The Second Edition of Applied Optimization with MATLAB Programming enables readers to harness all the features of MATLAB to solve optimization problems using a variety of linear and nonlinear design optimization techniques. By breaking down complex mathematical concepts into simple ideas and offering plenty of easy-to-follow examples, this text is an ideal introduction to the field. Examples come from all engineering disciplines as well as science, economics, operations research, and mathematics, helping readers understand how to apply optimization techniques to solve actual problems.
Applied Optimization with MATLAB Programming develops all necessary mathematical concepts, illustrates abstract mathematical ideas of optimization using MATLAB's rich graphics features, and introduces new programming skills incrementally as optimization concepts are presented. This valuable learning tool:
Applied Optimization with MATLAB Programming, 2nd Edition.pdf
The book was made possible through support from John WileydSonsInc.dMathWorks Inc. Sincere thanks are owed to Bob Argentierisenior editor at Jobn Wiley for accepting the proposaland who allalong displayed a lot of patiencein getting the book movingforward.Same is due to Bob Hilbed-associate managing editor at JohnWiley for his impssivework at c1eaning up the manuscript. BrianSnapp New Media editor at John Wiley created companionweb site(www.wiley.comlvenkat) and w alsobe rnaintaining it. NaomiFemandesomMathWorks Inc. saw to itat1 had the latest version ofMATLAB as soon as it was available. My regard for Dr. Angelo MieleProfessor Emeritus at Rice University is more than can be expressedin these Iines. It was he who introduced me to 1hsubjeclofoptimization and demonstrated the effectiveness of simplepresenution-I will always regard him as a great teacher. Of coursemy family deserves special rnention for putting up with all the Notnow"Later"How about tornorrow?" din2debuggingecode. Sp ialksare toA ana dVinayak rny offspring titheir patience understanding andencouragement. The auorapologizes for any shortcornings on thepresentation and welcomes comments criticisms and suggestions forimprovement at all times.
Optimization has become a necessary part of design activity inall major disciplines. These disciplines are not restricted toengineering. The motivation to produce economically relevantproducts or services with embedded quality is the principal reasonforisinclusion. Improved production and design tools wiasynergistic thrust through inexpensive computational resourceshave aided the consideration of optimization methods in newdevelopments particularly engineering products. Even in the absenccof a tangible product optimization ideas provide the ability todefine and explore problems while focusing on solutionsatsubscribeto some measure of usefulness. Generally the use of the wordoptimization implisthc best result under thcircumstances.Thisincludes the particular set of constraints on the developmentrcsources current knowledge market conditions and so on. Every oneof us has probably used thc term at some time to describeeprimaryquality of our work or endeavor. It is probablyemosl used or abusedlerm in advertising and presentations. Neverthclesscability to makethe best choice is a perpelual desire among us all.
Optimization is frequently associated with design be it aproduct service or strategy. Aerospace design wasamongearliestdisciplines to embrace optimization in a significantway driven by a natura1 need to lower the tremendous costassociated with carrying unnecessyweight in aerospace vehicles.Minimum mass structures arl norm.Optimization forms part of thepsyche of every aerospace designer. Saving on fuel throughtrjectorydesign was another problem that suggested itself. Verysoon enti engineeringcommunity could recognizecneed to definesolutions based on merit. Recognizing the desire for optimizationand actua1ly implementing we twodifferent issues.
primarily relatc to :;crvIces or strategy. It is nowapparcntatevery activity except aSeticprovides the scope foroptimization.lisjustifies looking atestudy of ootimizonas atoolcanbe applied 10 a variety of disciplines. If so the myriad
justifieseprerequisi thatyou must be capable of dcsigning the0tif you are planning to applyetechniqucs of optimi tion.It is al80a good idea 10 recognizc that optimization is a procedurc forscarching 1he best design among candidatscach of which can producean acceptable product. The oced for thc objec1 or product is notquestioned here. but this may be due to a decision based 00optimization applied in another disciplinc.
implies youcannot obtain one of the consaintsfrom elementaryarithmetic operations on the rcmaining constraints. This serves 10ensure that the mathematical search for solution will not fail.These tcchniques are based on methods from Iincar algebra. In thestandard format for optimization problems theua1ityconsaintsarewritten with a 0 on the right-hand side. This meanseequalityconstraint in thc first example will be expssedas
Figure 1.2 shows a side view .of the beam carrying the load andFigure 1.3 a typical cr.oss s nofthben.The I-shaped cross sectionis symmetric. The symbols d tw. br and tr c.orrespond to thedep.ofthe bcam thickness of the web width of thflangeand thickness.of the flange respectively. Thesquantitiesare sufficient todefineEcr.oss section. A handbo.ok or textb.ook on strcngth ofmaterials can aidedvelopment.of thdesignfunctions. An importantassumption for this problem iswe will be w.orking within theelastic Iimit of thc material where there is a Iinear relati.onshipbetweenestress and the strain. AlI .of the variables identified inFigure 1.3 will s nglyaffect the soluti.on. They are designvariables. The applied f.orce F will also directly affect theproblem. So will its locati.on L. H.ow about material?Stecl isdcfinitely supcri.or 1.0 c.opper for the beam. Should F L and thematerial pr.operties be included as design variables?
