Introduction | p. xi |
List of Results | p. xix |
Basic Notation | p. xxv |
Basic Concepts | p. 1 |
Formal Settings | p. 1 |
Multifunctions and Derivatives | p. 2 |
Particular Locally Lipschitz Functions and Related Definitions | p. 4 |
Generalized Jacobians of Locally Lipschitz Functions | p. 4 |
Pseudo-Smoothness and D[degree] f | p. 4 |
Piecewise C[superscript 1] Functions | p. 5 |
NCP Functions | p. 5 |
Definitions of Regularity | p. 6 |
Definitions of Lipschitz Properties | p. 6 |
Regularity Definitions | p. 7 |
Functions and Multifunctions | p. 9 |
Related Definitions | p. 10 |
Types of Semicontinuity | p. 10 |
Metric, Pseudo-, Upper Regularity; Openness with Linear Rate | p. 12 |
Calmness and Upper Regularity at a Set | p. 13 |
First Motivations | p. 14 |
Parametric Global Minimizers | p. 15 |
Parametric Local Minimizers | p. 16 |
Epi-Convergence | p. 17 |
Regularity and Consequences | p. 19 |
Upper Regularity at Points and Sets | p. 19 |
Characterization by Increasing Functions | p. 19 |
Optimality Conditions | p. 25 |
Linear Inequality Systems with Variable Matrix | p. 28 |
Application to Lagrange Multipliers | p. 30 |
Upper Regularity and Newton's Method | p. 31 |
Pseudo-Regularity | p. 32 |
The Family of Inverse Functions | p. 34 |
Ekeland Points and Uniform Lower Semicontinuity | p. 37 |
Special Multifunctions | p. 43 |
Level Sets of L.s.c. Functions | p. 43 |
Cone Constraints | p. 44 |
Lipschitz Operators with Images in Hilbert Spaces | p. 46 |
Necessary Optimality Conditions | p. 47 |
Intersection Maps and Extension of MFCQ | p. 49 |
Intersection with a Quasi-Lipschitz Multifunction | p. 49 |
Special Cases | p. 54 |
Intersections with Hyperfaces | p. 58 |
Characterizations of Regularity by Derivatives | p. 61 |
Strong Regularity and Thibault's Limit Sets | p. 61 |
Upper Regularity and Contingent Derivatives | p. 63 |
Pseudo-Regularity and Generalized Derivatives | p. 63 |
Contingent Derivatives | p. 64 |
Proper Mappings | p. 64 |
Closed Mappings | p. 64 |
Coderivatives | p. 66 |
Vertical Normals | p. 67 |
Nonlinear Variations and Implicit Functions | p. 71 |
Successive Approximation and Persistence of Pseudo-Regularity | p. 72 |
Persistence of Upper Regularity | p. 77 |
Persistence Based on Kakutani's Fixed Point Theorem | p. 77 |
Persistence Based on Growth Conditions | p. 79 |
Implicit Functions | p. 82 |
Closed Mappings in Finite Dimension | p. 89 |
Closed Multifunctions in Finite Dimension | p. 89 |
Summary of Regularity Conditions via Derivatives | p. 89 |
Regularity of the Convex Subdifferential | p. 92 |
Continuous and Locally Lipschitz Functions | p. 93 |
Pseudo-Regularity and Exact Penalization | p. 94 |
Special Statements for m = n | p. 96 |
Continuous Selections of Pseudo-Lipschitz Maps | p. 99 |
Implicit Lipschitz Functions on R[superscript n] | p. 100 |
Analysis of Generalized Derivatives | p. 105 |
General Properties for Abstract and Polyhedral Mappings | p. 105 |
Derivatives for Lipschitz Functions in Finite Dimension | p. 110 |
Relations between Tf and [partial differential]f | p. 113 |
Chain Rules of Equation Type | p. 115 |
Chain Rules for Tf and Cf with f [set membership] C[superscript 0,1] | p. 115 |
Newton Maps and Semismoothness | p. 121 |
Mean Value Theorems, Taylor Expansion and Quadratic Growth | p. 131 |
Contingent Derivatives of Implicit (Multi-) Functions and Stationary Points | p. 136 |
Contingent Derivative of an Implicit (Multi-)Function | p. 137 |
Contingent Derivative of a General Stationary Point Map | p. 141 |
Critical Points and Generalized Kojima-Functions | p. 149 |
Motivation and Definition | p. 149 |
KKT Points and Critical Points in Kojima's Sense | p. 150 |
Generalized Kojima-Functions - Definition | p. 151 |
Examples and Canonical Parametrizations | p. 154 |
