Uncertainty Quantification in Variational Inequalities: Theory, Numerics, and Applications

405 Pages·2021·8.456 MB·English

by  Joachim Gwinner, Baasansuren Jadamba, Akhtar A. Khan, Fabio Raciti

Uncertainty
Quantification in
Variational Inequalities

Uncertainty
Quantification in
Variational Inequalities
Theory, Numerics, and
Applications
Joachim Gwinner
University of the Federal Army Munich, Germany
Baasansuren Jadamba
Rochester Institute of Technology, USA
Akhtar A. Khan
Rochester Institute of Technology, USA
Fabio Raciti
University of Catania, Italy
First edition published 2022
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Library of Congress Cataloging‑in‑Publication Data
Names: Gwinner, Joachim, author. | Jadamba, Baasansuren, author. | Khan,
Akhtar A., author. | Raciti, Fabio, author.
Title: Uncertainty quantification in variational inequalities : theory,
numerics, and applications / Joachim Gwinner, University of the Federal
Army Munich, Germany, Baasansuren Jadamba, Rochester Institute of
Technology, USA, Akhtar A. Khan, Rochester Institute of Technology, USA,
Fabio Raciti, University of Catania, Italy.
Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2022.
| Includes bibliographical references and index.
Identifiers: LCCN 2021028860 (print) | LCCN 2021028861 (ebook) | ISBN
9781138626324 (hardback) | ISBN 9781032148496 (paperback) | ISBN
9781315228969 (ebook)
Subjects: LCSH: Measurement uncertainty (Statistics) | Variational
inequalities (Mathematics) | Research--Methodology.
Classification: LCC QA276.8 .G85 2022 (print) | LCC QA276.8 (ebook) | DDC
003/.54--dc23
LC record available at https://lccn.loc.gov/2021028860
LC ebook record available at https://lccn.loc.gov/2021028861
ISBN: 978-1-138-62632-4 (hbk)
ISBN: 978-1-032-14849-6 (pbk)
ISBN: 978-1-315-22896-9 (ebk)
DOI: 10.1201/9781315228969
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Typeset in Nimbus Roman
by KnowledgeWorks Global Ltd.
To
Franco Giannessi
On the occasion of his 85th birthday

