VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
INSTITUTE OF MATHEMATICS
NGUYEN THANH QUI
CODERIVATIVES OF NORMAL CONE MAPPINGS
AND APPLICATIONS
Speciality: Applied Mathematics
Speciality code: 62 46 01 12
SUMMARY
DOCTORAL DISSERTATION IN MATHEMATICS
HANOI - 2014
The dissertation was written on the basis of the author’s research works carried
at Institute of Mathematics, Vietnam Academy of Science and Technology
Supervisors:
1. Prof. Dr. Hab. Nguyen Dong Yen
2. Dr. Bui Trong Kien
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The dissertation is publicly available at:
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• The Library of Institute of Mathematics
Introduction
Motivated by solving optimization problems, the concept of derivative was

development of nonlinear analysis and optimization theory.
In the mid 1970s, to avoid the above-mentioned convexity limitations of
the Clarke concepts, B. S. Mordukhovich introduced the notions of limiting
normal cone and limiting subdifferential which are based entirely on dual-
space constructions. His dual approach led to a modern theory of generalized
differentiation with a variety of applications. Long before the publication
of his books (2006), Mordukhovich’s contributions to Variational Analysis
had been presented in the well-known monograph of R. T. Rockafellar and
R. J B. Wets (1998).
The limiting subdifferential is generally nonconvex and smaller than the
Clarke subdifferential. Similarly, the limiting normal cone to a closed set in
a Banach space is nonconvex in general and usually smaller than the Clarke
normal cone. Therefore, necessary optimality conditions in nonlinear pro-
gramming and optimal control in terms of the limiting subdifferential and
limiting normal cone are much tighter than that given by the corresponding
Clarke’s concepts. Furthermore, the Mordukhovich criteria for the Lipchitz-
like property (that is the pseudo-Lipschitz property in the original terminol-
ogy of J P. Aubin, or the Aubin continuity as suggested by A. L. Dontchev
and R. T. Rockafellar) and the metric regularity of multifunctions are remark-
able tools to study stability of variational inequalities, generalized equations,
and the Karush-Kuhn-Tucker point sets in parametric optimization prob-
lems. Note that if one uses Clarke’s theory then only sufficient conditions
for stability can be obtained. Meanwhile, Mordukhovich’s theory provides
one with both necessary and sufficient conditions for stability. Another ad-
vantage of the latter theory is that its system of calculus rules is much more
developed than that of Clarke’s theory. So, the wide range of applications
and bright prospects of Mordukhovich’s generalized differentiation theory are
understandable.
As far as we understand, Variational Analysis is a new name of a math-
ematical discipline which unifies Nonsmooth Analysis, Set-Valued Analysis

where f(¯x, w
1
) = ∇
x
ϕ(¯x, w
1
) denotes the partial derivative of ϕ with respect
to x at (¯x, w
1
) and
N(¯x; Θ(w
2
)) = {x
∗
∈ X
∗
| x
∗
, x − ¯x ≤ 0, ∀x ∈ Θ(w
2
)},
with X
∗
being the dual space of X, stands for the normal cone of Θ(w
2
).
This means that ¯x is a solution of the following generalized equation
0 ∈ f(x, w
1
) + F(x, w

2
⇒ X
∗
. Such a computation has been done
by Dontchev and Rockafellar (1996) for the case Θ(w
2
) is a fixed polyhedral
convex set in IR
n
, and by Yao and Yen (2010) for the case where Θ(w
2
) is a
fixed smooth-boundary convex set. The problem is rather difficult if Θ(w
2
)
depends on w
2
.
J C. Yao and N. D. Yen (2009a,b) first studied the case Θ(w
2
) = Θ(b) :=
{x ∈ IR
n
| Ax ≤ b} where A is an m × n matrix, b is a parameter. Some argu-
ments from these papers have been used by R. Henrion, B. S. Mordukhovich
and N. M. Nam (2010) to compute coderivatives of the normal cone mappings
to a fixed polyhedral convex set in Banach space. Nam (2010) showed that
the results of Yao and Yen on normal cone mappings to linearly perturbed
polyhedra can be extended to an infinite dimensional setting. N. T. Q. Trang
(2012) proposed some developments and refinements of the results of Nam.

