Summary Of Mathematics Doctoral Thesis: Optimality conditions for Vector equilibrium problems in terms of contingent derivatives

MINISTRY OF EDUCATION AND
TRAINING

VIETNAM ACADEMY
OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY
............***............

TRAN VAN SU

OPTIMALITY CONDITIONS FOR VECTOR
EQUILIBRIUM PROBLEMS IN TERMS OF
CONTINGENT DERIVATIVES

Major: Applied Mathematics
Code: 62 46 01 12

SUMMARY OF MATHEMATICS DOCTORAL THESIS

Hanoi - 2018


This thesis is completed at: Graduate University of Science and
Technology - Vietnam Academy of Science and Technology

Supervisors 1: Assoc. Prof. Dr. Do Van Luu
Supervisors 2: Dr. Nguyen Cong Dieu

First referee

1: . . . . . .

Second referee 2: . . . . . .
Third referee 3: . . . . . .

The thesis is to be presented to the Defense Committee of the Graduate University of Science and Technology - Vietnam Academy of Science
and Technology on . . . . . . . . . . . . 2018, at . . . . . . . . . . . . o’clock . . . . . . . . . . . .

The thesis can be found at:
- Library of Graduate University of Science and Technology
- Vietnam National Library


Introduction
The vector equilibrium problem plays an important role in nonlinear
analysis and has attracted extensive attention in recent years because of
its widely applied areas, see, for example, Anh (2012, 2015), Ansari (2000,
2001a, 2001b, 2002), Bianchi (1996, 1997), Feng-Qiu (2014), Khanh (2013,
2015), Luu (2014a, 2014b, 2014c, 2015, 2016), Su (2017, 2018), Tan (2011,
2012, 2018a, 2018b), etc. The vector equilibrium problem is extended from
the scalar equilibrium problem which was first introduced by Blum-Oettli
(1994) and the optimality condition for its efficient solutions is a main subject which will be needed to study, see, for instance, Luu (2010, 2016, 2017),
Gong (2008, 2010), Long-Huang-Peng (2011), Jiménez-Novo-Sama (2003,
2009), Li-Zhu-Teo (2012), etc. Our thesis studies the first- and secondorder optimality conditions for vector equilibrium problems in terms of
contingent derivatives and epiderivatives in which the conditions of order
one using stable functions and two using arbitrary functions.
The contingent derivative plays a central role in analysis and applied
analysis, and it will be used to establish the optimality conditions. Aubin
(1981) first introduced a concept of a contingent derivative for set-valued
mapping and their applications to express the optimality conditions in

vector optimization problems like Aubin-Ekeland (1984), Corley (1988)
and Luc (1991). Jahn-Rauh (1997) provided a concept of a contingent
epiderivative for set-valued mapping and obtained the respectively optimality conditions. Chen-Jahn (1998) proposed a concept of a general contingent epiderivative for set-valued mapping and the result is applied to
the set-valued vector equilibrium problems. In the case of single-valued
optimization problems, we don’t need to move from set-valued results into
single-valued results which establishing the new results are sharper.


2

Based on the concept of Aubin (1981), Jiménez-Novo (2008) have proved
the good calculus rules of contingent derivatives with steady, stable, Hadamard
differentiable, Fréchet differentiable functions as well as their applications
for establishing optimality conditions in unconstrained vector equilibrium
problems. The author also derived the necessary and sufficient optimality conditions for multiobjective optimization problems involving equality
and inequality constraints with stable functions via contingent derivatives.
One limitation in the results of Jiménez-Novo (2008) is not considered
the Fritz John and Kuhn-Tucker necessary optimality conditions for local weakly efficient solutions of constrained vector equilibrium problem
including inequality, equality and set constraints with their applications.
Our thesis has contributed to solving the above mentioned open issues.
Rodríguez-Marín and Sama (2007a, 2007b) have investigated the existences, uniqueness and some properties of contingent epiderivatives and
hypoderivatives, the relationships between contingent epiderivatives/ hypoderivatives and contingent derivatives with both stable functions and
set-valued mappings in case the finite-dimensional image spaces. One limitation in the results of Rodríguez-Marín and Sama (2007a, 2007b) is not
considered the existences of contingent epiderivatives and hypoderivatives
for arbitrary single-valued functions with Banach image spaces. On optimality conditions, Jiménez-Novo and Sama (2009) only derived the sufficient and necessary optimality conditions for strict local minimums of
order one via the contingent epiderivatives and hypoderivatives with stable objective functions in multiobjective optimization problems. In case the
sufficient and necessary optimality conditions for weakly efficient, Henig
efficient, global efficient and superefficient solutions of vector equilibrium
problems in terms of contingent epiderivatives and hypoderivatives with
stable functions are not considered by Jiménez-Novo and Sama (2009) and

