a sense andisis true of all books that depend on particular software systems the book rnay appear out of date with respect toesoftware. This is not really so. While the usage ofMATLAB functions rnay change between versions there should be suitable wamings and instruction about edifferent usageAnimportant insurance is built into the book particularly to handle such changes if they arise. Sprinkled throughout the book is the habit ofeusingeonline help. In most instancesiswas primarily done to expose the reader to alternative use of the same command. Another significant change the manuscript had to endure waseloss of access to a UNIX version because of thuniversity'schange to distributed computing on PC clusters. The author's experience wieearly chapters suggs18that this should not mat1er. Except for external files and resources useof MATlAB as illustrated in the book is indifferent to the various operating sy(WindowsMac UNIX or Linux). The topics chosen for inclusion and the simplicity of presentation ofetopics are

directly related to the experience of teaching the course on optimization at the senior/graduate level ineDpartmentof Mechanical Engineering for over ten years. Experience provedatcomprehension irnproved with simple iIIustrations however complicated the mathematical ideas. On the other hand straighorwardmathernatical expressions elicited no interest or understanding. The emphasis on application was important to keep the students' attention. In this connection the author would like to 1kall his studen18 for pushing hirn for simple explanations for providing positive interaction and for their contribution of original and useful ideas in the discuss

PREFACE xvii

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The book was made possible through support from John Wiley dSonsInc.dMathWorks Inc. Sincere thanks are owed to Bob Argentieri senior editor at Jobn Wiley for accepting the proposaland who all along displayed a lot of patiencein getting the book moving forward.Same is due to Bob Hilbed-associate managing editor at John Wiley for his impssivework at c1eaning up the manuscript. Brian Snapp New Media editor at John Wiley created companionweb site (www.wiley.comlvenkat) and w alsobe rnaintaining it. Naomi FemandesomMathWorks Inc. saw to itat1 had the latest version of MATLAB as soon as it was available. My regard for Dr. Angelo Miele Professor Emeritus at Rice University is more than can be expressed in these Iines. It was he who introduced me to 1hsubjeclof optimization and demonstrated the effectiveness of simple presenution-I will always regard him as a great teacher. Of course my family deserves special rnention for putting up with all the Not now"Later"How about tornorrow?" din2debuggingecode. Spialksare to A ana dVinayak rny offspring titheir patience understanding and encouragement. The auorapologizes for any shortcornings on the presentation and welcomes comments criticisms and suggestions for improvement at all times.

P. VENKATARAMhN RochesteNewYork

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INTRODUCTION

Optimization has become a necessary part of design activity in all major disciplines. These disciplines are not restricted to engineering. The motivation to produce economically relevant products or services with embedded quality is the principal reason forisinclusion. Improved production and design tools wiasynergistic thrust through inexpensive computational resources have aided the consideration of optimization methods in new developments particularly engineering products. Even in the absencc of a tangible product optimization ideas provide the ability to define and explore problems while focusing on solutionsatsubscribe to some measure of usefulness. Generally the use of the word optimization implisthc best result under thcircumstances.This includes the particular set of constraints on the development rcsources current knowledge market conditions and so on. Every one of us has probably used thc term at some time to describeeprimary quality of our work or endeavor. It is probablyemosl used or abused lerm in advertising and presentations. Neverthclesscability to make the best choice is a perpelual desire among us all. Optimization is frequently associated with design be it a product service or

strategy. Aerospace design was amongearliestdisciplines to embrace optimization in a significant way driven by a natura1 need to lower the tremendous cost associated with carrying unnecessyweight in aerospace vehicles. Minimum mass structures arl norm.Optimization forms part of the psyche of every aerospace designer. Saving on fuel through trjectorydesign was another problem that suggested itself. Very soon enti engineeringcommunity could recognizecneed to define solutions based on merit. Recognizing the desire for optimization and actua1ly implementing we twodifferent issues.

2 INTRODUCTION

Until recently. for much of the time. optimization was usually attempted only in those situations where there were significant penalties for generic designs. The application of optimi tiondemanded large computational resources. In the nascent years of digital computation these were available only to large national laboratories and research programs. These resources were necessary to handle thnonlinearproblemsat associatedwith engineering optimi.tion.As a result of these constraints most of the everyday products were designed without regard to optimization. This includes everything you soundyou or use in your daily life. It is inconceivableatthe new generation of replacement products. likc the car. the house. the desk. orepencil. redesigned and manufactured today arc not designed optimally in one sense or anothcr. Today. you would defmitIyexplore procedures to optimize your inveslentsby

tailoring your portfolio. You would optimize your business travel time by appropriaIychoosing your dcstinations. You can optimize your commuting timbychoosing your time and route. You can optimize your necessary expenditure for Iiving by choosing your day and store for shopping. You can optimizc the useful time you connecttoIntemetby detcrmining your time of connection. You can buy software willoptimize your conncction to thc lntemct. You can buy books or read articles tcllyou how to perform these various optimizations. The above activitics primarily relatc to :;crvIces or strategy. It is now apparcntatevery activity except aSeticprovides the scope for optimization.lisjustifies looking atestudy of ootimizonas a toolcanbe applied 10 a variety of disciplines. If so the myriad ofoptimizationproblemnsso11IThisiseemphasisofthebook. The partnership between dsignand optimization activity is often found in

engineering. This book recognizes that connection and many of thc problems used for illustrations and practice are from enginringprimariIy mechanical civil and aerospace design. Ncvertheless the study of optimization. particularly applied optimization.is not m exclusive pmperty ofmy sptcinc discipurle.It involves the dECOVEryanddesignof utionsthrough appropriate techniques ociatedwith the formulation of the problem in a sp ificmanner. This can be done for example in economics chemistry and business man

1.1 OPTIMIZATION FUNDAMENTALS

Optimization can be applied to all disciplincs. Qualitatively this assertion implies multiple decision choices; implicitly recognizing the ncessityof choosing among a1tematives. This books deals with optirnization in a quantitative way. This means that an outcome of applying optimization toeproblem design. or service must yield numberatwill define the solution. or in other words numbers or valueswillcharacterize the particular design or service. Quantitative dcscription oflutionrequires a quantitative description of the problem itself. This description is called a

;:.~ ". "

1.1 OPTIMIZATION FUNDAMENTALS 3

mathematical model.bedi.icharactcrization. dits circumstances must be expressed mathematically. Consider the design activity in the following cases:

New consumcr research indicatthat people like 10 drink about 0.5 litcr of soda pop al a time during the summer months. The fabrication COSI of the redesigned soda can is proportional tosurfacearea. and can be estimatcd at $1.00 pcr square meter of the materiaI used. A circular cross section is the most plausible given current tooling available for manufacture. For aesthetic reasonseheight must be at least twice the diametcr. Studies indicateatholding comfort requires a diameter between 6 and 9 cm.

A cantilever beam nceds to be dcsigndto carry a point load F at the cnd of a beam of length L. The cross section of the beam wiII be in the shupe of thltterI(edas I-bm).The beam should meet prescribed failure criteria. ereis also a limit on idefItion.A beam of minimum mass is requir tobe designed.

MyPC Company has decidcd to invcst $12 million in acquiring several nWComponenl Placemnl Machines to manufacture differcnt kinds of mothcrboards for a ncw gcncration of personal compulers. Three models of thcsc machines are under consideration. Total numbcr of operators available is 100 becausc of the local labor market. A fIoor spacc constraint needs 10 bc satisfied because of the diffcrent dimcnsions of thcse machines. Additional information rclating to each of the machines is given in Table 1.1. The company wishes to determine how many of each kind is appropriate 10 maximize the number of boards manufactured per day.

