Advances in Mathematical Modeling, Optimization and Optimal Control

205 Pages·2016·2.962 MB·English

by  Jean-Baptiste Hiriart-Urruty, Adam Korytowski, Helmut Maurer, Maciej Szymkat (eds.)

Springer Optimization and Its Applications 109
Jean-Baptiste Hiriart-Urruty
Adam Korytowski
Helmut Maurer
Maciej Szymkat Editors
Advances in
Mathematical Modeling,
Optimization and
Optimal Control
Springer Optimization and Its Applications
VOLUME109
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor–CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
C.A.Floudas(TexasA&MUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(LehighUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimization has been expanding in all directions at an astonishing rate
during the last few decades. New algorithmic and theoretical techniques
havebeendeveloped,thediffusionintootherdisciplineshasproceededata
rapidpace,andourknowledgeofallaspectsofthefieldhasgrownevenmore
profound.Atthesametime,oneofthemoststrikingtrendsinoptimization
is the constantly increasing emphasis on the interdisciplinary nature of the
field.Optimizationhasbeenabasictoolinallareasofappliedmathematics,
engineering,medicine,economics,andothersciences.
The series Springer Optimization and Its Applications publishes under-
graduate and graduate textbooks, monographs and state-of-the-art exposi-
tory work that focus on algorithms for solving optimization problems and
alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered
include nonlinear optimization (convex and nonconvex), network flow
problems, stochastic optimization, optimal control, discrete optimization,
multi-objectiveprogramming,descriptionofsoftwarepackages,approxima-
tiontechniquesandheuristicapproaches.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7393

Jean-Baptiste Hiriart-Urruty • Adam Korytowski
Helmut Maurer • Maciej Szymkat
Editors
Advances in Mathematical
Modeling, Optimization
and Optimal Control
123
Editors
Jean-BaptisteHiriart-Urruty AdamKorytowski
InstitutdeMathématiques DepartmentofAutomaticsandBiomedical
UniversitéPaulSabatier Engineering
Toulouse,France AGHUniversityofScienceandTechnology
Kraków,Poland
HelmutMaurer
InstituteofComputationalandApplied MaciejSzymkat
Mathematics DepartmentofAutomaticsandBiomedical
UniversityofMünster Engineering
Münster,Germany AGHUniversityofScienceandTechnology
Kraków,Poland
ISSN1931-6828 ISSN1931-6836 (electronic)
SpringerOptimizationandItsApplications
ISBN978-3-319-30784-8 ISBN978-3-319-30785-5 (eBook)
DOI10.1007/978-3-319-30785-5
LibraryofCongressControlNumber:2016939392
MathematicsSubjectClassification:49J15,49J20,49J21,49J30,49N90,26B25,49M29,97N20,47J35
©SpringerInternationalPublishingSwitzerland2016
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Contents
Introduction ...................................................................... 1
Jean-BaptisteHiriart-Urruty,AdamKorytowski,HelmutMaurer,
andMaciejSzymkat
Bregman Distances in Inverse Problems and Partial
DifferentialEquations........................................................... 3
MartinBurger
On Global Attractor for Parabolic Partial Differential
InclusionandItsTimeSemidiscretization .................................... 35
PiotrKalita
PassiveControlofSingularitiesbyTopologicalOptimization:
The Second-Order Mixed Shape Derivatives of Energy
FunctionalsforVariationalInequalities....................................... 65
GünterLeugering,JanSokołowski,andAntoniZ˙ochowski
Optimal Control for Applications in Medical and
RehabilitationTechnology:ChallengesandSolutions....................... 103
KatjaMombaur
Second-OrderOptimalityConditionsforBrokenExtremals
andBang-BangControls:TheoryandApplications ......................... 147
NikolaiP.OsmolovskiiandHelmutMaurer
v
Introduction
Jean-BaptisteHiriart-Urruty,AdamKorytowski,HelmutMaurer,
andMaciejSzymkat
Thisbookconstitutesacollectionofdevelopedversionsofplenarypaperspresented
(withoneexception)atthe16thFrench–German–PolishConferenceonOptimiza-
tion, held in Kraków in 2013. They are authored by researchers of international
reputeinthefieldofoptimizationandoptimalcontrol.Thebookincludesanumber
of new theoretical results and applications in biomechanics, medical technology,
imageprocessing,robotcontrol,etc.
