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Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications

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The book establishes links between regularity and derivative concepts of nonsmooth analysis and studies of solution methods and stability for optimization, complementarity and equilibrium problems. In developing necessary tools, it presents, in particular:
an extended analysis of Lipschitz functions and the calculus of their generalized derivatives, including regularity, successive approximation and implicit functions for multivalued mappings;
a unified theory of Lipschitzian critical points in optimization and other variational problems, with relations to reformulations by penalty, barrier and NCP functions;
an analysis of generalized Newton methods based on linear and nonlinear approximations;
the interpretation of hypotheses, generalized derivatives and solution methods in terms of original data and quadratic approximations;
a rich collection of instructive examples and exercises.Audience: Researchers, graduate students and practitioners in various fields of applied mathematics, engineering, OR and economics. Also university teachers and advanced students who wish to get insights into problems, future directions and recent developments.

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Contents
Introduction List of Results Basic Notation 1Basic Concepts 1.1Formal Settings 1.2Multifunctions and Derivatives 1.3Particular Locally Lipschitz Functions and Related Definitions Generalized Jacobians of Locally Lipschitz Functions PseudoSmoothness and D°f Piecewise Functions NCP Functions 1.4RegularityDefinitions of Definitions of Lipschitz Properties Regularity Definitions Functions and Multifunctions 1.5Related Definitions Types of Semicontinuity Metric, Pseudo,Regularity; Openness with Linear Rate Upper Calmness and Upper Regularity at a Set 1.6First Motivations Parametric Global Minimizers Parametric Local Minimizers EpiConvergence 2Regularity and Consequences 2.1Upper Regularity at Points and Sets Characterization by Increasing Functions Optimality Conditions Linear Inequality Systems with Variable Matrix Application to Lagrange Multipliers Upper Regularity and Newton’s Method
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xi xix xxv 1 1 2 4 4 4 5 5 6 6 7 9 10 10 12 13 14 15 16 17
19 19 19 25 28 30 31
PseudoRegularity 2.2.1The Family of Inverse Functions 2.2.2Ekeland Points and Uniform Lower Semicontinuity 2.2.3Special Multifunctions Level Sets of L.s.c. Functions Cone Constraints Lipschitz Operators with Images in Hilbert Spaces NecessaryOptimalityConditions 2.2.4Intersection Maps and Extension of MFCQ Intersection with a QuasiLipschitz Multifunction Special Cases Intersections with Hyperfaces
Characterizations of Regularity by Derivatives 3.1Strong Regularity and Thibault’s Limit Sets 3.2Upper Regularity and Contingent Derivatives 3.3PseudoRegularity and Generalized Derivatives Contingent Derivatives Proper Mappings Closed Mappings Coderivatives Vertical Normals
2.2
5Closed Mappings in Finite Dimension 5.1Closed Multifunctions in Finite Dimension 5.1.1Summary of Regularity Conditions via Derivatives 5.1.2Regularity of the Convex Subdifferential 5.2Continuous and Locally Lipschitz Functions 5.2.1PseudoRegularity and Exact Penalization 5.2.2forSpecial Statements 5.2.3PseudoLipschitz MapsContinuous Selections of 5.3Implicit Lipschitz Functions on
4
Nonlinear Variations and Implicit Functions 4.1PseudoRegularitySuccessive Approximation and Persistence of 4.2Persistence of Upper Regularity Persistence Based on Kakutani’s Fixed Point Theorem Persistence Based on Growth Conditions 4.3Implicit Functions
Contents
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Analysis of Generalized Derivatives 6.1General Properties for Abstract and Polyhedral Mappings 6.2Derivatives for Lipschitz Functions in Finite Dimension 6.3Relations between and 6.4Chain Rules of Equation Type 6.4.1Chain Rules for and