The above list rcpr entsthree problemsatwiII be used 10 definc formal elements of an optimization problem. Each problemquiresinformation from the specific arca or discipline to which it refers. To rccognize or design these problems assumesatdesigncris conversant wieparticular subj tmatter.leproblems are kept simple to focus on optimization issucs. Problems of such variely are important today. Recent advertisements in general consumer magazines i1IustrateatAlcoa (an aluminum m ufaclurer)is quithappyto have rcducedLhc weight of the standard soda pop can by over 30% in recenl years. A similar c1aim by the plastics industry with respect to the standard milk jug (gaIlon) is al50 evident in these magazines although iniscase the number is 409.Roof coUapses incNortheast due 10 excessive snow

Table 1.1 Component Placement Machir

Machine Operablc Model BoardTypes BoardslHour Opemtors/Shift HoursIDay CostIMachine

A 10 55 18 400000 B 20 50 2 18 600000 C 18 50 2 21 700000

4 INTRODUCTION

during the 1998-1999 winter will have structura1 designers and homebuilders exploring secondproblem. Thvibrantstock market in these times has made ba1ancing the investment portfolio more challenging. Thc third case may suggest a mathematica1 model appropriate for such decision making.

1.1.1 Elements of Problem Formulatlon

In this section we will in duceeforma1 elements of thc optimization problem. In this book the tcrm product a1sofersto a servicora strategy. It should be understoodatopzationpresupposes the knowlcdge ofedesign rules for the sp ificproblem primarily the ability to describe the design in mathematica1 terms. These termscJudedesign variables design parameters and design functions. Traditiona1 design practiatis design without regard to optimization incJudes a11 of these elementsugheywere not formally recognized as such. This also justifieseprerequisi thatyou must be capable of dcsigning the 0tif you are planning to applyetechniqucs of optimi tion.It is al80 a good idea 10 recognizc that optimization is a procedurc for scarching 1he best design among candidatscach of which can produce an acceptable product. The oced for thc objec1 or product is not questioned here. but this may be due to a decision based 00 optimization applied in another disciplinc.

Deslgn Varlables: Design variables are entities that identify a particular design. In esearch foroptimaldesigneseentities will change over a prcscribed range. The values of a complete set of these variables charactcrize a specific design. Thnumber and type of cntities bclonging to this set are vcry important in identifying and setting upequ titativedesign problem. It is essentialischoice captueessence of objectbeing desied datesamc time provide a quantitative characterization of the design problem. In applicd mathematical terminology. design variables serve as the unknowns of the problem being solvcd. Borrowing an ana10gy from the area of system dynami andcontrol theoryeyare equiva1ent to defining the state of the systcm iniscase thc statc of design. Typically design variables Can be associatcd with describingobject'ssize such as its length and height. In other cesthey may represen1 the number of items. The choice of design variables is the responsibility ofedesigner guidcd by intuition expeseand knowledge.le isa fundamenta1 requirement to be mct byisset of design variables namelyeymust be Iinearly independent. This meansa1you cannot cstablish thvalueof onc ofedesign variables fromevalues oferemaining variables through basic arithmetic (scaling or addition) operations. For example in a design having a rcctangular cross section you cannot haveevariables reprentingelcngwidth and area. If the first two arprescribedethird is automatica1ly established. In complex designs these relationships may not be very apparent. Nevertheless the choi ofthe set of design variables must meet criterionof linear independence for applying the techniques of optimization. From a practicnl pcrspcc1ivc thc property of Iinear independence identifies a minimum set of variables that cun complctely describe the

r::.-: l

'"

1.1 OPTIMIZATION FUNDAMENTALS 5

design. This is significant because the effort in obtnining the solution varies as an integer power of the number of variables and this power is 1ypica11y greateran2. Mtingthe requirement ensures reduced difficulty in mathematically exploring thc solution. leset of design variables is identified as the design vector.1Isv torwill be

consider acolumn vector in this book. In fac1 a11 vectors are column vectorsEtext. The length ofisvector which is n is the number of design varilesioEproblem. The design variables can cxpress different dimensional qu titiesinproblem but in the malhematical model they are distinguished bycharac1erx. AlI of the techniques of optimization in this book nre bascd oneabstract mathematical model.lesubscript on x for cxamplc X3rcpresentsthe third design variable which may be the height of an objec1 inechucterizationof the product. This abstract model is nessaryfor mathematical convenience. This book will refer toedesign variables in one offollowingways:

(1) [-refcingto the vcc10r 01' dsignvariablcs (2) X or x-referring to thc vcctor again omilling 1hc square brackets for convcnicncc if appropriate

(3) [xj x2'...XIl]T_indicatingthc vcclor lhrough its elements. Noleesuperscript transposition symbol to identify it as a colunm vector.

(4) X;. i = 1 2. . . . n-referring to all of the elements of the design vctor.

Thc above notationa1 convenience is extcndcd to all vcctors in the book.

Deslgn Parameters: Inisbookeseidentify constan1will not changedierentdesigns are compared. Many 1exts use termdesign parameters 10 prescntdcsignvariables we defined earlier and do not formally recognize design parameters as defincd hc.The principnl reason is that parameters have no role 10 play detenningoptima1design. They are significant in the discussion of modeling issues. Examples of parameters include matcria1 propcyapplied loads. and choicc of shape. The paramctcrs in the abstract mathematical model are repsentcdin a similar form as the design vec1or excep1 twe use the charactcr p. Therefore. [P] P [P"P2'...Pq1 repsenteparameters of the problem. Noteelengoftheparnervector is q. Except incdiscussion of modelingeparneterswill not be explicitly rredto as they are primarily predetrmincdconstants indesign.

Des'nFunctlons: Design functions define meaningful information about the design. They are evaluated using the design variables and design parameters discussed earlier. They establish the mathematical model of the design problem. These functions can represent design objective(s) andlor constraillts. As iname implies design 0ectivedrives the search for the optimal design. The satisfaction of the trailltsestablishes the validity of the design. If not explicit1y statededesigner is responsiblc for idntifyingthc objectivc and constraints. Minimize the mass of the

6 INTRODUCTION

structure will translate to objectiveU1ction.The sin the material must be less than the yieId strength wi1l translate to a constraintfunction. In many problems it is possible for thc same function to switch roles to provide diffcrcnt design scenarios.

ObJectlve Function Thetraditional design optimization problem is dcfindusing a single objective function.leformat of this statemcnt is usually to minimizc or maximize some quantity iscalculated using somc dcsign function. This function must depend explicitly or implicitly on the design variables. In the literaturcisproblcm is cxpssedexc1usively without loss of generaIity as a minimum problem. A maximum problem can brccastas a minimization problcm using the ncgativc orreciprocalof thc function used for the objectivc function. ln the first example in ducedearliercoectiveis to minimize cost.lereforedesi functionrepresgcost wilI be the objective function. Inesecond seeobjective is to minimize mass. In the third case the objective is to maximizc machine utilization.lear ofsingle objective desi isconsidered mature today. No new solution techniques for classical problcms havbecnadvanced for somc time now. Today. much of the work in applied optimitionis dirccted at expanding applications to practical problems. ln many cases. this has involved crcative use of the solution techniques. In abstractmathematical model. the objective function is reprosented by the symbolf To indicate its dependence on the design variables. it is fquentlycxpressed f(x"x2.. . . xn). A more conciscprcsentationis j{X). Single objective problcms have only one function dcnoted by f It is a scaIar (not a V tor).Notcatalthough the objcctive function depends on P (parameter vector). it is not explicitly inc1uded in the format. Multiobjective and multidisciplinary dsignsarc important developments today.

Multiobjtive design or multiple objecti ve design. refersusing sevcraI dierentdesign functions toeshfor opnaldign. GeneraIlyeyare expecdbeconflicting 0cctives.They could also beopcrating objectives.lecurrent approach toesolution of these problems involves standard optimization procedapplicdto a single (onstructedobjective optimization problcm based on the different multiple objectives. A poparapproach is to use a suitably weighdlinrcombination ofmultiplc objectives. A practicallimitation wiisapproach is.echoi ofweights usin the model. This approach has not been cmbraced widely. An altemative approach of recognizing a prcnectiveand solving a singlc objectivc problem with additional constraints based onningovenctionscan usually geneaccep lesolution. In multiobjective problen

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? ?

??

