Stochastic Optimization Methods: Applications in Engineering and Operations Research

368 Pages·2015·3.34 MB·English

by  Kurt Marti

Kurt Marti
Stochastic
Optimization
Methods
Applications in Engineering and
Operations Research
Third Edition
Stochastic Optimization Methods

Kurt Marti
Stochastic Optimization
Methods
Applications in Engineering
and Operations Research
Third edition
123
KurtMarti
InstituteforMathematicsandComputer
Science
FederalArmedForcesUniversityMunich
Neubiberg/Munich
Germany
ISBN978-3-662-46213-3 ISBN978-3-662-46214-0 (eBook)
DOI10.1007/978-3-662-46214-0
LibraryofCongressControlNumber:2015933010
SpringerHeidelbergNewYorkDordrechtLondon
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Preface
Optimization problems in practice depend mostly on several model parameters,
noise factors, uncontrollableparameters,etc., which are not givenfixed quantities
attheplanningstage.Typicalexamplesfromengineeringandeconomics/operations
research are: material parameters(e.g., elasticity moduli,yield stresses, allowable
stresses, moment capacities, specific gravity), external loadings, friction coeffi-
cients, moments of inertia, length of links, mass of links, location of center of
gravityoflinks,manufacturingerrors,tolerances,noiseterms,demandparameters,
technological coefficients in input–output functions, cost factors, interest rates,
exchange rates, etc. Due to the several types of stochastic uncertainties (physical
uncertainty,economicuncertainty,statisticaluncertainty,modeluncertainty),these
parameters must be modeled by random variables having a certain probability
distribution.Inmostcases,atleastcertainmomentsofthisdistributionareknown.
In order to cope with these uncertainties, a basic procedure in the engi-
neering/economic practice is to replace first the unknown parameters by some
chosen nominal values, e.g., estimates, guesses, of the parameters. Then, the
resulting and mostly increasing deviation of the performance (output, behavior)
of the structure/system from the prescribed performance (output, behavior), i.e.,
the “tracking error”, is compensated by (online) input corrections. However, the
onlinecorrectionofasystem/structureisoftentime-consumingandcausesmostly
increasing expenses (correction or recourse costs). Very large recourse costs may
ariseincaseofdamagesorfailuresoftheplant.Thiscanbeomittedtoalargeextent
by taking into account already at the planning stage the possible consequences
of the tracking errors and the known prior and sample information about the
random data of the problem. Hence, instead of relying on ordinary deterministic
parameteroptimizationmethods—basedon some nominalparametervalues—and
applyingthenjustsomecorrectionactions,stochasticoptimizationmethodsshould
be applied: Incorporating stochastic parameter variations into the optimization
process, expensiveand increasingonline correctionexpensescan be omittedor at
leastreducedtoalargeextent.
Consequently, for the computation of robust optimal decisions/designs, i.e.,
optimaldecisionswhichareinsensitivewithrespecttorandomparametervariations,
v
vi Preface
appropriatedeterministicsubstituteproblemsmustbeformulatedfirst.Basedonthe
decisiontheoreticalprinciples,thesesubstituteproblemsdependonprobabilitiesof
failure/successand/oronmoregeneralexpectedcost/lossterms.Sinceprobabilities
and expectations are defined by multiple integrals in general, the resulting often
nonlinear and also nonconvexdeterministic substitute problems can be solved by
approximativemethodsonly.Twobasictypesofdeterministicsubstituteproblems
occurmostlyinpractice:
• Expected primary cost minimization subject to expected recourse (correction)
costconstraints
Minimization of the expected primary costs subject to expected recourse
costconstraints(reliabilityconstraints)andremainingdeterministicconstraints,
e.g., boxconstraints. In case of piecewise constantcost functions,probabilistic
objectivefunctionsand/orprobabilisticconstraintsoccur;
• ExpectedTotalCostMinimizationProblems:
Minimization of the expected total costs (costs of construction, design,
recoursecosts,etc.)subjecttotheremainingdeterministicconstraints.
Themainanalyticalpropertiesofthesubstituteproblemshavebeenexaminedin
thefirsttwoeditionsofthebook,wherealsoappropriatedeterministicandstochastic
solutionprocedurescanbefound.
Afteranoverviewonbasicmethodsforhandlingoptimizationproblemswithran-
domdata,inthispresentthirdeditiontransformationmethodsforoptimizationprob-
lems with random parameters into appropriate deterministic substitute problems
are described for basic technical and economic problems. Hence, the aim of this
present third edition is to provide now analytical and numerical tools—including
theirmathematicalfoundations—fortheapproximatecomputationofrobustoptimal
decisions/designs/controlasneededinconcreteengineering/economicapplications:
stochastic Hamiltonian method for optimal control problems with random data,
the H-minimal control and the Hamiltonian two-point boundary value problem,
Stochastic optimal open-loop feedback control as an efficient method for the
constructionof optimal feedback controlsin case of random parameters, adaptive
optimal stochastic trajectory planning and control for dynamic control systems
under stochastic uncertainty, optimal design of regulators in case of random
parameters, optimal design of structures under random external load and with
randommodelparameters.
Finally,anewtoolispresentedfortheevaluationoftheentropyofaprobability
distribution and the divergence of two probability distributions with respect to
the use in an optimal decision problem with random parameters. Applications to
statisticsaregiven.
Realizing this monograph, the author was supported by several collaborators
fromtheInstituteforMathematicsandComputerSciences.Especially,Iwouldlike
to thank Ms Elisabeth Lößl for the excellent support in the LATEX-typesetting of
the manuscript.Withouther veryprecise and carefulwork, the completionof this
projectcouldnothavebeenrealized.Moreover,IthankMsInaSteinforproviding
several figures. Last but not least, I am indebted to Springer-Verlag for inclusion
Preface vii
ofthebookintotheSpringer-program.I wouldliketo thankespeciallythe Senior
EditorforBusiness/EconomicsofSpringer-Verlag,ChristianRauscher,forhisvery
longpatienceuntilthecompletionofthisbook.
Munich,Germany KurtMarti
October2014