other authors. Our thesis has studied the existence results of contingent
epiderivatives and hypoderivatives with arbitrary single-valued functions
in Banach spaces, the relationships between them and contingent derivatives, and obtaining the sufficient and necessary optimality conditions for
efficient solutions of vector equilibrium problems via the contingent epiderivatives with steady functions in Banach spaces, and providing, in ad-


3

dition, a sufficient optimality condition for weakly efficient solution of unconstrained vector equilibrium problem with stable functions as a basis for
extending the results to research the second order optimality conditions.
In a recent decade, the second-order optimality conditions for vector
equilibrium problems and its special cases via contingent derivatives and
epiderivatives has been intensively studied by many authors like JahnKhan-Zeilinger (2005), Durea (2008), Li-Zhu-Teo (2012), Khan-Tammer
(2013), etc. We see that the existence results of second order contingent
epiderivatives and hypoderivatives with arbitrary single-valued functions in
Banach spaces are not considerd, and the sufficient optimality conditions
for weakly efficient solutions via second-order composed contingent epiderivatives only studied to the unconstrained optimization problem. Our
dissertation has researched the existence results for second-order general
contingent epiderivatives and hypoderivatives with arbitrary single-valued
functions as well as constructed the sufficient, sufficient and necessary optimality conditions for efficient solutions of constrained vector equilibrium
problems in terms of contingent epiderivatives in Banach spaces.
The main purpose of this thesis is to study the first- and second-order
optimality conditions for efficient solutions of vector equilibrium problems
in terms of contingent derivatives and epiderivatives, and the results are:
1) Research optimality conditions for local weak efficient solution in vector equilibrium problem involving set, inequality and equality constraints
with stable functions via contingent derivatives in finite-dimensional spaces.
2) Research optimality conditions for weak, Henig, global and superefficient solutions in vector equilibrium problems with steady, Hadamard
differentiable, Fréchet differentiable functions in terms of contingent epiderivatives in Banach spaces.
3) Research second order optimality conditions for weak, Henig, global,
super-efficient solutions in vector equilibrium problems with arbitrary functions in terms of contingent epiderivatives in Banach spaces.

4) Application to vector variational inequalities, optimization problems.
Besides introductions, general conclusions and references, the content of
the thesis consists of four chapters and the main results of the dissertation
are contained in Chapters 2,3,4.


4

Chapter 1 introduces some concepts from efficient solutions to (CVEP),
contingent cones, contingent sets, contingent derivatives, epiderivatives
and hypoderivatives. Besides, it provides the concept of stable, steady,
Hadamard differentiable and Fréchet differentiable functions and several
contingent derivatives related fomulars. Finally, the concept of ideal and
Pareto efficient points with respect to a cone is also derived as well.
Chapter 2 studies the Fritz John and Karush-Kuhn-Tucker necessary
optimality conditions for local weak efficient solution of constrained vector equilibrium problems with stable functions via contingent derivatives
in finite-dimensional spaces and presents some its applications to vector
inequality variational problems, vector optimization problems. Besides, we
have proposed two constraint qualifications (CQ1) and (CQ2) for investigating Karush-Kuhn-Tucker and strong Karush-Kuhn-Tucker necessary
optimality conditions. Many examples to illustrate the results are derived.
Chapter 3 studies the existences of contingent epiderivatives as well as
the necessary and sufficient optimality condition for weak, Henig, global,
super-efficient solutions in vector equilibrium problems with stable functions via contingent epiderivatives in two cases the initial and final spaces
are Banach, the initial space is Banach and the final space is finite-dimensional.
The last part investigates constrained vector equilibrium problems based
on a constraint qualification of Kurcyusz-Robinson-Zowe (KRZ).
Chapter 4 studies the existences of second order contingent epiderivatives and second order sufficient optimality conditions for weakly efficient,
Henig efficient, global efficient and superefficient solutions in vector equilibrium problems with constraints with arbitrary functions via contingent
epiderivatives in Banach spaces. The last part of this chapter makes an
assumption 4.1 as a basis for studying second order optimality conditions.

