变分分析

书籍目录

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 Continuity

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.Continruty 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

Cbapter 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 R，elations

G.R；ecession 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

F.Aubin Property and Mordukhovich Criterion

G.Metric R，egularity and Openness

H.Semiderivatives and Strict Graphical Derivatives

I.Other Properties

J.R，ademacher's Theorem and Consequences

K.Mollifiers and Extremals

Commentary

Chapter 10.Subdifferential Calculus

A.Optimality and Normals to Level Sets

B.Basic Chain Rule

C.Parametric Optimality

D.R，escaling

E.Piecewise Linear—Quadratic Functions

F.Amenable Sets and Functions

G.Semiderivatives and Subsmoothness

H.Coderivative Calculus

I.Extensions

Commentary

Chapter 11.Dualization

A.Legendre—Fenchel Transform

B.Special Cases of Conjugacy

C.The R，ole of Differentiability

D.Piecewise Linear—Quadratic Functions

E.Polar Sets and Gauges

F.Dual Operations

G.Duality in Convergence

H.Dual Problems of Optimization

I.Lagrangian Functions

J.Minimax Problems

K.Augmented Lagrangians and Nonconvex Duality

L.Generalized Conjugacy

Commentary

Chapter 12.Monotone Mappings

A.Monotonicity Tests and Maximality

B.Minty Parameterization

C， Connections with Convex Functions

D.Graphical Convergence

E.Domains and Ranges

F.Preservation of Maximality

G.Monotone Variational Inequalities

H.Strong Monotonicity and Strong Convexity

I.Continuity and Differentiability

Commentary

Chapter 13.Second—Order Theory

A.Second—Order Differentiability

B.Second Subderivatives

C.Calculus Rules

D.Convex Functions and Duality

E.Second—Order Optimality

F.Prox—Regularity

G.Subgradient Proto—Differentiability

H.Subgradient Coderivatives and Perturbation

I.Further Derivative Properties

J.Parabolic Subderivatives

Commentary

Chapter 14.Measurability

A.Measurable Mappings and Selections

B.Preservation of Measurability

C.Limit Operations

D.Normal Integrands

E.Operations on Integrands

F.Integral Functionals

Commentary

References

Index of Statements

Index of Notation

Index of Topics

作者简介

本书从该理论的最初起源—积分函数的最小化开始，对该理论做了较深的讨论。变分观点的发展很大程度上和优化、平衡、控制这些理论是紧密相关的。书中在一个统一的框架之中，全面讲述了经典分析和凸分析之外的变分几何和次微积分知识。也讲述了集收敛、集值映射和epi收敛、对偶和正则被积函数。

目次：最大和最小；凸性；柱体；集合凸性；集值映射；变分几何；上境图极限；次梯度和次导数；Lipschitzian性质；次微积分；对偶化；单调映射；二阶理论；可测性。

读者对象：数学专业的研究生、老师和相关的科研人员。

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