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Variational Analysis and Generalized Differentiation I: Basic Theory

Publisher: 
Springer Verlag
Number of Pages: 
579
Price: 
99.00
ISBN: 
3-540-25437-4
Date Received: 
Monday, January 23, 2006
Reviewable: 
No
Include In BLL Rating: 
No
Reviewer Email Address: 
Boris S. Mordukhovich
Series: 
Grundlehren der mathematischen Wissenschaften 330
Publication Date: 
2006
Format: 
Hardcover
Category: 
Monograph

1 Generalized Differentiation in Banach Spaces . . . . . . . . . . . . . . 3

1.1 Generalized Normals to Nonconvex Sets . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Basic Definitions and Some Properties . . . . . . . . . . . . . . . 4

1.1.2 Tangential Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.3 Calculus of Generalized Normals . . . . . . . . . . . . . . . . . . . . 18

1.1.4 Sequential Normal Compactness of Sets . . . . . . . . . . . . . . 27

1.1.5 Variational Descriptions and Minimality . . . . . . . . . . . . . . 33

1.2 Coderivatives of Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . 39

1.2.1 Basic Definitions and Representations . . . . . . . . . . . . . . . . 40

1.2.2 Lipschitzian Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.2.3 Metric Regularity and Covering . . . . . . . . . . . . . . . . . . . . . 56

1.2.4 Calculus of Coderivatives in Banach Spaces . . . . . . . . . . . 70

1.2.5 Sequential Normal Compactness of Mappings . . . . . . . . . 75

1.3 Subdifferentials of Nonsmooth Functions . . . . . . . . . . . . . . . . . . . 81

1.3.1 Basic Definitions and Relationships . . . . . . . . . . . . . . . . . . 82

1.3.2 Fr´echet-Like ε-Subgradients

and Limiting Representations . . . . . . . . . . . . . . . . . . . . . . . 87

1.3.3 Subdifferentiation of Distance Functions . . . . . . . . . . . . . . 97

1.3.4 Subdifferential Calculus in Banach Spaces . . . . . . . . . . . . 112

1.3.5 Second-Order Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . 121

1.4 Commentary to Chap. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

2 Extremal Principle in Variational Analysis . . . . . . . . . . . . . . . . 171

2.1 Set Extremality and Nonconvex Separation . . . . . . . . . . . . . . . . . 172

2.1.1 Extremal Systems of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

2.1.2 Versions of the Extremal Principle

and Supporting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 174

2.1.3 Extremal Principle in Finite Dimensions . . . . . . . . . . . . . 178

2.2 Extremal Principle in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . 180

XVIII Contents

2.2.1 Approximate Extremal Principle

in Smooth Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.2.2 Separable Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