N(x; Θ(w
2
)) (respectively, of the lim-
iting normal cone mapping (x, w
2
) → N(x; Θ(w
2
))), are interesting, but very
difficult. All the above-mentioned normal cone mappings are special cases
of the last two normal cone mappings. It will take some time before signifi-
cant advances on these general problems can be done. Some aspects of this
question have been investigated by R. Henrion, J. Outrata, and T. Surowiec
(2009).
It is worthy to stress that coderivatives of normal cone mappings are noth-
ing else as the second-order subdifferentials of the indicator functions of the
set in question. The concepts of Fr´echet and/or limiting second-order sub-
differentials of extended-real-valued functions have been discussed by Mor-
dukhovich (2006), R. A. Poliquin and R. T. Rockafellar (1998), Mordukhovich
and Outrata (2001), N. H. Chieu, T. D. Chuong, J C. Yao, and N. D. Yen
(2011), N. H. Chieu and N. Q. Huy (2011), Chieu and Trang (2012), Mor-
dukhovich and Rockafellar (2012) from different points of views.
This dissertation studies some problems related to the generalized differ-
entiation theory of Mordukhovich and its applications. Our main efforts
concentrate on computing or estimating the Fr´echet coderivative and the
4
Mordukhovich coderivative of the normal cone mappings to: a) linearly per-
turbed polyhedra in finite dimensional spaces, as well as in infinite dimen-
sional reflexive Banach spaces; b) nonlinearly perturbed polyhedra in finite
dimensional spaces; c) perturbed Euclidean balls.
Applications of the obtained results are used to study the metric regularity

5
been obtained. Combining the formulas with the necessary and the sufficient
conditions for the local Lipschitz-like property of implicit multifunctions from
a paper by Lee and Yen (2011), we get new results on stability of the Karush-
Kuhn-Tucker point set maps of parametric trust-region subproblems. This
chapter also solves the two open questions of Lee and Yen (2014).
Except for Chapter 1, each chapter has several illustrative examples.
The results of Chapter 2 and Chapter 3 were published on the journals
Nonlinear Analysis [1], Journal of Mathematics and Applications [2], Acta
Mathematica Vietnamica [3], Journal of Optimization Theory and Applica-
tions [4]. Chapter 4 is written on the basis of a joint paper by N. T. Qui
and N. D. Yen, which has been accepted for publication on SIAM Journal on
Optimization [5].
These results were reported by the author of this dissertation at Seminar of
Department of Numerical Analysis and Scientific Computing of Institute of
Mathematics (VAST, Hanoi), Workshops “Optimization and Scientific Com-
puting” (Ba Vi, April 20-23, 2010; April 20-23, 2011), The 8
th
Vietnam-Korea
Workshop “Mathematical Optimization Theory and Applications” (Univer-
sity of Dalat, December 8-10, 2011), Summer Schools “Variational Analysis
and Applications” (Institute of Mathematics (VAST, Hanoi), June 20-25,
2011; Institute of Mathematics (VAST, Hanoi) and Vietnam Institute for
Advanced Study in Mathematics, May 28-June 03, 2012).
6
Chapter 1
Preliminary
This chapter reviews some background material of Variational Analysis. The
basic concepts of generalized differentiation of multifunctions and extended-
real-valued functions are taken from Mordukhovich (2006, Vols I and II).

∗
k
∈ F (x
k
), ∀k ∈ IN

.
Definition 1.1 Let Ω be a nonempty subset of a Banach space X.
(i) Given ¯x ∈ Ω and ε ≥ 0, we define the set of ε-normals to Ω at ¯x by

N
ε
(¯x; Ω) :=

x
∗
∈ X
∗



limsup
x
Ω
→¯x
x
∗
, x − ¯x
x − ¯x
≤ ε

Definition 1.2 Let F : X ⇒ Y be a multifunction between Banach spaces
X and Y .
(i) For any (¯x, ¯y) ∈ X × Y and ε ≥ 0, ε-coderivative of F at (¯x, ¯y) is the
multifunction