1.1 OPTIMIZATION FUNDAMENTALS 7

Constraint Functlons: As design functions these will be influenced by the design variables. The format of these functions requires them to be compared to some numerically limiting valueatis established by design requirement or the designer. This value remains constant during thc optimization of the problcm. A wcl1-describcd dcsign problem is expected to inc1ude scveral such functions. which can be represented as a vcctor. The comparison is usual1y set up using the threc standard onaloperators: = S and.Consider our first example. LetfimJ(X) represent efunctioncalculatesthe volume of the new soda can we are designing.leconstrnton the dsigncan bc expresscd as

IIJ(X)= 500 cmJ

In tlIe second example.letm2(X) bthefunction that calculates the deflection of the beam under the applied load. The constraint can be statcd as

ft11l2(X) :S; I mm

Thccons intfunctions can be c1assified as eqllality constraints [likefunl(X) aboveJ or inequality consnts[likefun2(X)]. Problems without constraints are tcrmcd unconstrained problems. If constraints are

present then meeting them is more pmountanoptimizalion. Constraint satisfaction is necessary beforc the dcsign established by currcntvalue of the desi variablesis considered valid and acceptable. If constraints are not ssfiedthen there is no solution. A feasible design is one in which all of the constraints are satisfied. An optimal solution is oneathas met the design objective. An optimal dcsign must be fcasiblc. The design spacc cncloscd by thc constraints is called the feasible domain. Dcsign space is described a few paragraphs bclow.

Equallty Constralnts: Equality constraints are mathematically neat and easy to handlc. NumericallyeyJ1uiremore effort to satisfy. They are also more restrictive on tlIe designthey limit the region from which the solution can be obtained. The symbol representing equality constraints in the abstract model is h. There may be more than one equality constraint in the design problcm. A vector representation forualityconstraints is introduced through the following representation. [8] [hl h2 . . . . h]. and hk: k = 12 . . . .1 are ways of identifyingeequality constraints.ledependence on the design variables X is omitted for convcnience. Notcatthe length of the vector is 1. An importantonfor distinguishing the equality and inequality constraints isatthey arc mnnipulated differently in the senrch for the optimal solution. The number n of design variables incproblcm must be greatcr thenumber of equality constraints 1 for optimization to take place. lf n is cqual to 1. then problemwill be solved without reference to the objective. ln mathematical terms the number of equations matchesenumber of unknowns. If n is less than 1 then you have an overdetermined scl of relations which could result in an inconsistent problcm definition. The sct of cquality constraints must be linearly indcpendent. Broadly this

8 INTAODUCTION

implies youcannot obtain one of the consaintsfrom elementary arithmetic operations on the rcmaining constraints. This serves 10 ensure that the mathematical search for solution will not fail. These tcchniques are based on methods from Iincar algebra. In the standard format for optimization problems theua1ityconsaintsare written with a 0 on the right-hand side. This meanseequality constraint in thc first example will be expssedas

hl(X):funl(X) -500 = 0

In practica1 problems equality constraints are rarIyencountcred.

Inequallty Constraints: Inua1ityconsaintsapp rmore naturally in problcm formulation. Inequality constraints also provide more flexibility in dcsign selection. The symbol represcnting inequality cons'aintsin the abstract model is 8. There may be more oncinua1ityconstraint in the design problem. The vector reprsentationfor inequality constraints is similar to what we havseenbefore. Thus [G) [8182 . . .811/]and:j = 1 2 . . . m arc ways of identifying Ihe inequality consaints.m reprentsenumber of inequa1ity constraints. AIl design functions explicitly or implicitly depend on the design (or independent) variable X. 8 is used to descriho bo lossthun or cqual 10)dgrealcr 01"equlll 10 ()constralnts. Strictly greater(>)and strictly Icss th(

10 INTROOUCT10N

xlsxjSx i=12... n Exploiting vector notation the mathematical model is

Minimize j(X) [X].

Subject to: [h(X)]1 = 0

(X)]tnS 0

X10wSX S Xup

(1.8)

(1.9)

(1.10)

( 1.11)

(1.12)

The above mathematical model expresses followingstandard format of the optimizonproblem expressed in naturallanguage:

Minimize the objective functionj subject to 1 equaity constraints m incquality constraints withen design variables Iying between prcribed lower and upper limits.

letechniqucs inisbook will apply to thc problem describcd incabovc format. To solve any specific design problem it is requircd 10 refunnulatc the problcm in the above manner so methodscan be applied directly. Also in this book the techniques are developed progressively by considering the standard model widucedelements. For example thunconstrainedproblem wil be explodfirst. The equality cons intsa considerednext followed by the inequality constraints. dfinally the complete model. This represents a natural progression as prior knowledge is used to develop additional conditions needto be satisfied by the solution in einstances.

1.1.2 Mathematlcal Modellng

In this sectionethree design problems i'Oducedearlier will beanslatedtoestandard format. The mathematical mode1 will first be identified. The second problem requires information from a course in mechanics and materials. This should be within .epurview of most engineering students.

Example 1.1 New consumer research indicatesatpeople like to drink about 0.5 liter of soda pop at a time during the summer months. The fabrication cost of the redesigned soda can is proportional 10 the surface area and can be estimaled al $1.00 per square meter of lhe matrialused. A circular cross section is the most plausible given current looling available for manufaclure. For aesthetic reasons tl1e height must be at least twice tl1e diameter. Studies indicateatholding comfort requires a diameter between 6 and 9 cm. Figurc 1.1 shows a sketch of tl1e can. In most product designs particularly in

engineringthe model is easier to develop using a figure. The diameter d and the height h are sufficient to dcscribe tl1e soda can. What about the tl1ickness t ofe

~

p J :

12 INTRODUCTION ????? 1.1 OPTIMIZATION FUNDAMENTALS 13

Intuitively therc is some concem with the problem as expressed by Equations (1.13(1.15)even thoughdescriptionis valid. How can the value of the design variable t be estab1ished? The variation in t does not a'ectthe design. Changing the value of t does not changef hl or glHenceit cannot be a design variable. (Note: the variation in t may affect the value of c.) lfiswerc a practical design problcm then canshave to be designed for impact and stacking strength. InCet will probably be a critical design variable. This will require severadditionals cluralconstraints in the problem. It could serve as an interesting extension to this problem for homework or projectleoptimization problem after dropping t dexpressing [d. h] [XIX2]becomes:

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? ? ? ? ?L ?

? ? ?gM.h.t): 2d-h SO

6Sds9; 5s/ts20; O.OOIStsO.Ol (1.15)

Flgure 1.2 Example 1.2-Canlllever beam.

Minimize f(XIX2): c: ( 1.16)

(1.17)

gl(lXU:2x1-X2S0 (1.18)

lntuitivelyelarger the value of F. the greater iscros8-Secllionaalare=anecsvt.0 handleit. 11 canbeilyconcluded atifFw adsibselat its lower limit a8 lhe mass of the beam will be direclly proportional 10earea of croS8 8cc(Iun. No techniques are ruiredf.or lhis conclusi.on. By .our definition F is a go.od candidate as a parneterrather asa design variable. Similarly the larger the value of Legateris the momentFgenerates about the cantilevered end.liswill require an increase inearea .of cross-section. Therefore L is not a good choice f.oredesign variable either as the best choice f.or it will bc il.owerIimit. To represcnt material as a design variable. we necd to use its SlruClural propcrties ineformulalion ofthe problem. Typically. a malerial is characlerized by its specc

Subjecllo: hl(xlXv:ru/4-500=0

6 SX) S 9; 5 SX2 S 20

The problempresenledby Equati.ons (1.16)(1.18)is mathematicalmode f.or thdesignproblem expressed in the slandard format. For this problem no extranc.ous informali.on was ndto set upeproblem except for some geometrical relations.