Thepurposeofthebookwastogivetheauthorsanopportunitytopresenttheir
new results to a wider audience than it was possible at the conference, and in an
extended, more comprehensive form. The motivation was that the topics of the
articlesarerelatedtoareasoftheoryandapplicationsthatareofmostvividinterest
tothescientificcommunity,suchasimageprocessing,partialdifferentialinclusions,
shape optimization, optimal control in medical and rehabilitation technology, or
sufficientconditionsofoptimality.
Thefirstpaper,byMartinBurger,providesanoverviewofrecentdevelopments
relatedtoBregmandistances.Approachesininverseproblemsandimageprocessing
based on Bregman distances are discussed, which have evolved to a standard tool
J.-B.Hiriart-Urruty
InstitutdeMathématiques,UniversitéPaulSabatier,31062ToulouseCedex09,France
A.Korytowski((cid:2))
DepartmentofAutomaticsandBiomedicalEngineering,AGHUniversityofScienceand
Technology,30-059Kraków,Poland
e-mail:[e-mail]
H.Maurer
UniversityofMünster,InstituteofComputationalandAppliedMathematics,
48149Münster,Germany
M.Szymkat
AGHUniversityofScienceandTechnology,DepartmentofAppliedComputerScience,
30-059Kraków,Poland
©SpringerInternationalPublishingSwitzerland2016 1
J.-B.Hiriart-Urrutyetal.(eds.),AdvancesinMathematicalModeling,Optimization
andOptimalControl,SpringerOptimizationandItsApplications109,
DOI10.1007/978-3-319-30785-5_1
2 J.-B.Hiriart-Urrutyetal.
inthesefieldsinthelastdecade.Relatedissuesintheanalysisofnonlinearpartial
differentialequationswithavariationalstructurearealsoconsidered.
The paper by Piotr Kalita studies the operator version of a first order in time
partial differential inclusion and its time discretization by implicit Euler scheme.
Thesemidiscretetrajectoriesareprovedtoconvergetothesolutionoftheoriginal
problem. It is shown that, as times goes to infinity, all trajectories are attracted
towards the so-called global attractors. It is also proved that the semidiscrete
attractors converge upper-semicontinuously to the global attractor of the time
continuousproblem.
InthepaperbyGünterLeugering,JanSokołowski,andAntoniZ˙ochowski,non-
smoothshapeoptimizationproblemsforvariationalinequalitiesareconsidered.The
variational inequalities model elliptic boundary value problems with the Signorini
type unilateral boundary conditions. The shape functionals are given by the first
order shape derivatives of the elastic energy. The topological optimization is used
for passive control of singularities of weak solutions. The Hadamard directional
differentiability is employed to sensitivity analysis. The topological derivatives of
nonsmooth integral shape functionals for variational inequalities are derived. The
obtained expressions for derivatives prove useful in numerical optimization for
contactproblems.
Thenextpaper,byKatjaMombaur,isdevotedtoapplicationsofoptimalcontrol
and inverse optimal control in the field of medical and rehabilitation technology,
in particular in human movement analysis, therapy and improvement by means of
medical devices. Efficient methods for the solution of optimal control and inverse
optimalcontrolproblemsarediscussed.Exampleapplicationsofthesemethodsare
considered in the development of mobility aids for geriatric patients, the design
of exoskeletons, the analysis of running motions with prostheses, the optimal
functionalelectricalstimulationofhemiplegicpatients,aswellasstabilityanalysis.
The last paper, by Nikolai Osmolovskii and Helmut Maurer, provides a survey
on no-gap second-order optimality conditions in the calculus of variations and
optimalcontrol,andadiscussionoftheirfurtherdevelopment.Suchconditionsare
formulated for discontinuous controls in optimal control problems with endpoint
andmixedstate-controlconstraints,andafreecontroltime.Forproblemswiththe
control appearing linearly in the Pontryagin function, it is shown that the second-
order sufficient condition for the Induced Optimization Problem together with the
so-calledstrictbang-bangpropertyensuresecond-ordersufficientconditionsforthe
originalcontrolproblem.Thetheoreticalresultsareillustratedbythreeapplications:
to optimal control of chemotherapy of HIV, time-optimal control of robots, and
controloftheRayleighequation.
Bregman Distances in Inverse Problems
and Partial Differential Equations
MartinBurger
Abstract The aim of this paper is to provide an overview of recent development
relatedtoBregmandistancesoutsideitsnativeareasofoptimizationandstatistics.
WediscussapproachesininverseproblemsandimageprocessingbasedonBregman
distances, which have evolved to a standard tool in these fields in the last decade.