Preface to the First Edition
Optimization problems in practice depend mostly on several model parameters,
noise factors, uncontrollableparameters,etc., which are not givenfixed quantities
attheplanningstage.Typicalexamplesfromengineeringandeconomics/operations
research are: material parameters(e.g., elasticity moduli,yield stresses, allowable
stresses, moment capacities, specific gravity), external loadings, friction coeffi-
cients, moments of inertia, length of links, mass of links, location of center of
gravityoflinks,manufacturingerrors,tolerances,noiseterms,demandparameters,
technological coefficients in input–output functions, cost factors, etc. Due to the
severaltypesofstochasticuncertainties(physicaluncertainty,economicuncertainty,
statistical uncertainty, model uncertainty), these parameters must be modeled by
random variables having a certain probability distribution. In most cases, at least
certainmomentsofthisdistributionareknown.
In order to cope with these uncertainties, a basic procedure in the engi-
neering/economic practice is to replace first the unknown parameters by some
chosen nominal values, e.g., estimates, guesses, of the parameters. Then, the
resulting and mostly increasing deviation of the performance (output, behavior)
of the structure/system from the prescribed performance (output, behavior), i.e.,
the “tracking error,” is compensated by (online) input corrections. However, the
onlinecorrectionofasystem/structureisoftentime-consumingandcausesmostly
increasing expenses (correction or recourse costs). Very large recourse costs may
ariseincaseofdamagesorfailuresoftheplant.Thiscanbeomittedtoalargeextent
by taking into account already at the planning stage the possible consequences
of the tracking errors and the known prior and sample information about the
random data of the problem. Hence, instead of relying on ordinary deterministic
parameteroptimizationmethods—basedon some nominalparametervalues—and
applyingthenjustsomecorrectionactions,stochasticoptimizationmethodsshould
be applied:incorporatingthe stochastic parametervariationsinto the optimization
process, expensiveand increasingonline correctionexpensescan be omittedor at
leastreducedtoalargeextent.
Consequently, for the computation of robust optimal decisions/designs, i.e.,
optimaldecisionswhichareinsensitivewithrespecttorandomparametervariations,
ix