Optimization Concepts and Applications in Engineering

479 Pages·2011·2.052 MB·English

by  Belegundu A., Chandrupatla T.

Contents ix
5.7 Convexity 212
5.8 SensitivityofOptimumSolutiontoProblemParameters 214
5.9 Rosen’sGradientProjectionMethodforLinearConstraints 216
5.10Zoutendijk’sMethodofFeasibleDirections(Nonlinear
Constraints) 222
5.11TheGeneralizedReducedGradientMethod(Nonlinear
Constraints) 232
5.12SequentialQuadraticProgramming(SQP) 241
5.13FeaturesandCapabilitiesofMethodsPresentedinthisChapter 247
6 PenaltyFunctions,Duality,andGeometricProgramming . . . . . . . . . 261
6.1 Introduction 261
6.2 ExteriorPenaltyFunctions 261
6.3 InteriorPenaltyFunctions 267
6.4 Duality 269
6.5 TheAugmentedLagrangianMethod 276
6.6 GeometricProgramming 281
7 DirectSearchMethodsforNonlinearOptimization . . . . . . . . . . . . . . 294
7.1 Introduction 294
7.2 CyclicCoordinateSearch 294
7.3 HookeandJeevesPatternSearchMethod 298
7.4 Rosenbrock’sMethod 301
7.5 Powell’sMethodofConjugateDirections 304
7.6 NelderandMeadSimplexMethod 307
7.7 SimulatedAnnealing(SA) 314
7.8 GeneticAlgorithm(GA) 318
7.9 DifferentialEvolution(DE) 324
7.10Box’sComplexMethodforConstrainedProblems 325
8 MultiobjectiveOptimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
8.1 Introduction 338
8.2 ConceptofParetoOptimality 339
8.3 GenerationoftheEntireParetoCurve 343
8.4 MethodstoIdentifyaSingleBestCompromiseSolution 345
9 IntegerandDiscreteProgramming. . . . . . . . . . . . . . . . . . . . . . . . . .359
9.1 Introduction 359
9.2 Zero–OneProgramming 361
9.3 BranchandBoundAlgorithmforMixedIntegers(LP-Based) 368
x Contents
9.4 GomoryCutMethod 372
9.5 Farkas’MethodforDiscreteNonlinearMonotone
StructuralProblems 377
9.6 GeneticAlgorithmforDiscreteProgramming 380
10 DynamicProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
10.1Introduction 385
10.2TheDynamicProgrammingProblemandApproach 387
10.3ProblemModelingandComputerImplementation 392
11 OptimizationApplicationsforTransportation,Assignment,
andNetworkProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
11.1Introduction 400
11.2TransportationProblem 400
11.3AssignmentProblems 408
11.4NetworkProblems 413
12 FiniteElement-BasedOptimization . . . . . . . . . . . . . . . . . . . . . . . . . 424
12.1Introduction 424
12.2DerivativeCalculations 427
12.3Sizing(i.e.,Parameter)OptimizationviaOptimality
CriteriaandNonlinearProgrammingMethods 432
12.4TopologyOptimizationofContinuumStructures 437
12.5ShapeOptimization 441
12.6OptimizationwithDynamicResponse 449
Index 461
Preface
This book is a revised and enhanced edition of the first edition. The authors have
identifiedaclearneedforteachingengineeringoptimizationinamannerthatinte-
gratestheory,algorithms,modeling,andhands-onexperiencebasedontheirexten-
sive experience in teaching, research, and interactions with students. They have
strived to adhere to this pedagogy and reinforced it further in the second edition,
withmoredetailedexplanations,anincreasednumberofsolvedexamplesandend-
of-chapterproblems,andsourcecodesonmultipleplatforms.
The development of the software, which parallels the theory, has helped to
explain the implementation aspects in the text with greater insight and accuracy.