Example 1.2 A cantilever beam needs 10 be designed to crya point load F al the end .of the beam .of lenglh L. The cross section of the beam wiII be in the shape of the letter 1 (ferredto as 1bm).lebeam should meet prescribed failure criteria. Thcre is also a Iimit on its deflecti.on. A beam of minimum mass isquired10 be digncd. Figure 1.2 shows a side view .of the beam carrying the load and Figure 1.3 a typical

cr.oss snofthben.The I-shaped cross section is symmetric. The symbols d tw. br and tr c.orrespond to the dep.ofthe bcam thickness of the web width of thflangeand thickness .of the flange respectively. Thesquantitiesare sufficient to defineEcr.oss section. A handbo.ok or textb.ook on strcngth of materials can aidedvelopment.of thdesignfunctions. An important assumption for this problem iswe will be w.orking within the elastic Iimit of thc material where there is a Iinear relati.onship betweenestress and the strain. AlI .of the variables identified in Figure 1.3 will s nglyaffect the soluti.on. They are design variables. The applied f.orce F will also directly affect the problem. So will its locati.on L. H.ow aboutmaterial?Stecl is dcfinitely supcri.or 1.0 c.opper for the beam. Should F L and the material pr.operties be included as design variables?

bt I~ 'l

4tW-

Flguro 1.3 Example 1.2-Cross-sectlonal shape.

14 INTRODUCTION

weight y its value ofthe modulus of elasticity. E. its modulus of rigidity. G. its yield limit in tension and compr sion.yie'd.iyield in shyield.its ultimate strengin tension.ult. and its ultimate strength in shear.ullOptimi liontechniques identify diffentva1ues of the design variables. If E is the variable used topresentEmaterial design variable in a problem. then it is possible the solution canquirea materia1atdoes not existatis. a va1ue of E still undiscoverd.For a11 intents and pposes.isis beyond the scope of the designer. Again. materia1 as a parameter makes a lot of.sense andeoptimization problem should be reinvestigated if For the material changes. Concluding fromisdiscussion on modeling. the following Iist can be established for this example:

Design p1eters F.L.{y.G. Y'eld'1:yield'u11'1:EDesign variables: d tw' br'If

In the devlopmentof the mathematical model standard technical definitionsused. Much of the information can be obtained from a mechanical engineering handbook. Thfirstdesign function isweightof the beam. This is the product of y L and Ac where Ac isearea of cross section. The maximum slress due to bending ""nd can be calculated as the product of FLdI21cwhereisthc momenl of inertia about thc ccntroid of the cross section along the axis parallel to the f1ange. The m imumshear stress incrosssection is expressed as FQcllc1w wherc Qc is first moment of area about the cenidparallcl to thc flange. The maximum deflection (~) of the beam wi11 be at the end of the beam calculated by the expression FL3/3Elc' Ac of the cross section of the beam is 2brtr + Iw (d -2tf). The first momcnt of area Qc can bc calculated as 0.5bftf (d -tr) + 0.51w (0.5d -tf)2. Finally.cmomcnt of inertiais obtained from (b112) ((br-tw )(d -2tf)3/12). In followingfor convcnience we will continue to use Ae Qc' andinstcad of detai1ing their dependence onedesign va1ues. Associating x with d X2 with tw' X3 with br and X4 with lr soatX=[xX2X3'x41eproblem in standard format is

Minimize .ftX): YLAC ( 1.19)

Subject to: g(X): FLx/21c-yie'd ~ 0 g2(x):FQJ1.x:x2-1:yieeg3(X):FL3'/3Elc 3m ~O0.0 1 ~ x ~ 0.25; 0.00 1 ~ X2 ~ 0.05; 0.0 1 ~ X3 ~ 0.25; 0.00 1 ~ X4 ~ 0.05

(1.20)

To solveisproblem F must bc prescribed (10000 N) L must be givcn (3 m). Matcria1 must bc selected (stee1 := 7860 kglm3; O"yield = 250 E+06 N/m2;yicld = 145 E+06 N/m2). A1so the maximum dct1ection is prcscribcd (amn~ = 0.005 m).

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????????

1.1 OPTIMIZA TION FUNDAMENTALS 15

This is an example with four design variables three inequality constraints. and no equality constraints. Othcr vcrsions of this problem can easily be formulated. It can be reduced to a two-variablc problcm. Standard failure criteria with respect to combined stresscs or principa1 strcsses can bc formulated. If the cantilevered end is bolted then additiona1 design functions regarding bolt failure need to be exarned.

Example 1.3 MyPC Company has decided to invest $12 million in acquiring severa1 ncw Component P1acement Machines to manufacture dierentkinds of moeoardsfor a new generation of personal computers. Three models of these machines are under consideration. Tota1 number of operators available is 100 becausc ofelocal labor market. A floor space constraint needs to be satisfied because ofedifferent dimensions of semachines. Additiona1 information relating to each ofemachines is given in Table 1.1.ecompany wishes to determine how m yofeach kind is appropriate to maximizc the number of boards manufactudperday. lt is difficu1t to use a figure in this problem to sct up our mathematical model. The

number of machines of cach model necds to be determined. This will serve as our design variables. Let XI represent thnumberof Component Placement Machines of Model A. Similarly X2 wi1l be associatcd with Model B and X3 with Model C.

??

D igllvariubles: xX2X3The infonnation in Table 1.1 is used to set up the design functions in terms ofedesign variables. An assumplion is made that all machines are run for three shs.The cost of acquisition of the machines is the sum ofecost per machine mu1tiplied by numberof machines (g 1)' Thmachinesmust satisfyefloor spacc constraint (82) which is defined without details in Eq. (1.23). The constraint on the number of opemtors is three timcs sumofeproduct of the number of machines of chmodel and the operators per shift (g3)' The utilization of each machine is the number of boards per hour times the number of hours the machine operates per day. Thc optimization problem can be assembled infollowingform:

f

s

Maimizef(X):18 x55xxl+18x50x~+2Ix50xx3 (1.21)

or

T Minimize f(X):-18 x 55 x x -18 x 50 X x2 -21 X 50 xX3 Suctto: g(X): 400OOOx + 600.000X2 + 700000X3 ~ 120 000 (1.22)

g2(X): 3x -X2 + X3 ~ 30 (1.23) 83(X): 3xI + 6X2 + 6x3 ~ 100 (1.24)

XI ~ 0; X2 ~ 0; X) ~ 0

? ? ? ? ? ?

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-61'.1.Y:.iK.~17

9

OPTrMIZATION FUNDAMENTALS

6

1.1

5

INTRODUCTION

Equations (1.21)(1.24)express mathematicalmodel of the problem. Note tinthis problem there is no product bcing dcsigned. Here a strategy for placing order for numberof machines is being dctermined. Equation (1.21) illusthetranslation of the m imizationobjective into anuivalentminimizing one. The inequality constraints in Equations (1.22)(1.24)are different from the previous two exnples.He theright-hand side of the S operator is nonzero. Similarly the side constraints are bound on the lower side only. This is done deliberately. There is a significanl difference betweenisproblem and the previous two. All of the design functions here are linear. This problem is c1assified as a linear prograrnming problem. The solutions 10 these typcs of problems are very different than thc solution 10 thc first two cxamplswhich are recognized as nonlinarprogramming problms.

1.1.3 Nature of Solution

leecxarnpIsincprevious section arc used to discussenature of solutions the optimization problem. Exarnples 1.1 and 1.2 dcscribeenonlinrprogramming problcm which forms the major part ofisbook. Exarnple 1.3 is a lincar programming problem which is very significant in dccision scicnccs but ra inproduct design. Its inc1usion here while necessary for complctcness is also important for understanding contemporary optimization techniques for nonlinear programming problcms. Exarnples 1.1 and 1.2a dealtwith ftrSt. Between the lwo we notice that Exarnple 1.1 is quite simple relative to Exarnple 1.2. Second the four variables in Example 1.2 make it difficult to use i1lustration to establish some of thc discussion.