2.2.3 Extremal Characterizations of Asplund Spaces . . . . . . . . 195

2.3 Relations with Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 203

2.3.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 204

2.3.2 Subdifferential Variational Principles . . . . . . . . . . . . . . . . . 206

2.3.3 Smooth Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 210

2.4 Representations and Characterizations in Asplund Spaces . . . . 214

2.4.1 Subgradients, Normals, and Coderivatives

in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

2.4.2 Representations of Singular Subgradients

and Horizontal Normals to Graphs and Epigraphs . . . . . 223

2.5 Versions of Extremal Principle in Banach Spaces . . . . . . . . . . . . 230

2.5.1 Axiomatic Normal and Subdifferential Structures . . . . . . 231

2.5.2 Specific Normal and Subdifferential Structures . . . . . . . . 235

2.5.3 Abstract Versions of Extremal Principle . . . . . . . . . . . . . . 245

2.6 Commentary to Chap. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

3 Full Calculus in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

3.1 Calculus Rules for Normals and Coderivatives . . . . . . . . . . . . . . . 261

3.1.1 Calculus of Normal Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 262

3.1.2 Calculus of Coderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

3.1.3 Strictly Lipschitzian Behavior

and Coderivative Scalarization . . . . . . . . . . . . . . . . . . . . . . 287

3.2 Subdifferential Calculus and Related Topics . . . . . . . . . . . . . . . . . 296

3.2.1 Calculus Rules for Basic and Singular Subgradients . . . . 296

3.2.2 Approximate Mean Value Theorem

with Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

3.2.3 Connections with Other Subdifferentials . . . . . . . . . . . . . . 317

3.2.4 Graphical Regularity of Lipschitzian Mappings . . . . . . . . 327

3.2.5 Second-Order Subdifferential Calculus . . . . . . . . . . . . . . . 335

3.3 SNC Calculus for Sets and Mappings . . . . . . . . . . . . . . . . . . . . . . 341

3.3.1 Sequential Normal Compactness of Set Intersections

and Inverse Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

3.3.2 Sequential Normal Compactness for Sums

and Related Operations with Maps . . . . . . . . . . . . . . . . . . 349

3.3.3 Sequential Normal Compactness for Compositions

of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

3.4 Commentary to Chap. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

4 Characterizations of Well-Posedness

and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

4.1 Neighborhood Criteria and Exact Bounds . . . . . . . . . . . . . . . . . . 378

4.1.1 Neighborhood Characterizations of Covering . . . . . . . . . . 378

Contents XIX

4.1.2 Neighborhood Characterizations of Metric Regularity

and Lipschitzian Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 382

4.2 Pointbased Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

4.2.1 Lipschitzian Properties via Normal

and Mixed Coderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

4.2.2 Pointbased Characterizations of Covering

and Metric Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

4.2.3 Metric Regularity under Perturbations . . . . . . . . . . . . . . . 399

4.3 Sensitivity Analysis for Constraint Systems . . . . . . . . . . . . . . . . . 406

4.3.1 Coderivatives of Parametric Constraint Systems . . . . . . . 406

4.3.2 Lipschitzian Stability of Constraint Systems . . . . . . . . . . 414

4.4 Sensitivity Analysis for Variational Systems . . . . . . . . . . . . . . . . . 421

4.4.1 Coderivatives of Parametric Variational Systems . . . . . . 422

4.4.2 Coderivative Analysis of Lipschitzian Stability . . . . . . . . 436

4.4.3 Lipschitzian Stability under Canonical Perturbations . . . 450

4.5 Commentary to Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

1 Generalized Differentiation in Banach Spaces . . . . . . . . . . . . . . 3

1.1 Generalized Normals to Nonconvex Sets . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Basic Definitions and Some Properties . . . . . . . . . . . . . . . 4

1.1.2 Tangential Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.3 Calculus of Generalized Normals . . . . . . . . . . . . . . . . . . . . 18

1.1.4 Sequential Normal Compactness of Sets . . . . . . . . . . . . . . 27

1.1.5 Variational Descriptions and Minimality . . . . . . . . . . . . . . 33

1.2 Coderivatives of Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . 39

1.2.1 Basic Definitions and Representations . . . . . . . . . . . . . . . . 40

1.2.2 Lipschitzian Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.2.3 Metric Regularity and Covering . . . . . . . . . . . . . . . . . . . . . 56

1.2.4 Calculus of Coderivatives in Banach Spaces . . . . . . . . . . . 70

1.2.5 Sequential Normal Compactness of Mappings . . . . . . . . . 75

1.3 Subdifferentials of Nonsmooth Functions . . . . . . . . . . . . . . . . . . . 81

1.3.1 Basic Definitions and Relationships . . . . . . . . . . . . . . . . . . 82

1.3.2 Fr´echet-Like ε-Subgradients

and Limiting Representations . . . . . . . . . . . . . . . . . . . . . . . 87

1.3.3 Subdifferentiation of Distance Functions . . . . . . . . . . . . . . 97

1.3.4 Subdifferential Calculus in Banach Spaces . . . . . . . . . . . . 112

1.3.5 Second-Order Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . 121

1.4 Commentary to Chap. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

2 Extremal Principle in Variational Analysis . . . . . . . . . . . . . . . . 171

2.1 Set Extremality and Nonconvex Separation . . . . . . . . . . . . . . . . . 172

2.1.1 Extremal Systems of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

2.1.2 Versions of the Extremal Principle

and Supporting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 174

2.1.3 Extremal Principle in Finite Dimensions . . . . . . . . . . . . . 178

2.2 Extremal Principle in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . 180

XVIII Contents

2.2.1 Approximate Extremal Principle

in Smooth Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.2.2 Separable Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