The Subdifferential of a Convex Maximum Function | p. 154 |
Complementarity Problems | p. 156 |
Generalized Equations | p. 157 |
Nash Equilibria | p. 159 |
Piecewise Affine Bijections | p. 160 |
Derivatives and Regularity of Generalized Kojima-Functions | p. 160 |
Properties of N | p. 160 |
Formulas for Generalized Derivatives | p. 164 |
Regularity Characterizations by Stability Systems | p. 167 |
Geometrical Interpretation | p. 168 |
Discussion of Particular Cases | p. 170 |
The Case of Smooth Data | p. 170 |
Strong Regularity of Complementarity Problems | p. 175 |
Reversed Inequalities | p. 177 |
Pseudo-Regularity versus Strong Regularity | p. 178 |
Parametric Optimization Problems | p. 183 |
The Basic Model | p. 185 |
Critical Points under Perturbations | p. 187 |
Strong Regularity | p. 187 |
Geometrical Interpretation | p. 189 |
Direct Perturbations for the Quadratic Approximation | p. 190 |
Strong Regularity of Local Minimizers under LICQ | p. 191 |
Local Upper Lipschitz Continuity | p. 193 |
Reformulation of the C-Stability System | p. 194 |
Geometrical Interpretation | p. 196 |
Direct Perturbations for the Quadratic Approximation | p. 197 |
Stationary and Optimal Solutions under Perturbations | p. 198 |
Contingent Derivative of the Stationary Point Map | p. 199 |
The Case of Locally Lipschitzian F | p. 200 |
The Smooth Case | p. 202 |
Local Upper Lipschitz Continuity | p. 203 |
Injectivity and Second-Order Conditions | p. 205 |
Conditions via Quadratic Approximation | p. 208 |
Linearly Constrained Programs | p. 209 |
Upper Regularity | p. 210 |
Upper Regularity of Isolated Minimizers | p. 211 |
Second-Order Optimality Conditions for C[superscript 1,1] Programs | p. 215 |
Strongly Regular and Pseudo-Lipschitz Stationary Points | p. 217 |
Strong Regularity | p. 217 |
Pseudo-Lipschitz Property | p. 220 |
Taylor Expansion of Critical Values | p. 221 |
Marginal Map under Canonical Perturbations | p. 222 |
Marginal Map under Nonlinear Perturbations | p. 225 |
Formulas under Upper Regularity of Stationary Points | p. 225 |
Formulas under Strong Regularity | p. 227 |
Formulas in Terms of the Critical Value Function Given under Canonical Perturbations | p. 229 |
Derivatives and Regularity of Further Nonsmooth Maps | p. 231 |
Generalized Derivatives for Positively Homogeneous Functions | p. 231 |
NCP Functions | p. 236 |
Descent Methods | p. 237 |
Newton Methods | p. 238 |
The C-Derivative of the Max-Function Subdifferential | p. 241 |
Contingent Limits | p. 243 |
Characterization of C [partial differential subscript c]f for Max-Functions: Special Structure | p. 244 |
Characterization of C [partial differential subscript c]f for Max-Functions: General Structure | p. 251 |
Application 1 | p. 253 |
Application 2 | p. 254 |
Newton's Method for Lipschitz Equations | p. 257 |
Linear Auxiliary Problems | p. 257 |
Dense Subsets and Approximations of M | p. 260 |
Particular Settings | p. 261 |
Realizations for locPC[superscript 1] and NCP Functions | p. 262 |
The Usual Newton Method for PC[superscript 1] Functions | p. 265 |
Nonlinear Auxiliary Problems | p. 265 |
Convergence | p. 267 |
Necessity of the Conditions | p. 270 |
Particular Newton Realizations and Solution Methods | p. 275 |
Perturbed Kojima Systems | p. 276 |
Quadratic Penalties | p. 276 |
Logarithmic Barriers | p. 276 |
Particular Newton Realizations and SQP-Models | p. 278 |
Basic Examples and Exercises | p. 287 |
Basic Examples | p. 287 |
Exercises | p. 296 |
Appendix | p. 303 |
Ekeland's Variational Principle | p. 303 |
Approximation by Directional Derivatives | p. 304 |
Proof of TF = T(NM) = NTM + TNM | p. 306 |
Constraint Qualifications | p. 307 |
Bibliography | p. 311 |
Index | p. 325 |
Table of Contents provided by Ingram. All Rights Reserved. |