Studentshaveintegratedtheoptimizationprogramswithsimulationcodesintheir
theses. The programs can be tried out by researchers and practicing engineers as
well. Programs on the CD-ROM have been developed in Matlab, Excel VBA,
VBScript, and Fortran. A battery of methods is available for the user. This leads
to effective solution of problems since no single method can be successful on all
problems.
The book deals with a variety of optimization problems: unconstrained, con-
strained, gradient, and nongradient techniques; duality concepts; multiobjective
optimization; linear, integer, geometric, and dynamic programming with applica-
tions; and finite element–based optimization. Matlab graphics and optimization
toolbox routines and the Excel Solver optimizer are presented in detail. Through
solved examples, problem-solving strategies are presented for handling problems
wherethenumberofvariablesdependsonthenumberofdiscretizationpointsina
meshandforhandlingtime-dependentconstraints.Chapter8dealsexclusivelywith
treatmentoftheobjectivefunctionitselfasopposedtomethodsforminimizingit.
Thisbookcanbeusedincoursesatthegraduateorsenior-undergraduatelevel
andasalearningresourceforpracticingengineers.Specifically,thetextcanbeused
xi
xii Preface
in courses on engineering optimization, design optimization, structural optimiza-
tion,andnonlinearprogramming.Thebookmaybeusedinmechanical,aerospace,
civil, industrial, architectural, chemical, and electrical engineering, as well as in
appliedmathematics.Indecidingwhichchaptersaretobecoveredinacourse,the
instructor may note the following. Chapters 1, 2, 3.1–3.5, and 8 are fundamental.
Chapters4,9,and11focusonlinearproblems,whereasChapters5–7focusonnon-
linearproblems.Evenifthefocusisonnonlinearproblems,Sections4.1–4.6present
important concepts related to constraints. Chapters 10 and 12 are specialized top-
ics.Thus,forinstance,acourseonstructuraloptimization(i.e.,finiteelement-based
optimization)maycoverChapters1–2,3.1–3.5,4.1–4.6,5,6,7.7–7.10,8,and12.
Wearegratefultothestudentsatourrespectiveinstitutionsformotivatingus
todevelopthisbook.Ithasbeenapleasureworkingwithoureditor,PeterGordon.
OPTIMIZATION CONCEPTS AND APPLICATIONS
IN ENGINEERING
SecondEdition

1
Preliminary Concepts
1.1 Introduction
Optimization is the process of maximizing or minimizing a desired objective func-
tion while satisfying the prevailing constraints. Nature has an abundance of exam-
pleswhereanoptimumsystemstatusissought.Inmetalsandalloys,theatomstake
positions of least energy to form unit cells. These unit cells define the crystalline
structureofmaterials.Aliquiddropletinzerogravityisaperfectsphere,whichis
the geometric form of least surface area for a given volume. Tall trees form ribs
near the base to strengthen them in bending. The honeycomb structure is one of
themostcompactpackagingarrangements.Geneticmutationforsurvivalisanother
exampleofnature’soptimizationprocess.Likenature,organizationsandbusinesses
have also strived toward excellence. Solutions to their problems have been based
mostlyonjudgmentandexperience.However,increasedcompetitionandconsumer
demandsoftenrequirethatthesolutionsbeoptimumandnotjustfeasiblesolutions.
A small savings in a mass-produced part will result in substantial savings for the
corporation. In vehicles, weight minimization can impact fuel efficiency, increased
payloads, or performance. Limited material or labor resources must be utilized to
maximizeprofit.Often,optimizationofadesignprocesssavesmoneyforacompany
bysimplyreducingthedevelopmentaltime.
In order for engineers to apply optimization at their work place, they must
haveanunderstandingofboththetheoryandthealgorithmsandtechniques.This
isbecausethereisconsiderableeffortneededtoapplyoptimizationtechniqueson
practicalproblemstoachieveanimprovement.Thiseffortinvariablyrequirestuning
algorithmicparameters,scaling,andevenmodifyingthetechniquesforthespecific
application. Moreover, the user may have to try several optimization methods to
findonethatcanbesuccessfullyapplied.Todate,optimizationhasbeenusedmore
asadesignordecisionaid,ratherthanforconceptgenerationordetaileddesign.In
1
2 PreliminaryConcepts
thissense,optimizationisanengineeringtoolsimilarto,say,finiteelementanalysis
(FEA).
