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