16

G~hlIrepresentation 01 Example 1.1.

boofthesc functions.leualityconstraint gl is lincar. ln Equation (1.18)isis evidentbause the desi variablesappcar byemselveswithout being raised 10 a powcr o1.As we developetechniques for applying optimizationisdistinction is important to keep in mind. It is also significant graphicalrepresentation is typically restricted to two variables. Forreevariables we need a fourth dimension 10 resolve the informalIon and three-dimensional conlour plots are not easy to iIIustrate. The power of imagination is necessary to overcomesehurdles. 1problemreprcsented in Figure 1.5 providcs opportunityto identify the graphical

solution. First the feasible region is identified. In Figure 1.5isisEualityconsintabove the incquality consint.Bothfeible and oplimal must occur from this :gion.Any solution fromisregion is an acceplable (sible)solution. There are usually a large number of such solutions infinilesolutions. If optimization is involved thcn thesc choices must be rcduced to the best one withspcctto some criteria theobjeclive. 1n this problemlfigure the smallcst value offis desired. The lowest valuc of f is just less than the contour value of 308. lt is ateintersection of the two constraints. While the value of f needs to be calculated the optimal values of the design variables read from the figure are about6.75 and 13.5 respectively. Another significant item of information obtained from Figure 1.5 is glis an active constraint. While there are infinite feasible solutions 10 thc problem the optimal solution is Ultique.

Flgure 1.4

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???

[~ 5

SOLUT10N TO EXAMPLE 1.1 Thc simplcst determination of nonlinearity is through a graphical Psentationofcdcsign functions involved in thc problem. liscan be donc silyfor ODe or two variables. If the fUDCtiOD does not plot as a straight line or a planeenit is nonlincar. Figu 1.4shows a three-dimcnsional plot of Examplc 1.1. The figure is obtained using MATLAB. Chapter 2 will providdctailcdinstructions for drawing such plots for graphical optimization. The threc-dimcnsional pr entationof Exarnple 1.1 in Figure 1.4 does not real1y enhance our understanding of the problem or the solution. Figure 1.5 is an altemate repscntationof the problem. This is thoneencouraged in this book. Thc horizontal isrepscntsthe diameter (XI) and verticalaxiseheight (xu. 1n Figure 1.5 the equaliconstraintis markcd appropriately as h 1. Any prof valucs onisline will give a volume of 500 cm3 The inequa/ity constraint is shown as g 1. The pair of values on this line exact1y satisfies thc aeseticrequiment.Hash lines on the side of the inequality constraint cstablish the disallowedgionfor the design variables.leconstraints are drawn thicker for emph is.The scaled 0'eclivenClionis reprendthrough sevcral labelcd contours. Each contour is associated with axedvalue of the objcctive function and these values are shown on the figure.lerange of the two axes establishes the side COllStraints. The objective functionf and the equalily constraint h 1 'cnonlinear sincc they do not plot IIS straight lines. Rcferring to Equations (1.16) and (1.17) this is substantiated by the pl'OduCIS of the two unknowns (design varillbles) in

19

An interesting observation in the above set of equations isatthere are only two design variables but three constraints. This is a valid problem. If the constraints were equality constraintsenthis would not be an acceptable problem definition as it would violateerelationship between the number of variables and the number of equality constraints. Therefore the number of inequaliconstraints are not related to the numbrof design variables used. This fact also applies to nonlinear constrnints. Figure 1.6 is a scaled plot ofcfunctions (1.25)(1.28).Choosing the first quadrant satisfies the side constraints. 'he linearity of all of the functions is indicated by the straight-line plots on the figure. Again several contours offare shown to aid the recognition ofthe solution. All of the inequalities are drawn with hash marks indicating the nonfeasib1e side of the constraint. The feasible region is enclosed by the two axes along with constraints g2 and g3' The objective is to increasthevalue off(which is indicated by the contour levels) without leaving the feasible region. The solution can be

(1.28)

1.1 OPTIMIZATION FUNOAMENTALS

83(X): 3xI + 6x2 ~ 70

X2~0 X1 ~O;

????????

????

INTROOUCTION 18

20

? ? ? ?

??

25 20

Conlour plol 01 modllled Example 1.3.

1S number 01 Model A

10

Flgure 1.6

5

9

SOLUTION TO EXAMPLE 1.3 In contrast to Example 1.1 all of the relationships in mathematicalmodel expressed by Equations (1.21)(1.24)are linear. The word equation is generally used to describe the equivalence of two quantities on either side of the = sign. In optirnization we come across mostly inequalities.eword equatioll will beedto include both of these situations. In Example 1.1enaof solution was explained using Figu1.5.In Example 1.3 the psenceof thrdesign variables deniesisapproach. Instead of designing a new problem suppose thecompany psidenthas d idedto buy five machines of the third type. The mathematical model with two design variables Xl and X2' is reconstructed using the information that X3 is now a parameter with the value of 5. Substituting for x3 in the model for Example 1.3 and cleaning up the first cons int:

M imize

Subject to:

(1.25)

(1.26)

( 1.27)

H137 8.5

j(X): 990 XXI + 9xX2+5250

Conlour plol 01 Example 1.1.

gl):O.4xl + 0.6x2 ~ 8.5

7.5 dlameler (cm)

g2(X): 3xI -X2 ~ 25

7

Agure 1.5

6.5 5 6

20 lNTRODUCT10N

spottcd at the intersection of constraints g2 dg3. The values of the design variIcsat the solution can be read off from thc plot. It is not over yet. Before proceeding further il is neccssary to acknowledgc that a solution of 6.5 machines for Model B is unacceptable. The solution must be adjusted to the ncarest integer values. In aClual practice. this adjustmcnt has to bc made without violating any of the constraints. The need for integer variables is an important consideration in dcsign. Vablesatarquiredto have only integer values belong toetype of variables called discrete variables.'his book most1y considers continuous variables. Almost a11 of the mathematics an engin rencounters especially academically belong toEdomain of continuous variables. Thc assumption of continuity providcs very fast and cfficicnt techniques for the search of constrained optimum solutions as we will develop in this text. Many real-Iifc designs especially those that use off-the-shelf materials and components determine discrete models. Discrete programming and integer programming are disciplines that address these models. The nature ofsearch methodsatis methods used for searchingeoptimal solution in discrctc prograrnming is very different fromatin continuous programming. Many of thmethods are based on the scanning and replacement of the solutions through exhauslive search of the design space in the design region. In this book disctcproblems arc typically handled as if they were continuous problems. The conversion toediscrcte solution is the final part of the design effort and is usually performed by the designer outsidofany optimization technique we develop in this book. Chapter 8 discusses discrete optimization with examples.

Getting back toelinear programming problem just discussed it should also be noted that only the boundary of the feasible region will affect the solution. He isa way that will makeisapparcnt. Note that the objctivefunction contours arstraightlines. A different objective function will be displayed by parallellines with a different slope. Imagine these Iines on the figure instead of the actllal objective. Noteatthe solution always appears to lie at thintcrsectionof the constraints that is thcornersofthedsignspacc. Solution can nevcr comfrominside of the fcasible gionunlike in the nonlinear programming problem which accommodates solution on the boundary and from within the feasible region. Recognizing thc natu ofsolutions is important in developing a search procedure. For example in linear prograrnming problems a technique employcd to search for the optimal dsignneed only search thc boundaries particularly the intcrsection of the boundaries whereas nonlinearproblems the search procedure cannot ignore interior values.

1.1.4 Characterlstlcs of the Search Procedure

lesearch for the optimal solution will depend onenature of the problem being solved. For nonlinear problems it must consider the fact that solution could lie inside the feasible region. Example 1.1 throllgh Figurc 1.5 will provideecontext for discussing the scarch procedure for nonlincar problems. Simi1arIy the characteristics

1.1 OPT1M1ZAT10N FUNDAMENTALS 21

of the search process for linear problcms will be explored using Ex lple1.3 and Figure 1.6.

????? ??

?

? ? ?

? ? ? ? ?

??

? ?? ? ?

? ??

?