The result of the thesis is presented in:
• The 4th National Conference on Applied Mathematics, National Economics University, Hanoi 23-25/12/2015;
• The 14th Workshop on Optimization and Scientific Computing, Bavi
- Hanoi 21-23/04/2016;
• Seminar of Optimal Group, Faculty of Mathematics and Informatics,
Thang Long University, Hanoi.


Chapter 1
Some Knowledge of Preparing
Chapter 1 of the thesis introduces the basic knowledge to serve for the
presentation of research results achieved in the next chapters and exactly:
Section 1.1 deals with several concepts such as: tangent sets, stable
functions, contingent derivatives, epiderivatives and hypoderivatives.
• In section 1.1.1 presents the concepts of contingent cone, adjacent
cone, interior tangent cone, sequential interior tangent cone, normal cone,
second order contingent set, second order adjacent set, second order interior
tangent set and some its properties.
• In section 1.1.2 presents the definitions of first and second order contingent derivatives.
• In section 1.1.3 presents the definitions of Hadamard derivative, stable
function, steady function and some properties related.
• In section 1.1.4 presents the definitions of ideal and Pareto minimal
(maximal) points of a set with respect to a cone and its properties; the
concepts of first and second order contingent epiderivatives along with
some results on its existences.
Section 1.2 deals with general vector equilibrium problem and some its
special cases.
• In section 1.2.1 presents several vector equilibrium problems such as
(VEP), (VEP1 ), (CVEP) and (CVEP1 ), and constructions of the concepts
of (CVEP) in weakly efficient, local weakly efficient, Henig efficient, global

efficient and superefficient solutions are addressed.
•• Some the definitions for efficient solutions of (CVEP)
Let X, Y, Z and W be real Banach spaces in which C be a nonempty


6

subset of X; Q and S be convex cones in Y and Z, respectively; F :
X × X → Y be a vector bifunction; g : X → Z and h : X → W be
constraints functions, and denote K = {x ∈ C : g(x) ∈ −S, h(x) = 0}
instead of the feasible set of vector equilibrium problems.
The vector equilibrium problem with constraints is denoted by (CVEP),
which can be stated as follows: Finding a vector x ∈ K such that
F (x, y) ∈ −intQ (∀ y ∈ K).

(1.1)

Vector x is called a weakly efficient solution of problem (CVEP). If there
exists a neighborhood U of x such that (1.1) holds for every y ∈ K ∩ U
then x is called a local weakly efficient solution of problem (CVEP). If the
problem (CVEP) with a set constraint (in short, (VEP)), and called the
unconstrained vector equilibrium problem. If X = Rn , Y = Rm , Z = Rr ,
r
W = Rl and the cones Q = Rm
+ , S = R+ , then the problem (CVEP) is
said to be (CVEP1 ) and the problem (VEP) is said to be (VEP1 ).
Let Y ∗ be the topological dual space of Y. Let us denote Q+ be the dual
cone of Q ⊂ Y, which means that
Q+ = {y ∗ ∈ Y ∗ : y ∗ , y ≥ 0 ∀ y ∈ Q}.
We denote the quasi-interior of Q+ by Q , i.e.

Q = {y ∗ ∈ Y ∗ : y ∗ , y > 0 ∀ y ∈ Q \ {0}}.
Let B be a base of cone Q. Set
Q∆ (B) = {y ∗ ∈ Q : ∃ t > 0 such that y ∗ , b ≥ t ∀ b ∈ B}.
Making use of the seperation theorem of disjoint convex sets {0} and B,
it yields that there exists y ∗ ∈ Y ∗ \ {0} satisfying
r = inf { y ∗ , b : b ∈ B} > y ∗ , 0 = 0.
Let us consider an open absolutely convex neighborhood VB of zero in Y
be of the form
r
VB = {y ∈ Y : | y ∗ , y | < }.
2
The notion VB will be used throughout this dissertation. It is evident that
r
inf { y ∗ , y : y ∈ B + VB } ≥ ,
2


7

and for any convex neighborhood U of zero with U ⊂ VB , it holds that
B + U is a convex set and 0 ∈ cl(B + U ). Thus, cone(B + U ) is a pointed
convex cone satisfying Q \ {0} ⊂ int cone(U + B).
Based on the preceding illustrations, Gong (2008, 2010) has constructed
the concept for globally efficient, Henig efficient and super-efficient solutions of problem (CVEP), which can be illustrated as follows.
Definition 1.1 A vector x ∈ K is called a globally efficient solution to the
(CVEP) if there exists a pointed convex cone H ⊂ Y with Q \ {0} ⊂ intH
such that
F (x, K) ∩ (−H) \ {0} = ∅.
Definition 1.2 A vector x ∈ K is called a Henig efficient solution to the
(CVEP) if there exists some absolutely convex neighborhood U of 0 with

