变分分析

Chapter 1. Max and Min

A. Penalties and Constraints

B. Epigraphs and Semicontinuity

C. Attainment of a Minimum

D. Continuity, Closure and Growth

E. Extended Arithmetic

F. Parametric Dependence

G. Moreau Envelopes

H. Epi-Addition and Epi-Multiplication

I*. Auxiliary Facts and Principles

Commentary

Chapter 2. Convexity

A. Convex Sets and Functions

B. Level Sets and Intersections

C. Derivative Tests

D. Convexity in Operations

E. Convex Hulls

F. Closures and Contimuty

G.* Separation

H* Relative Interiors

I* Piecewise Linear Functions

J* Other Examples

Commentary

Chapter 3. Cones and Cosmic Closure

A. Direction Points

B. Horizon Cones

C. Horizon Functions

D. Coercivity Properties

E* Cones and Orderings

F* Cosmic Convexity

G* Positive Hulls

Commentary

Chapter 4. Set Convergence

A. Inner and Outer Limits

B. Painleve-Kuratowski Convergence

C. Pompeiu-Hausdorff Distance

D. Cones and Convex Sets

E. Compactness Properties

F. Horizon Limits

G* Contimuty of Operations

H* Quantification of Convergence

I* Hyperspace Metrics

Commentary

Chapter 5. Set-Valued Mappings

A. Domains, Ranges and Inverses

B. Continuity and Semicontimuty

C. Local Boundedness

D. Total Continuity

E. Pointwise and Graphical Convergence

F. Equicontinuity of Sequences

G. Continuous and Uniform Convergence

H* Metric Descriptions of Convergence

I* Operations on Mappings

J* Generic Continuity and Selections

Commentary .

Chapter 6. Variational Geometry

A. Tangent Cones

B. Normal Cones and Clarke Regularity

C. Smooth Manifolds and Convex Sets

D. Optimality and Lagrange Multipliers

E. Proximal Normals and Polarity

F. Tangent-Normal Relations

G* Recession Properties

H* Irregularity and Convexification

I* Other Formulas

Commentary

Chapter 7. Epigraphical Limits

A. Pointwise Convergence

B. Epi-Convergence

C. Continuous and Uniform Convergence

D. Generalized Differentiability

E. Convergence in Minimization

F. Epi-Continuity of Function-Valued Mappings

G. Continuity of Operations

H* Total Epi-Convergence

I* Epi-Distances

J* Solution Estimates

Commentary

Chapter 8. Subderivatives and Subgradients

A. Subderivatives of Functions

B. Subgradients of Functions

C. Convexity and Optimality

D. Regular Subderivatives

E. Support Functions and Subdifferential Duality

F. Calmness

G. Graphical Differentiation of Mappings

H* Proto-Differentiability and Graphical Regularity

I* Proximal Subgradients

J* Other Results

Commentary

Chapter 9. Lipschitzian Properties

A. Single-Valued Mappings

B. Estimates of the Lipschitz Modulus

C. Subdifferential Characterizations

D. Derivative Mappings and Their Norms

E. Lipschitzian Concepts for Set-Valued Mappings

……

Chapter 10. Subdifferential Calculus

Chapter 11. Dualization

Chapter 12. Monotone Mappings

Chapter 13. Second-Order Theory

Chapter 14. Measurability

In this book we aim to present， in a unified framework， a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization， equilibrium， control， and stability of linear and nonlinear systems. The title Variational Analysis refiects this breadth.
　　For a long time， variational problems have been identified mostly with the 'calculus of variations'. In that venerable subject， built around the minimization of integral functionals， constraints were relatively simple and much of the focus was on infinite-dimensional function spaces. A major theme was the exploration of variations around a point， within the bounds imposed by the constraints， in order to help characterize solutions and portray them in terms of 'variational principles'. Notions of perturbation， approximation and even generalized differentiability were extensively investigated， Variational theory progressed also to the study of so-called stationary points， critical points， and other indications of singularity that a point might have relative to its neighbors， especially in association with existence theorems for differential equations.