D
∗
ε
F (¯x, ¯y) : Y
∗
⇒ X
∗
defined by

D
∗
ε
F (¯x, ¯y)(y
∗
) =

x
∗
∈ X
∗


(x
∗
, −y

∗
given by
D
∗
F (¯x, ¯y)(¯y
∗
) = Limsup
(x,y)→(¯x,¯y)
y
∗
w
∗
→ ¯y
∗
, ε↓0

D
∗
ε
F (x, y)(y
∗
).
If (¯x, ¯y) ∈ gphF , we put D
∗
F (¯x, ¯y)(y
∗
) = ∅ for all y
∗
∈ Y
∗

ϕ(¯x, ¯y)(u) := (D
∗
∂ϕ)(¯x, ¯y)(u), ∀u ∈ X
∗∗
,
is called the limiting second-order subdifferential of ϕ at ¯x relative to ¯y.
8
The indicator function of Ω is the function δ(· ; Ω) : X → IR defined by
δ(x; Ω) = 0 if x ∈ Ω and δ(x; Ω) = ∞ if x ∈ Ω. If F : X ⇒ X
∗
given by
F (x) = N(x; Ω) for all x ∈ X and (¯x, ¯x
∗
) ∈ gphF , then we have
D
∗
F (¯x, ¯x
∗
)(u) =

D
∗
∂δ(· ; Ω)

(¯x, ¯x
∗
)(u) = ∂
2
δ(· ; Ω)(¯x, ¯x
∗

(i) F is locally metrically regular around (¯x, ¯y).
(ii) D
∗
F
−1
(¯y, ¯x)(0) = {0}.
9
Chapter 2
Linear Perturbations of Polyhedral
Normal Cone Mappings
In this chapter, we differentiate the normal cone mappings to linearly per-
turbed polyhedral convex sets and apply the results to solution stability of
affine variational inequalities. We will answer two open questions stated by
Yao and Yen (2009a). This chapter is written on the basis of the results in
[1], [2], and [3].
2.1 The Normal Cone Mapping F(x, b)
Let X be a Banach space with its dual X
∗
and T = {1, 2, . . . , m} be an index
set. Consider a vector system {a
∗
i
∈ X
∗
| i ∈ T }, and a polyhedral convex set
Θ(b) =

x ∈ X| a
∗
i

≤ 0
(resp., b
I
≥ 0, b
I
= 0) if b
i
≤ 0 (resp., b
i
≥ 0, b
i
= 0) for all i ∈ I.
The multifunction F : X × IR
m
⇒ X
∗
defined by setting
F(x, b) = N(x; Θ(b)), ∀(x, b) ∈ X × IR
m
,
is said to be the linearly perturbed polyhedral normal cone mapping to the
10
perturbed polyhedron Θ(b). Following Nam (2010), we have
F(x, b) = pos

a
∗
i
| i ∈ I(x, b)


, λ
i
≥ 0 ∀i ∈ I

,
I
1
(x, b, x
∗
) =

i ∈ I


λ
i
= 0 for some (λ
j
)
j∈I
∈ Ξ(x, b, x
∗
)

,
H(x, b, x
∗
) =

(x

a
∗
i
, b
∗
¯
I
= 0, b
∗
I
1
≤ 0

,
where I
1
:= I
1
(x, b, x
∗
) and b
∗
= (b
∗
1
, . . . , b
∗
m
) ∈ IR
m

⇒ X
∗
× IR
m
of
F(·) at (¯x,
¯
b, ¯x
∗
) ∈ gphF is computed by

D
∗
F(¯x,
¯
b, ¯x
∗
)(v) =

(x
∗
, b
∗
) ∈ X
∗
× IR
m


x

∗
×

T (¯x; Θ(
¯
b)) ∩ {¯x
∗
}
⊥


, ∀v ∈ X
∗∗
,
where I := I(¯x,
¯
b) and I
1
:= I
1
(¯x,
¯
b, ¯x
∗
).
2.3 The Mordukhovich Coderivative of F(x, b)
Following Henrion, Mordukhovich, and Nam (2010), for any sets P , Q with
P ⊂ Q ⊂ T , we put
A
Q,P