2.2.3 Extremal Characterizations of Asplund Spaces . . . . . . . . 195

2.3 Relations with Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 203

2.3.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 204

2.3.2 Subdifferential Variational Principles . . . . . . . . . . . . . . . . . 206

2.3.3 Smooth Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 210

2.4 Representations and Characterizations in Asplund Spaces . . . . 214

2.4.1 Subgradients, Normals, and Coderivatives

in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

2.4.2 Representations of Singular Subgradients

and Horizontal Normals to Graphs and Epigraphs . . . . . 223

2.5 Versions of Extremal Principle in Banach Spaces . . . . . . . . . . . . 230

2.5.1 Axiomatic Normal and Subdifferential Structures . . . . . . 231

2.5.2 Specific Normal and Subdifferential Structures . . . . . . . . 235

2.5.3 Abstract Versions of Extremal Principle . . . . . . . . . . . . . . 245

2.6 Commentary to Chap. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

3 Full Calculus in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

3.1 Calculus Rules for Normals and Coderivatives . . . . . . . . . . . . . . . 261

3.1.1 Calculus of Normal Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 262

3.1.2 Calculus of Coderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

3.1.3 Strictly Lipschitzian Behavior

and Coderivative Scalarization . . . . . . . . . . . . . . . . . . . . . . 287

3.2 Subdifferential Calculus and Related Topics . . . . . . . . . . . . . . . . . 296

3.2.1 Calculus Rules for Basic and Singular Subgradients . . . . 296

3.2.2 Approximate Mean Value Theorem

with Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

3.2.3 Connections with Other Subdifferentials . . . . . . . . . . . . . . 317

3.2.4 Graphical Regularity of Lipschitzian Mappings . . . . . . . . 327

3.2.5 Second-Order Subdifferential Calculus . . . . . . . . . . . . . . . 335

3.3 SNC Calculus for Sets and Mappings . . . . . . . . . . . . . . . . . . . . . . 341

3.3.1 Sequential Normal Compactness of Set Intersections

and Inverse Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

3.3.2 Sequential Normal Compactness for Sums

and Related Operations with Maps . . . . . . . . . . . . . . . . . . 349

3.3.3 Sequential Normal Compactness for Compositions

of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

3.4 Commentary to Chap. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

4 Characterizations of Well-Posedness

and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

4.1 Neighborhood Criteria and Exact Bounds . . . . . . . . . . . . . . . . . . 378

4.1.1 Neighborhood Characterizations of Covering . . . . . . . . . . 378

Contents XIX

4.1.2 Neighborhood Characterizations of Metric Regularity

and Lipschitzian Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 382

4.2 Pointbased Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

4.2.1 Lipschitzian Properties via Normal

and Mixed Coderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

4.2.2 Pointbased Characterizations of Covering

and Metric Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

4.2.3 Metric Regularity under Perturbations . . . . . . . . . . . . . . . 399

4.3 Sensitivity Analysis for Constraint Systems . . . . . . . . . . . . . . . . . 406

4.3.1 Coderivatives of Parametric Constraint Systems . . . . . . . 406

4.3.2 Lipschitzian Stability of Constraint Systems . . . . . . . . . . 414

4.4 Sensitivity Analysis for Variational Systems . . . . . . . . . . . . . . . . . 421

4.4.1 Coderivatives of Parametric Variational Systems . . . . . . 422

4.4.2 Coderivative Analysis of Lipschitzian Stability . . . . . . . . 436

4.4.3 Lipschitzian Stability under Canonical Perturbations . . . 450

4.5 Commentary to Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

1 Generalized Differentiation in Banach Spaces . . . . . . . . . . . . . . 3

1.1 Generalized Normals to Nonconvex Sets . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Basic Definitions and Some Properties . . . . . . . . . . . . . . . 4

1.1.2 Tangential Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.1.3 Calculus of Generalized Normals . . . . . . . . . . . . . . . . . . . . 18

1.1.4 Sequential Normal Compactness of Sets . . . . . . . . . . . . . . 27

1.1.5 Variational Descriptions and Minimality . . . . . . . . . . . . . . 33

1.2 Coderivatives of Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . 39

1.2.1 Basic Definitions and Representations . . . . . . . . . . . . . . . . 40

1.2.2 Lipschitzian Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.2.3 Metric Regularity and Covering . . . . . . . . . . . . . . . . . . . . . 56