89 89 89 92 93 94 96 99 100
105 105 110 113 115 115
71 72 77 77 79 82
61 61 63 63 64 64 64 66 67
32 34 37 43 43 44 46 47 49 49 54 58
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Contents
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6.4.2 Newton Maps and Semismoothness121 6.5Mean Value Theorems, Taylor Expansion and Quadratic Growth131 6.6Contingent Derivatives of Implicit (Multi) Functions and Sta tionary Points136 6.6.1 Contingent Derivative of an Implicit (Multi)Function137 6.6.2 Contingent Derivative of a General Stationary Point Map141 Critical Points and Generalized Kojima–Functions149 7.1Motivation and Definition149 KKT Points and Critical Points in Kojima’s Sense150 Generalized Kojima–Functions – Definition151 7.2Examples and Canonical Parametrizations154 The Subdifferential of a Convex Maximum Function154 Complementarity Problems156 Generalized Equations157 Nash Equilibria159 Piecewise Affine Bijections160 7.3Derivatives and Regularity of Generalized Kojima–Functions160 Properties ofN160 Formulas for Generalized Derivatives164 Regularity Characterizations by Stability Systems167 Geometrical Interpretation168 7.4Discussion of Particular Cases170 7.4.1 The Case of Smooth Data170 7.4.2Strong Regularity of Complementarity Problems175 7.4.3 Reversed Inequalities177 7.5Pseudo–Regularity versus Strong Regularity178
Parametric Optimization Problems 8.1The Basic Model 8.2Critical Points under Perturbations 8.2.1 Strong Regularity Geometrical Interpretation Direct Perturbations for the Quadratic Approximation Strong Regularity of Local Minimizers under LICQ 8.2.2 Local Upper Lipschitz Continuity Reformulation of the CStability System Geometrical Interpretation Direct Perturbations for the Quadratic Approximation 8.3Stationary and Optimal Solutions under Perturbations 8.3.1 Contingent Derivative of the Stationary Point Map The Case of Locally Lipschitzian F The Smooth Case 8.3.2 Local Upper Lipschitz Continuity Injectivity and SecondOrder Conditions
183 185 187 187 189 190 191 193 194 196 197 198 199 200 202 203 205
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8.4
Contents
Conditions via Quadratic Approximation Linearly Constrained Programs 8.3.3Upper Regularity Upper Regularity of Isolated Minimizers SecondOrder Optimality Conditions for Programs 8.3.4Strongly Regular and PseudoLipschitz Stationary Points Strong Regularity PseudoLipschitz Property Taylor Expansion of Critical Values 8.4.1Marginal Map under Canonical Perturbations 8.4.2Marginal Map under Nonlinear Perturbations Formulas under Upper Regularity of Stationary Points Formulas under Strong Regularity Formulas in Terms of the Critical Value Function Given under Canonical Perturbations
Derivatives and Regularity of Further Nonsmooth Maps 9.1Generalized Derivatives for Positively Homogeneous Functions 9.2NCP Functions Case (i): Descent Methods Case (ii): Newton Methods 9.3The CDerivative of the MaxFunction Subdifferential Contingent Limits Characterization offor MaxFunctions: Special Structure Characterization offor MaxFunctions: General Structure Application 1 Application 2
10Newton’s Method for Lipschitz Equations 10.1Linear Auxiliary Problems 10.1.1Dense Subsets and Approximations of M 10.1.2Particular Settings 10.1.3and NCP FunctionsRealizations for 10.2FunctionsThe Usual Newton Method for 10.3Nonlinear Auxiliary Problems 10.3.1Convergence 10.3.2Necessity of the Conditions
11Particular Newton Realizations and Solution Methods 11.1Perturbed Kojima Systems Quadratic Penalties Logarithmic Barriers 11.2Particular Newton Realizations and SQPModels
208 209 210 211 215 217 217 220 221 222 225 225 227
229
231 231 236 237 238 241 243 244 251 253 254
257 257 260 261 262 265 265 267 270
275 276 276 276 278
Contents
12Basic Examples and Exercises 12.1Basic Examples 12.2Exercises Appendix Ekeland’s Variational Principle Approximation by Directional Derivatives Proofof= T(NM) = NTM + TNM TF Constraint Qualifications
Bibliography Index
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