P ⊂ I(x, b)


P = ∅, x
∗
∈ pos{a
∗
i
| i ∈ P }

,
J (x, b, x
∗
) =

P ∈ I(x, b, x
∗
)


a
∗
i
, i ∈ P, are linearly independent

,

I(x, b, x
∗

, x < b
i
∀i ∈ T \Q

.
Now, let (x, b, x
∗
) ∈ gphF, I = I(x, b), J = I\I
1
(x, b, x
∗
), I = I(x, b, x
∗
),
and

I =

I(x, b, x
∗
). Define
Σ(x, b, x
∗
) =

P ⊂Q⊂I, P ∈

I

(x

Q\P
≤ 0

,
(2.2)
Σ
0
(x, b, x
∗
) =

P ⊂Q⊂I, P ∈I
F
Q
(b)=∅

(x
∗
, b
∗
, v)



(x
∗
, v) ∈ A
Q,P
× B
Q,P

¯
b, ¯x
∗
) ⊂ N

(¯x,
¯
b, ¯x
∗
); gphF

⊂ Σ(¯x,
¯
b, ¯x
∗
), (2.4)
where Σ(¯x,
¯
b, ¯x
∗
) and Σ
0
(¯x,
¯
b, ¯x
∗
) are given respectively by (2.2) and (2.3),
hold. Besides, if ¯x
∗
= 0, then

∗
)(v) ⊂ D
∗
F(x, b, x
∗
)(v) ⊂ Ω(x, b, x
∗
)(v), ∀v ∈ X
∗∗
,
where
Ω(x, b, x
∗
)(v) :=

(u
∗
, η
∗
) ∈ X
∗
× IR
m


(u
∗
, η
∗
, −v) ∈ Σ(x, b, x

.
12
2.4 AVIs under Linear Perturbations
Let X = IR
n
and consider S : IR
m
× IR
n
⇒ IR
n
defined by
S(b, q) =

x ∈ IR
n
| q ∈ Mx + F(x, b)

, (2.6)
where M ∈ IR
n×n
is fixed, (b, q) ∈ IR
m
× IR
n
are parameters. Note that
S(b, q) can be rewritten as the solution set of a parametric affine variational
inequality (AVI):
S(b, q) =



, q

) ∈ IR
m+n



(−x

, b

, q

) ∈ M

v

× {0
IR
m+n
}
−{0
IR
n+m
} × {v

} +

D




(−x

, b

, q

) ∈ M

v

× {0
IR
m+n
}
−{0
IR
n+m
} × {v

} + Ω(¯x,
¯
b, ¯x
∗
)(v

) × {0
IR

(i) If S(·) is locally Lipschitz-like around (
¯
b, ¯q, ¯x), then

K
M,¯q
(0) = {0}.
(ii) If L
M,¯q
(0) = {0}, then S(·) is locally Lipschitz-like around (
¯
b, ¯q, ¯x).
(iii) If F(·) is graphically regular at (¯x,
¯
b, ¯x
∗
), then S(·) is locally Lipschitz-like
around (
¯
b, ¯q, ¯x) if and only if

K
M,¯q
(0) = {0}.
13
Chapter 3
Nonlinear Perturbations of Polyhedral
Normal Cone Mappings
This chapter is devoted to the estimation of the Fr´echet and the limiting
normal cones to the graphs of the normal cone mappings to nonlinearly per-

, . . . , b
m
), we call
I(x, A, b) =

i ∈ T | A
i
x = b
i

.
the active index set of Θ(A, b) at x. For any Γ := {i
1
, . . . , i
r
} ⊂ T , we denote
the column vector
a
Γ,j
=



a
i
1
j
.
.
.

where M ∈ IR
n×n
is fixed, and A ∈ IR
m×n
, b ∈ IR
m
, q ∈ IR
n
are subject to
change. Let S(A, b, q) be the solution set of (3.2). Then we have
S(A, b, q) = {x ∈ IR
n
| 0 ∈ Mx − q + F(x, A, b)}, (3.3)
where F(x, A, b) is given by (3.1).
3.2 Estimation of the Fr´echet Normal Cone to gphF
For any (x, A, b, ξ
∗
) ∈ gphF, we put
Ξ(x, A, b, ξ
∗
) :=

