變分分析

內(nèi)容概要

本書(shū)從該理論的最初起源—積分函數(shù)的最小化開(kāi)始，對(duì)該理論做了較深的討論。變分觀點(diǎn)的發(fā)展很大程度上和優(yōu)化、平衡、控制這些理論是緊密相關(guān)的。書(shū)中在一個(gè)統(tǒng)一的框架之中，全面講述了經(jīng)典分析和凸分析之外的變分幾何和次微積分知識(shí)。也講述了集收斂、集值映射和epi收斂、對(duì)偶和正則被積函數(shù)。
目次：最大和最??；凸性；柱體；集合凸性；集值映射；變分幾何；上境圖極限；次梯度和次導(dǎo)數(shù)；Lipschitzian性質(zhì)；次微積分；對(duì)偶化；單調(diào)映射；二階理論；可測(cè)性。
讀者對(duì)象：數(shù)學(xué)專(zhuān)業(yè)的研究生、老師和相關(guān)的科研人員。

書(shū)籍目錄

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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