Contents
ListofFigures xi
ListofTables xiii
SymbolDescription xv
Preface xvii
I VariationalInequalities 1
1 Preliminaries 3
1.1 ElementsofFunctionalAnalysis . . . . . . . . . . . . . . . . . . . 3
1.2 FundamentalsofMeasureTheoryandIntegration . . . . . . . . . . 9
1.3 EssentialsofOperatorTheory . . . . . . . . . . . . . . . . . . . . 15
1.4 AnOverviewofConvexAnalysisandOptimization . . . . . . . . 20
1.5 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 25
2 Probability 27
2.1 ProbabilityMeasure . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 ConditionalProbabilityandIndependence . . . . . . . . . . . . . . 29
2.3 RandomVariablesandExpectation . . . . . . . . . . . . . . . . . 30
2.4 Correlation,Independence,andConditionalExpectation . . . . . . 33
2.5 ModesofConvergenceofRandomVariables . . . . . . . . . . . . 36
2.6 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 37
3 ProjectionsonConvexSets 39
3.1 ProjectionsonConvexSetsinHilbertSpaces . . . . . . . . . . . . 40
3.2 ProjectionsonConvexSetsinBanachSpaces . . . . . . . . . . . . 52
3.2.1 MetricProjection . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.2 GeneralizedProjection . . . . . . . . . . . . . . . . . . . . 53
3.3 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 55
vii
viii Contents
4 VariationalandQuasi-VariationalInequalities 57
4.1 IllustrativeExamples . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 LinearVariationalInequalities . . . . . . . . . . . . . . . . . . . . 60
4.2.1 SomeBasicResults . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 RegularizationforLinearVariationalInequalities . . . . . . 66
4.2.3 RecessionApproachforLinearVariationalInequalities . . . 69
4.3 NonlinearVariationalInequalities . . . . . . . . . . . . . . . . . . 72
4.3.1 ExistenceResultsandStabilityofSolutions . . . . . . . . . 72
4.3.2 VariationalInequalitieswithMonotone-TypeMaps . . . . . 78
4.3.3 SomeGeometricandHypercircleEstimates . . . . . . . . . 84
4.3.4 RegularizationandConvergenceRates. . . . . . . . . . . . 88
4.4 Quasi-VariationalInequalities . . . . . . . . . . . . . . . . . . . . 99
4.5 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 105
5 NumericalMethodsforVariationalandQuasi-VariationalInequalities 107
5.1 ProjectionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.1.1 ProjectionMethodswithaConstantStep-Length . . . . . . 108
5.1.2 ProjectionMethodswithaVariableStep-Length . . . . . . 111
5.1.3 AProjectionMethodwithHypercircleErrorBounds . . . . 122
5.2 ExtragradientMethods . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.1 TheExtragradientMethodofKorpelevichandItsVariants . 126
5.2.2 HyperplaneExtragradientMethods . . . . . . . . . . . . . 131
5.2.3 SubgradientExtragradientMethods . . . . . . . . . . . . . 132
5.2.4 ProjectedReflectedGradientMethod . . . . . . . . . . . . 134
5.3 GapFunctionsandDescentMethods . . . . . . . . . . . . . . . . 139
5.3.1 Auchmuty’sGapFunctions . . . . . . . . . . . . . . . . . 139
5.3.2 TheRegularizedGapFunction . . . . . . . . . . . . . . . . 142
5.3.3 TheD-GapFunction . . . . . . . . . . . . . . . . . . . . . 148
5.4 TheAuxiliaryProblemPrinciple . . . . . . . . . . . . . . . . . . . 151
5.5 RelaxationMethodforVariationalInequalities . . . . . . . . . . . 155
5.6 ProjectionMethodsforQuasi-VariationalInequalities . . . . . . . 160
5.7 ConvergenceofRecursiveSequences . . . . . . . . . . . . . . . . 167
5.8 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 171
II UncertaintyQuantification 173
PrologueonUncertaintyQuantification 175
6 AnLp-ApproachforVariationalInequalitieswithUncertainData 177
6.1 LinearVariationalInequalitieswithRandomData . . . . . . . . . 177
6.1.1 AProbabilisticApproximationScheme . . . . . . . . . . . 183
6.1.2 AnIllustrativeExample . . . . . . . . . . . . . . . . . . . 191
6.2 NonlinearVariationalInequalitieswithRandomData . . . . . . . . 194
Contents ix
6.2.1 AProbabilisticApproximationScheme . . . . . . . . . . . 197
6.3 RegularizationofVariationalInequalitieswithRandomData . . . . 199
6.4 VariationalInequalitieswithMean-ValueConstraints . . . . . . . . 203
6.4.1 AProbabilisticApproximationScheme . . . . . . . . . . . 205
6.5 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 209
7 ExpectedResidualMinimization(ERM) 213
7.1 ERMforLinearComplementarityProblems . . . . . . . . . . . . 213
7.2 ERMforNonlinearComplementarityProblems . . . . . . . . . . . 222
7.3 ERMforVariationalInequalities . . . . . . . . . . . . . . . . . . . 224
7.4 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 230
8 StochasticApproximationApproach 233
8.1 StochasticApproximation.AnOverview . . . . . . . . . . . . . . 233
8.2 GradientandSubgradientStochasticApproximation . . . . . . . . 235
8.3 StochasticApproximationforVariationalInequalities . . . . . . . . 245
8.4 StochasticIterativeRegularization . . . . . . . . . . . . . . . . . . 251
8.5 StochasticExtragradientMethod . . . . . . . . . . . . . . . . . . . 258
8.6 IncrementalProjectionMethod . . . . . . . . . . . . . . . . . . . 268
8.7 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 273
III Applications 277
9 UncertaintyQuantificationinElectricPowerMarkets 279
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.2 TheModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.2.1 EquilibriumConditionsfortheConsumers . . . . . . . . . 281
9.2.2 EquilibriumConditionsforthePowerSuppliers . . . . . . 281
9.2.3 EquilibriumConditionsforthePowerGenerators . . . . . . 282
9.3 CompleteSupplyChainEquilibriumConditions . . . . . . . . . . 283
9.4 NumericalExperiments . . . . . . . . . . . . . . . . . . . . . . . 289
9.5 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 290
10 UncertaintyQuantificationinMigrationModels 293
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.2 ASimpleModelofPopulationDistributions . . . . . . . . . . . . 294
10.3 AMoreRefinedModel . . . . . . . . . . . . . . . . . . . . . . . . 297
10.4 NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . 301
10.5 CommentsandBibliographicalNotes . . . . . . . . . . . . . . . . 305