Nonllnear Prob/ems: Except for a few c1asses of problcms solutions to non1inear problems are obtained through numerical analysis. Through computr code numerica1 analysis becomes numcrical techniques. The methods or techniques for finding the solution to optimization prob1ems are called search methods. In applied mathematics and numerical thniquesthese referredto as iterative techniques. Implicit1y this means seveml tries will be necessary beforc the solution can be obtained. A1so implied is that cach try or search is executed in a consistent manner. Information from previousiteration is utilized in the computation of values in the present sequence. This consistcnt manner or press by which the search is carried out and solutiondetermined is ca1ed algorithm.This term a1so refers to the tr slationof the particular search procedure into an ordered suenceof step-bystepactions. The advantage is derived mostly in the conversion of these steps through some computer 1anguage into codeatwi1l execute onecomputcr. DevelopingEa1gorithm and trans1ating it into code that wil execute in MATLAB wil be a standard approach inisbook. The search process is started typica1y by trying to guess thinitialdesign solution.

The designer can base this selection on his experience. On sevcm1 occasions the success of optimization may hinge on his ability to choose a good initial solution. The search method even when used consistent1y responds different1y to the particular problematis being solved. The degree and type of nonlinearity may frequent1y causeemethod to fail assurning there are no inconsistencies in problem formu1ation. Oftennumberof design variablesenumber of cons intsas we1 as a bad initial guess p1ay a part in unexpected behavior under standard imDlementation of the method. This has led to the creation of seveml differcnt techniques for optimization. Practica1 experience bas a1so suggestedata class of problems respond well to certain a1gorithms. The mcthods foresearch or itmtivemethods. are current1y the only way to

implement numerical solutions to nonlinear problems irrespective of the discipline in which these problems app r.1eproblems are p1entifu1 in design optimization. optimal contro1. structural and t1uid mechanics. quantum physics. and astronomy. Whi1e optimization techniques are appliedallof the noted disciplines. in ditionaldesign

23

at a distan sS..This point satisfies the constraint h.. AirdsYwould be to choosc X.Thislies on the constraint boundary of g.. lt is to be noted that allEchoices are not feasible. Which point is chosen is dccided by the rules of a particular algorithm. Assume point X) is the next point inisdiscussion. Point X) now becomes newpoint xoAnewhdircction ~ is dennin.

Point X2 is 10d.Anoth Suenceof calculations locat X3 lenext iteration yields esolution. The essence of updatingedesi duringsucssiveiteration lies in two major computations thc seahdirectioll 8/ and stepsize.lCchange inedesign vector definIas .6X which representsveclordiffcrence for exnpleX2 -X. is obtained asS/. For each iteration the following suenceof activity can be identified:

The above is the basic structure of a typical optimization algorithm. A complete one will indicate m nerin which S and are computed. Il wil1 incJude tests which will determinc if solutionhas bcen reached (convergence) or if the procedure needs to be suspended or restarted. The diffcrent techniques to bexplorcdin this book mostly differ ineway S is established. Take a look at Figure 1.7 again. At Xo there are infinite choices for the search direction. To reach the solution in onabletime an acceptable algorithm will try not to search along directionsdonot improve the current design.lereare many ways to accomplishisstrategy.

1.1 OPTIMIZATION FUNDAMENTALS

ChooseXo Identify S Determine AX=S XMW= Xo+.6X SetXo Xnen GoToStepO

8tep 0: Step 1:

:r;-:.

INTRODUCTION

FJgure 1.7

g 8.5

A typical ssarch method.

7.5 dlamoler (cm)

22

20

5 6

? ? ? ?

Linear Programming Problem: It was previously mentionedthesolution to seproblems would be on the boundary of feasiblegioo(also called the jibledomain). In Figure 1.6e esand some of the cons intsdetennine this region. The lalter is distinguished also byepoints of interscction of the constrainLs among themselveseorigin andeintersection of the binding (active) coostraints with the axes. These are vertices or corners of the quadrilateral. Essentially the design improves by moving from one of these vertices to the next one through improving feasible designs. To do that you have to start at a feasiblcomer.This will be explored more fully in the chapter on linear programming. It is to be emphasizedatonlywith two variables can we actually see the geometric iIIustration of the techniqueisthe selection of vertices as the solution converges.

This part of Chapter 1 has dealt with the introduction of the optionproblem. While it was approached from an engineering design perspcctive the mathematical model in abstract terms is not sensitive to any pticulardiscipline. A few design rules with resp tto specific examples were examined. TIle standard mathematical model for optimization problems was established. Two broad cJasses of problems

initial startingessisepoint marked xoRemember whensolvingoptimization problem. such a figure does notist.If it didenwe can easily obtain the solution by inspection.lisstatement is espccially true ifereare more than two variables. In the event th aretwo variables isno reason not to obtain a graphical solution (Chapter 2). Thc mcthods in optimization rely on thc dircct ex polationof the concepts and ideas develodfor two variables to any number of design variables. That is the reason we will use two variables to discuss and design most of the algorithms and methods. Moreover using two variablesconceptscan also be reinfordusing geometry. The design at the point Xo is not feasible. lt is not useful to discuss whether it is an optimal solution. At least one more iteration is necessary to obnthe solution. Considerenext solution at XI. Most methods in optimization will attempt to move to X I by first identifying a search direction S 1Thisis a vector (same dimension as design variable vcctor X) poimXoloXIeactual distance from xo to XIcallit XoX.. is some multiple of SI' s SIX10nceagain is not feasible altbough it is an improvement on xo since it satisEincqualily constrainl gl (lying in lhe inlerior ofeconstraint). Altemately using lhe same scal'ch dircction S 1 another techniqueJmeodmay choose for its next point X 1'

.1""':-......1: n' '

24 INTRODUCTION

linear and nonlinear. were introduc.Thegeomeicalcharacteristics of the solution for thesc problems were shownroughgraphicaldescription ofeproblems. An overview ofesearch techniques for obtaining the solution to these problems w also iIIustrated. It was a1so mentionedatesolutions were to be obtained numerical.primarily through digital computation. The numerical methods used inisarea are iterative methods. Numerical solutions

involve two components: algorithmthat willstablish iterativeset of calculations. and second. the translation of the algorithm into computer codes. using aprogramming lang 'ge.This finished code is referred to software.Until recently in engineering education as well as practice. the compurlanguage of choice was predominantly FORTRAN with a Iittle Pascal and C. The computational environment in which it was delivered and practiced wemai'amecomputing systems. Today escene is very different. Individual personal computersCs)running Windows Macintosh LINUX or other flavors of UNIX operating systems are now the means for technical computing. They possess more power than the old mainamecomputers and they are inexpensive. The power of the graphical user interface (GUI)eway euser interacts withecomputer andesoftware h revolutionizedthe creation of software today. New programming paradigmsa11y0'ecloriented programming have led to new and improved ways to develop software. Today cnginccring softwarc is crcatcd by Buch gcncl'al programming languagcs IlS C C++ Java Pascal and Visual Basic. Many software vendors develop their own extensiol1s to the standard language providing a comple1environmentfor software devlopmentbyeuser. This makes it very convenient for users who are not necessarily programmers to geteirjob done in a reasonab1y short time. One such vendor is MaWoswitheirflagship product MA TLAB a10ng with a

complementary collection of toolboxesatdeliver specific discipline-oriented information. InisbkMATLAB is used exclusively for numerical support. No prior familiarity with MATLAB is expected though familiarity with the PC is required. Like all higher-1cvcllanguages effectiveness with this new programming resource comes with effort frequency of use. original work and a consistent manner of application. By the cnd of this book it is hoped theaderwill be very familiar with thc u

1.2 INTRODUCTION TO MA11.AII 25

1.2 INTRODUCTlON TO MATLAB

???

????

????

MATLAB is introduccd by MathWorks Inc. as the language for technical computing. Borrowing from the description in yone of its manuals MATLAB integrates mputation visualizatiodprogramming in an easy-touseenvironment where problems and solulions are expressed in farniliar mathematical notation. The typical uses for MATI..AD includ

Math and computation

A1goritllmdevelopment Modeling simulation and prototyping Data analysis exploration. and visualization Scientificand engineering graphics Application development including GUI building

The use ofMATI..AB has exploded recently and so have ifeatures. Tbis continuous enhancement makes MATI..AB an ideal vehicle for exploring problems in all disciplines that manipu1ate mathematical content. This abi1ity is multiplied by application-specific solutions through toolboxes. To learn more about the latest MATI..AB. its features the toolboxes collection of user-supported archives information about Usenet activities other forums information about books and 80 on visit MATLAIl a1 hup:/lwww.ma1hworks.com.

