Convex and Set-Valued Analysis

Selected Topics

Aram V. Arutyunov, Valeri Obukhovskii

• Publisher: De Gruyter
• Year: 2016
• eBook ISBN: 9783110460414
• Edition: 1
• Topic: Mathematics
• Subtopic: Optimization
• Index: Mathematics

This textbook is devoted to a compressed and self-contained exposition of two important parts of contemporary mathematics: convex and set-valued analysis. In the first part, properties of convex sets, the theory of separation, convex functions and their differentiability, properties of convex cones in finite- and infinite-dimensional spaces are discussed. The second part covers some important parts of set-valued analysis. There the properties of the Hausdorff metric and various continuity concepts of set-valued maps are considered. The great attention is paid also to measurable set-valued functions, continuous, Lipschitz and some special types of selections, fixed point and coincidence theorems, covering set-valued maps, topological degree theory and differential inclusions.

Contents:
Preface
Part I: Convex analysis
Convex sets and their properties
The convex hull of a set. The interior of convex sets
The affine hull of sets. The relative interior of convex sets
Separation theorems for convex sets
Convex functions
Closedness, boundedness, continuity, and Lipschitz property of convex functions
Conjugate functions
Support functions
Differentiability of convex functions and the subdifferential
Convex cones
A little more about convex cones in infinite-dimensional spaces
A problem of linear programming
More about convex sets and convex hulls
Part II: Set-valued analysis
Introduction to the theory of topological and metric spaces
The Hausdorff metric and the distance between sets
Some fine properties of the Hausdorff metric
Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps
A base of topology of the spaceH c ( X )
Measurable set-valued maps. Measurable selections and measurable choice theorems
The superposition set-valued operator
The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations
Special selections of set-valued maps
Differential inclusions
Fixed points and coincidences of maps in metric spaces
Stability of coincidence points and properties of covering maps
Topological degree and fixed points of set-valued maps in Banach spaces
Existence results for differential inclusions via the fixed point method
Notation
Bibliography
Index

Part I: Convex analysis

1Convex sets and their properties

First, for the reader’s convenience and the completeness of the presentation, let us recall the definitions of certain standard objects, such as real linear space, normed, Euclidean space and so on. But beforehand, let us make an important remark. Namely, without loss of understanding, the reader may suppose, for simplicity, that the spaces considered in most of this chapter are finite-dimensional arithmetical spaces.

Recall that a real linear space X is the commutative group with respect to addition and for each element (called vector) x ∈ X and a real number α ∈ ℝ the multiplication αx ∈ X is defined, satisfying the following axioms: for arbitrary α, β ∈ ℝ, x, y ∈ X we have

–distributivity with respect to the addition of vectors: α(x + y) = αx + αy;

–distributivity with respect to the addition of scalars: (α + β)x = αx + βx;

–associativity: (αβ)x = α(βx);

–the property of unity: 1 ⋅ x = x.

We may consider the following examples of real linear spaces.

(1)C [a, b], the space of continuous functions on the interval [a, b].

(2)ℝn, the set of all ordered collections of n real numbers (or n-dimensional arithmetical space).

(3)The space of rectangular n × m matrices.

(4)Ck[a, b], the space of functions continuous on the interval [a, b] and k times continuously differentiable on the interval (a, b).

A real linear spa...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• Contents
• Part I: Convex analysis
• Part II: Set-valued analysis
• Notation
• Bibliography
• Index