(λ
i
)
i∈I



ξ

∈ Ξ(x, A, b, ξ
∗
)

,
where I := I(x, A, b). Using I
1
:= I
1
(x, A, b, ξ
∗
), we construct the set
H(x, A, b, ξ
∗
)
=

(x
∗
, A
∗
, b
∗
, ξ)



x
∗
∈

j
< 0, a
∗
I
1
,j
≥ 0 if x
j
> 0,
a
∗
I
1
,j
= 0 if x
j
= 0, A
∗
¯
I
= 0, b
∗
I
= 0, b
∗
I
1
≤ 0

,

) ∈ gphF, we have

N

(¯x,
¯
A,
¯
b,
¯
ξ
∗
); gphF

⊂ H(¯x,
¯
A,
¯
b,
¯
ξ
∗
). (3.4)
15
3.3 Estimation of the Limiting Normal Cone to gphF
Given a matrix A ∈ IR
m×n
and subsets P, Q of T satisfying P ⊂ Q, following
Henrion, Mordukhovich, and Nam (2010) we put
A

, v ≤ 0 ∀i ∈ Q \ P

.
For each (x, A, b, ξ
∗
) ∈ gphF, we put
I(x, A, b, ξ
∗
) :=

P ⊂ I(x, A, b)


P = ∅, ξ
∗
∈ pos{A

i
| i ∈ P }

,
J (x, A, b, ξ
∗
) :=

P ∈ I


A


Σ(x, A, b, ξ
∗
) :=

P ⊂Q⊂I, P ∈

I

(x
∗
, A
∗
, b
∗
, ξ)


(x
∗
, ξ) ∈ A
Q,P
(A) × B
Q,P
(A),
x
∗
= −

i∈Q
b

j
}
j∈J
are called positively linearly independent if from conditions

j∈J
λ
j
v
j
= 0 and λ
j
≥ 0 for all j ∈ J it follows that λ
j
= 0 for all j ∈ J.
Theorem 3.2 Let (¯x,
¯
A,
¯
b,
¯
ξ
∗
) ∈ gphF and let I = I(¯x,
¯
A,
¯
b). If the vectors
{
¯

b,
¯
ξ
∗
)(ξ) =

(x
∗
, A
∗
, b
∗
)


(x
∗
, A
∗
, b
∗
, −ξ) ∈ Σ(¯x,
¯
A,
¯
b,
¯
ξ
∗
)

, b
∗
, −ξ) ∈ N

(¯x,
¯
A,
¯
b,
¯
ξ
∗
); gphF


,
we have
D
∗
F(¯x,
¯
A,
¯
b,
¯
ξ
∗
)(ξ) ⊂ Λ(¯x,
¯
A,

n
, we put
K( ¯w)(x
∗
) =

ξ∈IR
n

(A
∗
, b
∗
, q
∗
)



(−x
∗
, A
∗
, b
∗
, q
∗
) ∈ M

ξ × {(0

IR
n
}

,
L( ¯w)(x
∗
) =

ξ∈IR
n

(A
∗
, b
∗
, q
∗
)



(−x
∗
, A
∗
, b
∗
, q
∗

)(ξ) × {0
IR
n
}

,
where Λ(¯x,
¯
A,
¯
b,
¯
ξ
∗
)(ξ) is given by (3.6).
Theorem 3.3 Let ¯w = (
¯
A,
¯
b, ¯q, ¯x) ∈ gphS. If the vectors {
¯
A

i
| i ∈ I(¯x,
¯
A,
¯
b)}
are positively linearly independent, then the following assertions are valid:

The results presented below are taken from [5].
4.1 Linear Generalized Equations
The concept of generalized equation introduced in 1979 by Robinson has been
recognized as an efficient tool for dealing with various questions in optimiza-
tion theory. We consider the linear generalized equations of the form
0 ∈ Ax + b + N(x; E(α)), (4.1)
where symmetric n × n matrix A ∈ IR
n×n
, vector b ∈ IR
m
, and real number
α > 0 are parameters, E(α) := {x ∈ IR
n


x ≤ α}, and N(x; E(α)) is the
normal cone to E(α) at x. The solution set of (4.1) is denoted by S(A, b, α).
Let N : IR
n
× IR ⇒ IR
n
be defined by N (x, α) = N(x; E(α)) if α > 0,
and N(x, α) = ∅ if α ≤ 0. Thus N (·) is a multifunction with closed convex
values.
18
4.2 Formulas for Coderivatives
This section provides exact formulas for the Fr´echet and the Mordukhovich
coderivatives of N (·) at every point belonging to gphN in various cases.
Fix any point (x, α, v) ∈ gphN .
Theorem 4.1 If x = α and v = 0, then v = µx with µ = v · x

, if v

, x = 0
∅, if v

, x = 0
for every v

∈ IR
n
.
Theorem 4.2 If x = α and v = 0, then

D
∗
N (x, α, v)(v

) =




(x

, α

) ∈ IR
n
× IR


∗
N (x, α, v)(v

) = {(0
IR
n
, 0
IR
)}, ∀v

∈ IR
n
.
Theorem 4.3 If x = α and if v = 0, then we have
D
∗
N (x, α, v)(v

) =

D
∗
N (x, α, v)(v

)
=





.
Theorem 4.4 Suppose that x = α and v = 0. For every v

∈ IR
n
, the
following hold
(i) If v

, x = 0, then
D
∗
N (x, α, v)(v

) =




(x

, α

) ∈ IR
n+1


x

= −


) ∈ IR
n
× IR



x

= −
α

α
x, α

∈ IR

.
19
4.3 Necessary and Sufficient Conditions for Stability
Using the coderivative formulas of N (·), conditions for stability of the solution
map (A, b, α) → S(A, b, α) of (4.1) are obtained in this section.
By Martinez (1994), if x ∈ E(α) is a local minimum of the problem
min

f(x) =
1
2
x


b = 0, then S(·) is locally Lipschitz-like around
(
¯
A,
¯
b, ¯α, ¯x) if and only if detQ(
¯
A,
¯
b, ¯α, ¯x) = 0, where
Q(
¯
A,
¯
b, ¯α, ¯x) :=

¯
A + µI −
1
¯α
¯x
¯x

0

(4.2)
with µ being the unique Lagrange multiplier associated to ¯x.
Theorem 4.6 Let (
¯
A,

v

, ¯x ≥ 0
v

∈ IR
n
, α

≤ 0
=⇒



v

= 0
α

= 0.
(4.3)
(ii) If det
¯
A = 0, detQ
1
(
¯
A,
¯
b, ¯α, ¯x) = 0, where

graphs of the normal cone mappings to nonlinearly perturbed polyhedral
convex sets in finite dimensional spaces.
3. Exact formulas for the Fr´echet and the Mordukhovich coderivatives of
the normal cone mappings to perturbed Euclidean balls.
4. Conditions for the local Lipschitz-like property and local metric regularity
of the solution maps of parametric affine variational inequalities under
linear/nonlinear perturbations, and conditions for the local Lipschitz-like
property of the solution maps of a class of linear generalized equations
in finite dimensional spaces.
21
References
[1] N. T. Qui, Linearly perturbed polyhedral normal cone mappings and
applications, Nonlinear Anal., 74 (2011), pp. 1676–1689.
[2] N. T. Qui, New results on linearly perturbed polyhedral normal cone
mappings, J. Math. Anal. Appl., 381 (2011), pp. 352–364.
[3] N. T. Qui, Upper and lower estimates for a Fr´echet normal cone, Acta
Math. Vietnam., 36 (2011), pp. 601–610.
[4] N. T. Qui, Nonlinear perturbations of polyhedral normal cone mappings
and affine variational inequalities, J. Optim. Theory Appl., 153 (2012),
pp. 98–122.
[5] N. T. Qui and N. D. Yen, A class of linear generalized equations,
SIAM J. Optim., 24 (2014), pp. 210–231.
22