Nonsmooth Equations in Optimization

Regularity, Calculus, Methods and Applications

By: Diethard Klatte, B. Kummer

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Hardcover | 31 May 2002

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368 Pages

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Many questions dealing with solvability, stability and solution methods for va- ational inequalities or equilibrium, optimization and complementarity problems lead to the analysis of certain (perturbed) equations. This often requires a - formulation of the initial model being under consideration. Due to the specific of the original problem, the resulting equation is usually either not differ- tiable (even if the data of the original model are smooth), or it does not satisfy the assumptions of the classical implicit function theorem. This phenomenon is the main reason why a considerable analytical inst- ment dealing with generalized equations (i.e., with finding zeros of multivalued mappings) and nonsmooth equations (i.e., the defining functions are not c- tinuously differentiable) has been developed during the last 20 years, and that under very different viewpoints and assumptions. In this theory, the classical hypotheses of convex analysis, in particular, monotonicity and convexity, have been weakened or dropped, and the scope of possible applications seems to be quite large. Briefly, this discipline is often called nonsmooth analysis, sometimes also variational analysis. Our book fits into this discipline, however, our main intention is to develop the analytical theory in close connection with the needs of applications in optimization and related subjects. Main Topics of the Book 1. Extended analysis of Lipschitz functions and their generalized derivatives, including "Newton maps" and regularity of multivalued mappings. 2. Principle of successive approximation under metric regularity and its - plication to implicit functions.

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Non-Fiction Medicine Medicine in General Mathematics Optimisation Game Theory Linear Programming Computing & I.T.Computer Programming & Software Development Applied Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms

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
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Nonsmooth Equations in Optimization

Regularity, Calculus, Methods and Applications

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Hardcover

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