1.2.1 Why MA11.AB?

MATLAB is a standard tool for in'oductoryand advanced courses in mathematics engineering and science in many universities aroundworld.In industry it is a too1 of choice for reshdevelopment and analysis. Inisbook MATLAB'S basic array element is exploited to manipulate vectors and matrices that are natural to numerical techniques. In the next chapter its powerful visualization features are used for graphical optimization. Inerest of this book the interactive featurldbuilt-in support of MATLAB are utilized to translate a1gorithms into functioning code in a fraction of the time needed in other languages like C or FORTRAN. Towardeendwe will inoducethe use of its Optimi tionToolbox. Most books on optimization provide excellntcoverage on the ~hniques and

a1gorithms and some a1so provide a printed version of the code. The code isquidto be used in mainfmmEor legacy environments that must support a compiler in the languageecode is written which is predominantly FORTRAN. Each studentenanslatesthe orinted code into a forrn suitable for computation.lenext step is to compile the code debug compilation errors run programand troublhoot execution crrors before usefulsultscan be obtained. An important consideration is thestudent is expected to be conversant witheprogramming language. While FORTRAN was formally taught to enginrsseveral yrsagoisis no longer true.

7 -

26 INTRODUCTION

Many electrical engineering departments have started to embrace C and C++. PCs are great for the use of ready-made applications especially those with GUI but its use for programming in FORTRAN has yet to take off. MATI..AB suffers very few of these disadvantages. Its de is compact ditisinitive

to leam. To make problems work requires few lines of code compared to traditional languages. Instead of programming fromeground up stanrdMA TI..AB pieces of code canbeadedtogeerimplementthe new algorithm. It avoids the sndard:pru:mpilation and link suenceby using inte.reter.Another importantasoniscode po bility.ledewritten forePC version ofMATI.AB dsnot have to be changed for the Macintosh or the UNIX systcms.liswill not be true if system-dependent resources medinece.As an ilIustration of MATI.AB'S leaming agility boMATI.ABdapplied optimization arevered in this book something very diffictaomplishwisay FORTRAN and optimization or C++doptimization. FlermoreIA'..ABlike C or FORTRAN is a globally relevant product.

1.2.2 MATLAB Installalion Issues

This book does not Care if you are running MATLAB on an individual PC or Mac a networked PC or Mac or individual or networked Unix workstatioJls. MATLAB takcs care of this issu.The book assumes you have a working and appropriately Iicensed copy of MATLAB in the machine inontof you. On your personal machine for installing MATI.AB follow the instructions that accompanidthe box. Typicallyesystem administrator handles network installations. This book uses MATt.AB Version 5.2 (until Chapter 4) and Version 5.3 (erest).

Version 6 has been shipping since the fall of 2000. MAABversion 5.0 and above should work. If MATI.AB has changed any built-in commands in newer versionsthe oneedin the book it should provide information on the availability of newer version of commands(or lhrow an error a rare case). The MATLAB code and examples were created on an Inlel Pentium Pro 200 running the Windows NT 4 operating syslem. The earlier codes were also tested on a DEC Alpha server running OSFl V4. The MATI..AB version on the server was 5.1. In organizations and instilutions MATt.AB software is usually licensed on a

subscription basis. Users therefore havthe1atest version of the software in these situations. In such a case the new version (attimethe book is first published) will be Version 6 (or Release R 12). This book is based on Version 5 (Release RI1).lecode dprogramminginstructions in the book shou1d execute withoUl problem in the new version. In the new Release R12 on the PC p1atform the default appearance of MATI..AB is a composite of several windows presenting different pieces of information. This is shown in Figure 1.8. This c becustomized through the menu b byView Desktop LayoutCommand Window Only which also coponds toeprevious1ease.It is shown in Figure 1.9.lisview is simple and less distracting for getting familiar wiMATl..AB.NoteeMATt.AB command prompt is >>. This window delivers standard operating system commands for saving printing opening files and so on lhrough menus and a command bar. In order to

T

h' m.

f.

1.2 INTRODucnON TO MATl.AB 27

F1gure 1.8 MATLAB opening view.

create files (or code) another graphica1 window is neded.This iseMATLABMfileeditor/debugger window and is shown in Figu1.10.The editor uses co10r for enhancing code readability. Each of thesewindows spawn additiona1 dia10g boxes (if nec)fior.sctttin1gfieatureesduIriirworkspace browser. While this is not so much as an installation issue it wou1d be appropriale lo discusse

waykplaI1s to wriandexec MATI.ABC Gen yereare two ways to workMATI.AB:interactivelyand thro'pts.In interactive mode.rnn1andsareented dutedone at a time in the MATt.AB cOrnn1and window. Oncesess10nis over and MATt.AB exited there is no information 1tainedonset of commandswere used and what worked (unless a choice was made to save the session). Large seauensof cOrnn1aI1ds are usually dicu1u.okPackofinEeracUvsess10n.4o optifilein MATI.ABThisis called a script m-ftle. Most MATLAB fillledm-ftles. The m is a meextBIESion usually mserved for MA-I-LAB.Any code changEsmemade thmush eeditor. the changes to efile saved and e filcexecudin MATt.AB again. ExcutinMATt.ABtl ghscdptmleesispredom1lintorevertto interactive mode any time.

1.2.3 Using MATLAB the First Time

Eg

MATwil be used inte ive1yinissection. The au.thor verynyrecornn1ends a hands-on approach to understanding MATt.ABThismeans typmg tt1e codes yourself. It a1so inc1udes underslanding the syntaxrrorsand debugging and

28 INTRODUCTION

isis a Classtoom License fOt insttuctional use only. Reseatch and cometcialuse is ptohib1ted.

To qet statted select "HATLAB Help" ftom .the Help menu.

Flgure 1.9 MATLAB Command wlndow.

Optim1zation11~IATtAB 'iI John ~iley Dt.P. VenkataJ;amenl

F1gure 1.10 MaUab Editor/Debugger window.

1.2 INTRODUCTION TO MAlUB 29

rectiringecode. The author is not aware of a single example where programming was learned without the reader typing de.lnisbk MATLAB code segments are courier font withebold style lIsed for emphasizing commands or other pieces of information. Anything else is recommendations suggestions or exercises. The placeholder for command is in italic.

l

Before We Start: The folowing punctuation dspecial characters are worth noting:

Al text after the % sign is considered a comment. MATLAB ignores anything to the right of the % sign. Comments are used liberally in this book to indicate why's and what's of command or code usage. Do not overlook them. You do not have to code them in however probably two weeks later you wil not remember why you developed that particular piece of code. This is the default MATLAB prompt. A semicolon at the end of a line prevents MA ABfrom echoing the information you enter on the screen. A comma will cause the information to

A

. J

h. U

>>

????

?????

??

echo. Default is a comma. A succession of three periods at the end of the line informs MATLAB codewill continue the next line. You cannot split a variable name across two lines.You cannot continue a comment on another Iin.You can sto'pMATLAB execution and get back thcomm!mdpromptbytyping ^c (Cl-C)byh01dingdowntogeer.WilI display information on the various ways the command can be used. This is the quickest way to use help. Opens a help text window that provides more information on all of the MATLAB resources installed on your system. Provides help using a browser window.

?? ?? ? ? ? ?

?? ?????????

?

A c

>> help command_name

>> helpwin

>> helpdesk

30 INTAODUCTION

For those of you programming for the first time the equal to sign (=) has a very special meaning in all programming languages incJuding MATLAB. It is called the assignment operalor.levariable on thc Icft-hand side of the sign is assigned the value oferight-hand side. Thc actual eqllallO operation in MATLAB is usually accomplished by a double cqual to sign (==).