leecxarnpIsincprevious section arc used to discussenature ofsolutions the optimization problem. Exarnples 1.1 and 1.2dcscribeenonlinrprogramming problcm which forms the major partofisbook. Exarnple 1.3 is a lincar programming problem which isvery significant in dccision scicnccs but ra inproduct design. Itsinc1usion here while necessary for complctcness is also importantfor understanding contemporary optimization techniques fornonlinear programming problcms. Exarnples 1.1 and 1.2a dealtwithftrSt. Between the lwo we notice that Exarnple 1.1 is quite simplerelative to Exarnple 1.2. Second the four variables in Example 1.2make it difficult to use i1lustration to establish some of thcdiscussion.
bo ofthesc functions.leualityconstraint gl is lincar. lnEquation (1.18)isis evidentbause the desi variablesappcarbyemselveswithout being raised 10 a powcr o1.As wedevelopetechniques for applying optimizationisdistinction isimportant to keep in mind. It is also significantgraphicalrepresentation is typically restricted to two variables.Forreevariables we need a fourth dimension 10 resolve theinformalIon and three-dimensional conlour plots are not easy toiIIustrate. The power of imagination is necessary toovercomesehurdles.
1problemreprcsented in Figure 1.5 providcs opportunitytoidentify the graphical solution. First the feasible region isidentified. In Figure 1.5isisEualitycons intabove the incqualityconsint.Bothfeible and oplimal must occur from this :gion.Anysolution fromisregion is an acceplable ( sible)solution. There areusually a large number of such solutions infinilesolutions. Ifoptimization is involved thcn thesc choices must be rcduced to thebest one with spcctto some criteria theobjeclive. 1n thisproblemlfigure the smallcst value offis desired. The lowest valucof f is just less than the contour value of 308. lt isateintersection of the two constraints. While the value of f needsto be calculated the optimal values of the design variables readfrom the figure are about6.75 and 13.5 respectively. Anothersignificant item of information obtained from Figure 1.5 is glis anactive constraint. While there are infinite feasible solutions 10thc problem the optimal solution is Ultique.
SOLUT10N TO EXAMPLE 1.1 Thc simplcst determination ofnonlinearity is through a graphical P sentationofcdcsign functionsinvolved in thc problem. liscan be donc silyfor ODe or twovariables. If the fUDCtiOD does not plot as a straight line or aplaneenit is nonlincar. Figu 1.4shows a three-dimcnsional plot ofExamplc 1.1. The figure is obtained using MATLAB. Chapter 2 willproviddctailcdinstructions for drawing such plots for graphicaloptimization. The threc-dimcnsional pr entationof Exarnple 1.1 inFigure 1.4 does not real1y enhance our understanding of the problemor the solution. Figure 1.5 is an altemate repscntationof theproblem. This is thoneencouraged in this book. Thc horizontal isrepscntsthe diameter (XI) and verticalaxiseheight (xu. 1n Figure 1.5the equaliconstraintis markcd appropriately as h 1. Any prof valucsonisline will give a volume of 500 cm3 The inequa/ity constraint isshown as g 1. The pair of values on this line exact1y satisfies thcaeseticrequi ment.Hash lines on the side of the inequalityconstraint cstablish the disallowed gionfor the designvariables.leconstraints are drawn thicker for emph is.The scaled0'eclivenClionis reprendthrough sevcral labelcd contours. Eachcontour is associated with axedvalue of the objcctive function andthese values are shown on the figure.lerange of the two axesestablishes the side COllStraints. The objective functionf and theequalily constraint h 1 'cnonlinear sincc they do not plot IISstraight lines. Rcferring to Equations (1.16) and (1.17) this issubstantiated by the pl'OduCIS of the two unknowns (designvarillbles) in
This part of Chapter 1 has dealt with the introduction of theoptionproblem. While it was approached from an engineering designperspcctive the mathematical model in abstract terms is notsensitive to any pticulardiscipline. A few design rules with resptto specific examples were examined. TIle standard mathematicalmodel for optimization problems was established. Two broad cJassesof problems 2ff7e9595c
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