Thạc Sĩ Coderivatives of normal cone mappings and applications

Contents
Table of Notations vi
List of Figures viii
Introduction ix
Chapter 1. Preliminary 1
1.1 Basic Definitions and Conventions 1
1.2 Normal and Tangent Cones . 3
1.3 Coderivatives and Subdifferential 6
1.4 Lipschitzian Properties and Metric Regularity . 9
1.5 Conclusions 11
Chapter 2. Linear Perturbations of Polyhedral Normal Cone
Mappings 12
2.1 The Normal Cone Mapping F(x, b) . 12
2.2 The Fr´echet Coderivative of F(x, b) . 16
2.3 The Mordukhovich Coderivative of F(x, b) . 26
2.4 AVIs under Linear Perturbations 37
2.5 Conclusions 42
Chapter 3. Nonlinear Perturbations of Polyhedral Normal Cone
Mappings 43
3.1 The Normal Cone Mapping F(x, A, b) 43
3.2 Estimation of the Fr´echet Normal Cone to gphF 48
3.3 Estimation of the Limiting Normal Cone to gphF . 54
iv3.4 AVIs under Nonlinear Perturbations . 59
3.5 Conclusions 66
Chapter 4. A Class of Linear Generalized Equations 67
4.1 Linear Generalized Equations 67
4.2 Formulas for Coderivatives 69
4.2.1 The Fr´echet Coderivative of N (x, α) 70
4.2.2 The Mordukhovich Coderivative of N (x, α) 78
4.3 Necessary and Sufficient Conditions for Stability 83
4.3.1 Coderivatives of the KKT point set map 83
4.3.2 The Lipschitz-like property 84
4.4 Conclusions 91
General Conclusions 92
List of Author’s Related Papers 93
References 94
vTable of Notations
IN := {1, 2, .} set of positive natural numbers
∅ empty set
IR set of real numbers
IR++ set of x ∈ IR with x > 0
IR+ set of x ∈ IR with x ≥ 0
IRư set of x ∈ IR with x ≤ 0
IR := IR ∪ {±∞} set of generalized real numbers
|x| absolute value of x ∈ IR
IRn n-dimensional Euclidean vector space
kxk norm of a vector x
IRm×n
set of m × n-real matrices
detA determinant of a matrix A
A> transposition of a matrix A
kAk norm of a matrix A
X∗
topological dual of a norm space X
hx
∗
, xi canonical pairing
hx, yi canonical inner product
(\u, v) angle between two vectors u and v
B(x, δ) open ball with centered at x and radius δ
B¯(x, δ) closed ball with centered at x and radius δ
BX open unit ball in a norm space X
B¯
X closed unit ball in a norm space X
posΩ convex cone generated by Ω
spanΩ linear subspace generated by Ω
dist(x; Ω) distance from x to Ω
{xk} sequence of vectors
xk → x xk converges to x in norm topology
x
∗
k
w
∗
→ x
∗ x
∗
k
converges to x
∗
in weak* topology
vi∀x for all x
x := y x is defined by y
Nb(x; Ω) Fr´echet normal cone to Ω at x
N(x; Ω) limiting normal cone to Ω at x
f : X → Y function from X to Y
f
0
(x), ∇f(x) Fr´echet derivative of f at x
ϕ : X → IR extended-real-valued function
domϕ effective domain of ϕ
epiϕ epigraph of ϕ
∂ϕ(x) limiting subdifferential of ϕ at x
∂
2ϕ(x, y) limiting second-order subdifferential of ϕ at x
relative to y
F : X ⇒ Y multifunction from X to Y
domF domain of F
rgeF range of F
gphF graph of F
kerF kernel of F
Db∗F(x, y) Fr´echet coderivative of F at (x, y)
D∗F(x, y) Mordukhovich coderivative of F at (x, y)
viiList of Figures
4.1 The sequences {(xk, αk)}k∈IN, {zk}k∈IN, and {uk}k∈IN 7