Thisbookaimsatprovidingthereaderwithbasictheorycombinedwithdevel-
opment and use of numerical techniques. A CD-ROM containing computer pro-
grams that parallel the discussion in the text is provided. The computer programs
givethereadertheopportunitytogainhands-onexperience.Theseprogramsshould
be valuable to both students and professionals. Importantly, the optimization pro-
gramswithsourcecodecanbeintegratedwiththeuser’ssimulationsoftware.The
developmentofthesoftwarehasalsohelpedtoexplaintheoptimizationprocedures
inthewrittentextwithgreaterinsight.Severalexamplesareworkedoutinthetext;
many of these involve the programs provided. User subroutines to solve some of
theseexamplesarealsoprovidedontheCD-ROM.
1.2 HistoricalSketch
The use of a gradient method (requiring derivatives of the functions) for min-
imization was first presented by Cauchy in 1847. Modern optimization methods
were pioneered by Courant’s [1943] paper on penalty functions, Dantzig’s paper
on the simplex method for linear programming [1951]; and Karush, Kuhn, and
Tucker who derived the “KKT” optimality conditions for constrained problems
[1939, 1951]. Thereafter, particularly in the 1960s, several numerical methods to
solve nonlinear optimization problems were developed. Mixed integer program-
mingreceivedimpetusbythebranchandboundtechniqueoriginallydevelopedby
Land and Doig [1960] and the cutting plane method by Gomory [1960]. Methods
forunconstrainedminimizationincludeconjugategradientmethodsofFletcherand
Reeves[1964]andthevariablemetricmethodsofDavidon–Fletcher–Powell(DFP)
in [1959]. Constrained optimization methods were pioneered by Rosen’s gradient
projection method [1960], Zoutendijk’s method of feasible directions [1960], the
generalized reduced gradient method by Abadie and Carpenter [1969] and Fiacco
and McCormick’s SUMT techniques [1968]. Multivariable optimization needed
efficient methods for single variable search. The traditional interval search meth-
ods using Fibonacci numbers and Golden Section ratio were followed by effi-
cient hybrid polynomial-interval methods of Brent [1971] and others. Sequential
quadratic programming (SQP) methods for constrained minimization were devel-
opedinthe1970s.Developmentofinteriormethodsforlinearprogrammingstarted
withtheworkofKarmarkarin1984.HispaperandtherelatedUSpatent(4744028)
renewed interest in interior methods (see the IBM Web site for patent search:
http://patent.womplex.ibm.com/).
Also in the 1960s, alongside developments in gradient-based methods, there
were developments in nongradient or “direct” methods, principally Rosenbrock’s
method of orthogonal directions [1960], the pattern search method of Hooke and
1.2 HistoricalSketch 3
Jeeves[1961],Powell’smethodofconjugatedirections[1964],thesimplexmethod
of Nelder and Meade [1965], and the method of Box [1965]. Special methods that
exploitsomeparticularstructureofaproblemwerealsodeveloped.Dynamicpro-
grammingoriginatedfromtheworkofBellmanwhostatedtheprincipleofoptimal
policyforsystemoptimization[1952].Geometricprogrammingoriginatedfromthe
workofDuffin,Peterson,Zener[1967].Lasdon[1970]drewattentiontolarge-scale
systems.Paretooptimalitywasdevelopedinthecontextofmultiobjectiveoptimiza-
tion. More recently, there has been focus on stochastic methods, which are better
able to determine global minima. Most notable among these are genetic algo-
rithms [Holland 1975, Goldberg 1989], simulated annealing algorithms that origi-
natedfromMetropolis[1953],anddifferentialevolutionmethods[PriceandStorn,
http://www.icsi.berkeley.edu/∼storn/code.html].