U ⊂ VB such that
cone F (x, K) ∩ − int cone(U + B) = ∅.
Definition 1.3 A vector x ∈ K is called a superefficient solution to the
(CVEP) if for each neighborhood V of 0, there exists some neighborhood
U of 0 such that
cone F (x, K) ∩ U − Q ⊂ V.
Let L(X, Y ) be the space of all bounded linear mapping from X to Y.
We write h, x instead of the value of h ∈ L(X, Y ) at x ∈ X. The vector
variational inequality problem with constraints is denoted by (CVVI) and
given as F (x, y) = T x, y − x , where T is a mapping from X into L(X, Y ).
In this case, the concept of efficient solutions of (CVEP) is similar as the
concept of efficient solutions of (CVVI), respectively.
Similarly to the vector optimization problem with constraints (CVOP)
satisfying F (x, y) = f (y) − f (x) where f is a mapping from X to Y.
• In section 1.2.2 presents vector optimization problem concerning a
local weak minimum and a strict local minimum of order m (m ∈ N) as
well as the optimality condition for strict local minimum of order one via
contingent derivatives of multiobjective optimization problems is derived.
• In section 1.2.3 introduces vector variational inequality problem and
some related problems.


Chapter 2
Optimality Conditions for Vector
Equilibrium Problems in Terms of
Contingent Derivatives
This chapter studies the Fritz John and Karush-Kuhn-Tucker necessary
optimality conditions for local weakly efficient solutions of (CVEP1 ) and
some its applications to the vector variational inequality problem (CVVI1 ),
the vector optimization problem (CVOP1 ), the transportion - production

problem and the Nash-Cournot equilibria problem.
The chapter is written on the basis of the papers [1] and [5] in the list
of works has been published.

2.1. Fritz John type necessary optimality conditions for
local weak efficient solutions of (CVEP1 )
Let us consider problem (CVEP1 ) be given as in Chapter 1. Denote
I = {1, 2, . . . , r}, J = {1, 2, . . . , m} and L = {1, 2, . . . , l}. For each x ∈ K,
we set F = (F1 , F2 , . . . , Fm ), Fx (.) = F (x, .), Fk,x (.) = Fk (x, .) (∀ k ∈ J),
and then the feasible set of (CVEP1 ) is of the form:
K = {x ∈ C : gi (x) ≤ 0 (∀ i ∈ I), hj (x) = 0 (∀ j ∈ L)}.
Let us denote by
Ker∇h(x) = {v ∈ X : ∇h(x), v = 0},
I(x) = {i ∈ I : gi (x) = 0}.


9

Let us first make an assumption for obtaining optimality conditions to
(CVEP1 ).
Assumption 2.1 Fx (x) = 0; the functions Fx , g are continuous in a neighbourhood of x; the functions h1 , . . . , hl are Fréchet differentiable at x with
Fréchet derivatives ∇h1 (x), . . . , ∇hl (x) linearly independent.
Fritz John necessary optimality conditions for local weak efficient solution of (CVEP1 ) which can be stated as follows.
Theorem 2.1 Let x ∈ K be a local weak efficient solution of (CVEP1 ).
Assume that Assumption 2.1 holds, and the functions Fx , g steady at x.
Suppose, in addition, that for every v ∈ Ker∇h(x) ∩ IT (C, x), there exists
z ∈ Dc g(x)v such that zi < 0 (∀ i ∈ I(x)). Then, for every v ∈ Ker∇h(x)∩
IT (C, x) and for every (y, z) ∈ Dc (Fx , g)(x)v, there exist (λ, µ) ∈ Rm × Rr ,
λ ≥ 0, µ ≥ 0 with (λ, µ) = (0, 0) such that
λ, y + µ, z ≥ 0,

µi gi (x) = 0 (∀ i ∈ I).
Theorem 2.2 Let x ∈ K be a local weak efficient solution of (CVEP1 ).
Assume that Assumption 2.1 holds, and the functions Fx , g steady at x.
Suppose, furthermore, that for every v ∈ Ker∇h(x)∩IT (C, x), there exists
z ∈ Dc g(x)v such that zi < 0 (∀ i ∈ I(x)). Then,
(i) For every v ∈ IT (C, x), there exist λk ≥ 0 (∀ k ∈ J), µi ≥ 0 (∀ i ∈ I),
and γj ∈ R (∀ j ∈ L), not all zero, such that
0∈

λk Dc Fk,x (x)v +
i∈I

k∈J

γj ∇hj (x), v ,

µi Dc gi (x)v +

(2.1)

j∈L

µi gi (x) = 0 (∀ i ∈ I).