Moreover,wediscussrelatedissuesintheanalysisandnumericalanalysisofnon-
linear partial differential equations with a variational structure. For such problems
Bregmandistancesappeartobeofsimilarimportance,butarecurrentlyusedonlyin
aquitehiddenfashion.WetrytoworkoutexplicitlytheaspectsrelatedtoBregman
distances,whichalsoleadtonovelmathematicalquestionsandmayalsostimulate
furtherresearchintheseareas.
1 Introduction
Bregmandistancesfor(differentiable)convexfunctionals,originallyintroducedin
thestudyofproximalalgorithmsin[11]andnamedin[25],areawell-established
concept in continuous and discrete optimization in finite dimension. A classical
example is the celebrated Bregman projection algorithm for finding points in the
intersectionofaffinesubspaces(cf.,e.g.,[26]).Wereferto[26,53]forintroductory
andexhaustiveviewsonBregmandistancesinoptimization.
Althoughconvexfunctionalsplayaroleinmanyotherbranchesofmathematics,
e.g.,inmanyvariationalproblemsandpartialdifferentialequations,thesuitability
of Bregman distances in such fields was hardly investigated for several decades.
In mathematical imaging and inverse problems the situation changed with the
rediscovery and further development of Bregman iterations as an iterative image
restorationtechniqueinthecaseoffrequentlyusedregularizationtechniques such
astotalvariation(cf.[50]),whichledtosignificantlyimprovedresultscomparedto
standardvariationalmodelsandcouldeliminatesystematicerrorstoacertainextent
M.Burger((cid:2))
InstitutfürNumerischeundAngewandteMathematik,WestfälischeWilhelms-Universität
(WWU)Münster.Einsteinstr.62,D48149Münster,Germany
e-mail:[e-mail]
©SpringerInternationalPublishingSwitzerland2016 3
J.-B.Hiriart-Urrutyetal.(eds.),AdvancesinMathematicalModeling,Optimization
andOptimalControl,SpringerOptimizationandItsApplications109,
DOI10.1007/978-3-319-30785-5_2
4 M.Burger
(cf.[9,16]).AnotherkeyobservationincreasingtheinterestinBregmandistancesin
thesefieldswasthattheycanbeemployedforerrorestimation,inparticularfornot
strictlyconvexandnonsmoothfunctionals(cf.[14]),whichpreventnormestimates.
AlthoughtherearemanyobviouslinkstothemainrouteofresearchinBregman
distances and related optimization algorithms, there are several peculiar aspects
that deserve particular discussion. Besides missing smoothness of the considered
functionalsandthefactthatproblemsinimaging,inverseproblemsandpartialdif-
ferential equations are naturally formulated in infinite-dimensional Banach spaces
suchasthespaceoffunctionsofboundedvariationorSobolevspaces,whichhave
only been considered in few instances before, a key point is that the motivation
for using Bregman distances in these fields often differs significantly from those
in optimization and statistics. In the following we want to provide an overview of
suchquestionsandconsequentdevelopments,keepinganeyeonpotentialdirections
and questions for future research. We start with a section including definitions,
examples, and some general properties of Bregman distances, before we survey
aspectsofBregmandistancesininverseproblemsandimagingdevelopedinthelast
decade.ThenweproceedtoadiscussionofBregmandistancesinpartialdifferential
equations,whichislessexplicitandhencethemaingoalistohighlighthiddenuseof
Bregmandistancesandmaketheideamoredirectlyaccessibleforfutureresearch.
Finallyweconcludewithasectiononrelatedrecentdevelopments.
2 BregmanDistancesandTheir BasicProperties
WestartwithadefinitionofaBregmandistance.Intheremainderofthispaper,let
X beaBanachspaceandJ:X→R∪{+∞}beconvexfunctionals.Wefirstrecall
thedefinitionofsubdifferential,respectively,subgradients.
Definition1. ThesubdifferentialofaconvexfunctionalJisdefinedby
∂J(u)={p∈X∗|J(u)+(cid:6)p,v−u(cid:7)≤J(v)forallv∈X}. (1)
Anelementp∈∂J(u)iscalledsubgradient.
Having defined a subdifferential we can proceed to the definition of Bregman
distances,respectively,generalizedBregmandistancesaccordingto[44].
Definition2. The(generalized)BregmandistancerelatedtoaconvexfunctionalJ
withsubgradientpisdefinedby
Dp(v,u)=J(v)−J(u)−(cid:6)p,v−u(cid:7), (2)
J
wherep∈∂J(u).ThesymmetricBregmandistanceisdefinedby
Dp,q(u,v)=Dp(v,u)+Dq(u,v)=(cid:6)p−q,u−v(cid:7), (3)
J J J