Some Additionsl Features

MA ABis case sensitive. An a is different than A. This is difficult for persons who arc uscd to FORTRAN. All built-in MATLAB commands arc in lowercase.

MATLAB does not necd a type definition or a dimension statement to introduce variables. It automatically crtesone on first encounter in code. The type is assigned in context

Variablnamesstart with a letter and contain up to 31 cbaractcrs (only Icttrsdigits and underscorc).

MATLAB uscs some built-in variable names. Avoid using built-in variable nam.

All numbers arc storcd internally using the long format spccificd by IEEE f1oating-point standard. These numbers have roughly a precision of 16 decimal digits. They range roughly between 10E-308 and 10E+308. Howeveyaredisplayed differcntly depending on the context.

MATLAB uses conventional decimal notation for numbers using decimal points and leading signs optionally for example 1 -9 +9.0 0.001 99.9999.

Scientific notation is cxprcssed with the letter e for cxample 2.0e-03 1.07e23 1.732e+03.

lmaginary numbers use either i or ja suffix for example li -3.l4j 3e5i.

Operators: The following arc the arithmetic operators in MATLAB.

+ Addition (wben adding matriccs/arrays size must match) Subaction(same as above)

* Multiplication (the subscripts of arrays must be consistent when multiplying them)

/ Division Power Complcx cougattranspose(0 aytranspose)

In the case of arrays each of these operators can be used wiaperiod prefixed toeoperator for cxample ( . * ) or ( ^ ) or (1).hasa special meaning in MATLAB. It imp1ieslement-by-elementoperation. It is uscful for quick computation. It has no

"

1.2 INTAODUCTlON TO MATlAD 31

relevance in mathematics or anywhere else. We wiJ1 usc it a lot in the next chapter for gClIerating data for graphical opzation.

An Interactlve Sesslon: SMA AB.On PCprcss the Start button. Press the Program button. Prcss thc MATLAB 'gramGroup. Press MA TLAB. The MATLAB Command window opens up and after somc introductory messagesE willbe a MATLAB promptindidby double forward arrow.lisis indicationMATLABis ready to aeptyourmmands. On the networked stionstyping matJab at the window prompt should do the same. Note that the MATLAB program must be in the paasdefmed ine.Iogin or.h file.Ask the lab assistant for help. ThestandardUNIX proccdtrnat the end ofelinc or befoecomment. The commcnts rclatc to what is being typed dwhat is displayed in the MATLAB command window. FI1frccyourown variatiosS. In fact to undedandreinfocommandsit is commendedyou make up your own examples often.

>>a 1.0; b 2.0; c 3.0 d 4.0; e 5.0 >> whydid only c and e echo on the screen? >> who listsall the variables in the workspace >> a givesthe value stored in a >>A 1.5; variableA >> a A % case matters >>one a; two b; three C; >> assigningvalues to new variables >>four d; five e; six pi; valueof pi available >> A1 [a b c ;' d e f] A1is a 2 by 3 matrix >> % space or comma separates columns >> % semi-colon separates rows

>> A1(22) accessesthe Matrix element on the >> % second row and second column >> size (A1l givesyou the size of the matrix >> (rowcolumIlsl >>AA1 size(A1l Whatshould happen here? >> fromprevious satementthe size of A1 >> containstwo numbers organized as a row >> matrix.This is assigned to AA1 >> size(AA1l AA1is a one by two matrix >> A1' thistransposes the matrix A1 >>B1 A1' thetranspose of matrix A1 >> isassigned to B1. B1 is a three by two matrix >> C1 '" A1 * B1 SinceA1 and B1 are matrices this >> i8a matrix multiplication >> Shouldthis multiplication be allowed? >> considermatrix multiplication in C or

32 INTRODUCTION

>> % Fortran >> thepower of MATLAB derives from its >> abilityto handle matrices like numbers >>C2 81 * A1 Howabout this? >> C1 * C2 Whatabout this? >> % read the error message >> itis quite informative >>D1 [1 2]' D1is a column vector >>C3 [C1 D1] C1is augmented by an extra column >>C3 [C3 C2(3 :)] Note is an assignment >> meansdo the right hand side and overwrite the >> oldinformation in C3 with the result >> ofthe right >> handside calculation >> Onthe right you are adding a row to current >> matrixC3. This row has the value of the third >> rowof C2 -Notice the procedure of >> identifyingthe third row. The colon >> % represents all the columns >>C4 C2 * C3 permissiblemultiplication >> Notethe presence of a scaling factor >> inthe displayed output >>CS C2 * C3 seemsto multiplyl >> Isthere a difference between C4 and CS? >> % The * represents the product of each element >> ofC2 multiplied with the corresponding >> elementof C3 >>C6 inverse(C2) % find the inverse of C2 >> apparentlyinverse is not a command in MATLAB >> ifcommand name is known it is easy to obain>> help>> lookfor inverse thiscommand will find all files >> whereit comes across the word "inverse" in >> theinitial comment lines >> Thecommand we need appears to be INV which >> saysInverse of a Matrix >> % The actual command is in lower case. To find >> outhow to use it -Now >> help inv % shows how to use the command >> inv(C2) inverseof C2 >> for i '" 1:20

f(i) i"'2; end

>> Thisis an example of a for loop

1.2 INTRODUCnON TO MAn.AD 33

>> theindex ranges from 1 to 20 in steps of >> 1(default) >> theloop is terminated with "end" >> theprompt does not appear until "end" is >> entered>> plot(sin(0.01*f)cos(0.03f)) >> xlabel('sin(O.Ol*f)') stringsappear in single >> quotes>> ylabel('cos(0.03*f)') >> legend ('Example') >> title ('A Plot Example') >> grid >> Theprevious set of commands will create plot >> labelaxes write a legend title and grid the >> plot>> exit finishedwihMATLAB

This completes the first session with MATLAB. Additiona1 commands and features will be encountered throughout the book. Inissession it is evidentMA.ABalows easy manipulation of matrices definitely in relation to other programming languages. Plotting is not difficult either. These advantages arquitesubstantial in the subject of optimization. 1n nextsession we wil1 use the editor 10 I1ccomplishesame through scripts. This session introduced

MATLAB Command window and Workspace

Variableassignment Basic matrix operations

Accessing rows and columns

Suppressing echoes

who inverse commands . * multiplication

Basic plotting commands

1.2.4 Using the Editor

In this section we will use the editor to tedrun a MATLAB a script file. Norma1ly eeditor is used to generate two kinds ofMA.ABfiles.1efiles are termed scriptftles andfunctionftles. AIoughboofthsefiIes connMATLAB commands likeeoneswe have used the second type of fiIes needs to be organized in a specified format. Bofile tysshould have the extension .m. Although these files are ASCII text fiIegeneric .m should beedbecaeMATLABshesforisex sion.hisextension is quetoMA.AB.1cscript file contains a Iist of MA TLAB commandsareexedin sequence. This is differentmthe interactive session of the previous section where

34 INTRODUCTION

MA1.ABndedtochconunand immediately. The script file is more useful when there armanyconun dsneedto be exutedto acmplish some objective like running an optimization technique. It is important to remember that MATLAB aIlows you to switch back interactive mode at any time by just typing commands ineworkspacewindow like inepvioussection. MATLAB provides an editor for creating these files on the PC platform. This editor

uses color to identify MATLAB statements and elements. It providesecurrent vaIues ofthe variables (aftereyare available in the workspace) when the mouse is over the variable name in the editor. Thel twoways to accesseeditor through the MATLAB Command window on the PC. 5tart MATLAB. This will open a MATLAB Command or Workspace window. In this window the editor can be started by using the menu oretoolbar. On the File menu click on New and choose M-fiIc. Altemately click on the Ieftmost icon on the toolbar (etooltip reads New File). The icon for the editor can aIso be placed on the desktop in which case the editor can be started by double-clicking the icon. In this event a MATlAB Command window will not be opened. The editor provides its own window for enteringscriptstatements. At this point we are ready to use the editor. Make sure youadcommentsand