1.2.4 Calculus of Coderivatives in Banach Spaces . . . . . . . . . . . 70

1.2.5 Sequential Normal Compactness of Mappings . . . . . . . . . 75

1.3 Subdifferentials of Nonsmooth Functions . . . . . . . . . . . . . . . . . . . 81

1.3.1 Basic Definitions and Relationships . . . . . . . . . . . . . . . . . . 82

1.3.2 Fr´echet-Like ε-Subgradients

and Limiting Representations . . . . . . . . . . . . . . . . . . . . . . . 87

1.3.3 Subdifferentiation of Distance Functions . . . . . . . . . . . . . . 97

1.3.4 Subdifferential Calculus in Banach Spaces . . . . . . . . . . . . 112

1.3.5 Second-Order Subdifferentials . . . . . . . . . . . . . . . . . . . . . . . 121

1.4 Commentary to Chap. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

2 Extremal Principle in Variational Analysis . . . . . . . . . . . . . . . . 171

2.1 Set Extremality and Nonconvex Separation . . . . . . . . . . . . . . . . . 172

2.1.1 Extremal Systems of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

2.1.2 Versions of the Extremal Principle

and Supporting Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 174

2.1.3 Extremal Principle in Finite Dimensions . . . . . . . . . . . . . 178

2.2 Extremal Principle in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . 180

XVIII Contents

2.2.1 Approximate Extremal Principle

in Smooth Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 180

2.2.2 Separable Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

2.2.3 Extremal Characterizations of Asplund Spaces . . . . . . . . 195

2.3 Relations with Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 203

2.3.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 204

2.3.2 Subdifferential Variational Principles . . . . . . . . . . . . . . . . . 206

2.3.3 Smooth Variational Principles . . . . . . . . . . . . . . . . . . . . . . . 210

2.4 Representations and Characterizations in Asplund Spaces . . . . 214

2.4.1 Subgradients, Normals, and Coderivatives

in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

2.4.2 Representations of Singular Subgradients

and Horizontal Normals to Graphs and Epigraphs . . . . . 223

2.5 Versions of Extremal Principle in Banach Spaces . . . . . . . . . . . . 230

2.5.1 Axiomatic Normal and Subdifferential Structures . . . . . . 231

2.5.2 Specific Normal and Subdifferential Structures . . . . . . . . 235

2.5.3 Abstract Versions of Extremal Principle . . . . . . . . . . . . . . 245

2.6 Commentary to Chap. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

3 Full Calculus in Asplund Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

3.1 Calculus Rules for Normals and Coderivatives . . . . . . . . . . . . . . . 261

3.1.1 Calculus of Normal Cones . . . . . . . . . . . . . . . . . . . . . . . . . . 262

3.1.2 Calculus of Coderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

3.1.3 Strictly Lipschitzian Behavior

and Coderivative Scalarization . . . . . . . . . . . . . . . . . . . . . . 287

3.2 Subdifferential Calculus and Related Topics . . . . . . . . . . . . . . . . . 296

3.2.1 Calculus Rules for Basic and Singular Subgradients . . . . 296

3.2.2 Approximate Mean Value Theorem

with Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

3.2.3 Connections with Other Subdifferentials . . . . . . . . . . . . . . 317

3.2.4 Graphical Regularity of Lipschitzian Mappings . . . . . . . . 327

3.2.5 Second-Order Subdifferential Calculus . . . . . . . . . . . . . . . 335

3.3 SNC Calculus for Sets and Mappings . . . . . . . . . . . . . . . . . . . . . . 341

3.3.1 Sequential Normal Compactness of Set Intersections

and Inverse Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

3.3.2 Sequential Normal Compactness for Sums

and Related Operations with Maps . . . . . . . . . . . . . . . . . . 349

3.3.3 Sequential Normal Compactness for Compositions

of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

3.4 Commentary to Chap. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

4 Characterizations of Well-Posedness

and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

4.1 Neighborhood Criteria and Exact Bounds . . . . . . . . . . . . . . . . . . 378

4.1.1 Neighborhood Characterizations of Covering . . . . . . . . . . 378

Contents XIX

4.1.2 Neighborhood Characterizations of Metric Regularity

and Lipschitzian Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 382

4.2 Pointbased Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

4.2.1 Lipschitzian Properties via Normal

and Mixed Coderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

4.2.2 Pointbased Characterizations of Covering

and Metric Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

4.2.3 Metric Regularity under Perturbations . . . . . . . . . . . . . . . 399

4.3 Sensitivity Analysis for Constraint Systems . . . . . . . . . . . . . . . . . 406

4.3.1 Coderivatives of Parametric Constraint Systems . . . . . . . 406

4.3.2 Lipschitzian Stability of Constraint Systems . . . . . . . . . . 414

4.4 Sensitivity Analysis for Variational Systems . . . . . . . . . . . . . . . . . 421

4.4.1 Coderivatives of Parametric Variational Systems . . . . . . 422

4.4.2 Coderivative Analysis of Lipschitzian Stability . . . . . . . . 436

4.4.3 Lipschitzian Stability under Canonical Perturbations . . . 450

4.5 Commentary to Chap. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

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