In operations research and industrial engineering, use of optimization tech-
niquesinmanufacturing,production,inventorycontrol,transportation,scheduling,
networks,andfinancehasresultedinconsiderablesavingsforawiderangeofbusi-
nesses and industries. Several operations research textbooks are available to the
reader. For instance, optimization of airline schedules is an integer program that
can be solved using the branch and bound technique [Nemhauser 1997]. Shortest
path routines have been used to reroute traffic due to road blocks. The routines
mayalsobeappliedtoroutemessagesontheInternet.
The use of nonlinear optimization techniques in structural design was pio-
neeredbySchmit[1960].EarlyliteratureonengineeringoptimizationareJohnson
[1961], Wilde [1967], Fox [1971], Siddall [1972], Haug and Arora [1979], Morris
[1982], Reklaitis, Ravindran and Ragsdell [1983], Vanderplaats [1984], Papalam-
brosandWilde[1988],Banichuk[1990],HaftkaandGurdal[1991].Severalauthors
have added to this collection including books on specialized topics such as struc-
tural topology optimization [Bendsoe and Sigmund 2004], design sensitivity anal-
ysis [Haug, Choi and Komkov 1986], optimization using evolutionary algorithms
[Deb 2001] and books specifically targeting chemical, electrical, industrial, com-
puter science, and other engineering systems. We refer the reader to the bibliog-
raphy at end of this book. These, along with several others that have appeared in
thelastdecade,havemadeanimpactineducatingengineerstoapplyoptimization
techniques.Today,applicationsareeverywhere,fromidentifyingstructuresofpro-
tein molecules to tracing of electromagnetic rays. Optimization has been used for
decadesinsizingairplanewings.Thechallengeistoincreaseitsutilizationinbring-
ingoutthefinalproduct.
Widely available and relatively easy to use optimization software packages,
popular in universities, include the MATLAB optimization toolbox and the
EXCELSOLVER.AlsoavailableareGAMSmodelingpackages(http://gams.nist.
gov/) and CPLEX software (http://www.ilog.com/). Other resources include Web
sitesmaintainedbyArgonnenationallabs(http://www-fp.mcs.anl.gov/OTC/Guide/
4 PreliminaryConcepts
f
x Figure1.1. Maximizationoffisequivalent
x* tominimizationof−f.
−f
SoftwareGuide/) and by SIAM (http://www.siam.org/). GAMS is tied to a host of
optimizers.
Structural and simulation-based optimization software packages that can be
procured from companies include ALTAIR (http://www.altair.com/), GENESIS
(http://www.vrand.com/),iSIGHT(http://www.engineous.com/),modeFRONTIER
(http://www.esteco.com/), and FE-Design (http://www.fe-design.de/en/home.html).
OptimizationcapabilityisofferedinanalysiscommercialpackagessuchasANSYS
andNASTRAN.
1.3 TheNonlinearProgrammingProblem
Mostengineeringoptimizationproblemsmaybeexpressedasminimizing(ormax-
imizing)afunctionsubjecttoinequalityandequalityconstraints,whichisreferred
to as a nonlinear programming (NLP) problem. The word “programming” means
“planning.”Thegeneralformis
minimize f(x)
subject to g (x)≤0 i =1,...,m
i
(1.1)
and h (x)=0 j =1,...,(cid:2)
j
and xL ≤x≤xU
where x=(x ,x ,...,x )T is a column vector of n real-valued design variables. In
1 2 n
Eq.(1.1),f istheobjectiveorcostfunction,g’sareinequalityconstraints,andh’sare
equality constraints. The notation x0 for the starting point, x∗ for optimum, and xk
forthe(current)pointatthekthiterationwillbegenerallyused.
MaximizationversusMinimization
Notethatmaximizationoffisequivalenttominimizationof−f(Fig.1.1).
Problems may be manipulated so as to be in the form (1.1). Vectors xL, xU
representexplicitlowerandupperboundsonthedesignvariables,respectively,and