變分分析

《變分分析》是2013年世界圖書出版公司北京公司出版的圖書，作者是R.TyrrellRockafellar、RogerJ-BWets。

基本介紹

• 書名：變分分析
• 作者：R. Tyrrell Rockafellar、Roger J-B Wets
• ISBN：9787510061363
• 出版社：世界圖書出版公司北京公司
• 出版時間：2013年10月
• 裝幀：平裝
• 原作名：Variational Analysis

內容簡介

本書從該理論的最初起源—積分函式的最小化開始，對該理論做了較深的討論。變分觀點的發展很大程度上和最佳化、平衡、控制這些理論是緊密相關的。書中在一個統一的框架之中，全面講述了經典分析和凸分析之外的變分幾何和次微積分知識。也講述了集收斂、集值映射和epi收斂、對偶和正則被積函式。

目錄

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

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