(2.2)

(ii) For every v ∈ Ker∇h(x) ∩ IT (C, x), there exist λk ≥ 0 (∀ k ∈ J),
µi ≥ 0 (∀ i ∈ I) with (λ, µ) = (0, 0) such that
0∈

λk Dc Fk,x (x)v +
k∈J

µi Dc gi (x)v,
i∈I

µi gi (x) = 0 (∀ i ∈ I).

(2.3)


10

Remark 2.1 Theorem 2.2 is applied to establish the necessary optimality
conditions for local weak efficient solutions of the models of transportion–
production problem (Example 2.2) and Nash-Cournot equilibria problem
(Example 2.3).
Remark 2.2 Theorems 2.1 and 2.2 have solved the case of multiobjective
optimization problems with set constraint while the author Jiménez and
Novo (2008) have not been yet fully discovered. The author only studied
the optimality conditions for weak efficient solutions of problem (CVEP1 )
involving equality and inequality constraints. In addition, if C ≡ Rn then
Theorem 2.1 coincides with the result in Jiménez and Novo (2008).
In case C = Rn , Theorem 2.2 leads to the following direct consequence.
Corollary 2.1 Let C = Rn , and let x ∈ K be a local weak efficient solution
of (CVEP1 ). Assume that Assumption 2.1 holds, and the functions Fx , g
are steady x. Suppose, furthermore, that for every v ∈ Ker∇h(x), there
exists z ∈ Dc g(x)v such that zi < 0 (∀ i ∈ I(x)). Then,
(i) For every v ∈ Rn , there exist λk ≥ 0 (∀ k ∈ J), µi ≥ 0 (∀ i ∈ I), and
γj ∈ R (∀ j ∈ L), not all zero, such that

0∈

λk Dc Fk,x (x)v +
i∈I

k∈J

γj ∇hj (x), v ,

µi Dc gi (x)v +

(2.4)

j∈L

µi gi (x) = 0 (∀ i ∈ I).

(2.5)

(ii) For every v ∈ Ker∇h(x), there exist λk ≥ 0 (∀ k ∈ J), µi ≥ 0 (∀ i ∈ I)
with (λ, µ) = (0, 0) such that
0∈

λk Dc Fk,x (x)v +
k∈J

µi Dc gi (x)v,
i∈I

µi gi (x) = 0 (∀ i ∈ I).

In case Fk,x (k ∈ J) and gi (i ∈ I) are Hadamard differentiable at x,
we obtain an immediate consequence from Theorem 2.2 as follows.
Corollary 2.2 Let x ∈ K be a local weak efficient solution of (CVEP1 ).
Assume that Assumption 2.1 holds, and the functions Fx , g are Hadamard
differentiable and steady at x. Suppose, furthermore, that for every v ∈
Ker∇h(x) ∩ IT (C, x), dgi (x; v) < 0 (∀ i ∈ I(x)). Then,


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(i) For every v ∈ IT (C, x), there exist λk ≥ 0 (∀ k ∈ J), µi ≥ 0 (∀ i ∈ I),
and γj ∈ R (∀ j ∈ L), not all zero, such that
λk dFk,x (x; v) +
i∈I

k∈J

γj ∇hj (x), v = 0,

µi dgi (x; v) +

(2.6)

j∈L

µi gi (x) = 0 (∀ i ∈ I).

(2.7)

(ii) For every v ∈ Ker∇h(x) ∩ IT (C, x), there exist λk ≥ 0 (∀ k ∈ J),

µi ≥ 0 (∀ i ∈ I) with (λ, µ) = (0, 0) such that
λk dFk,x (x; v) +

µi dgi (x; v) = 0,
i∈I

k∈J

µi gi (x) = 0 (∀ i ∈ I).
Remark 2.3 The obtained results in this subsection are applied to the
constrained vector variational inequality problem (CVVI1 ) (Theorem 2.5),
the constrained vector optimization problem (CVOP1 ) (Theorem 2.8), the
models of transportion-production problem and Nash-Cournot equilibria
problem.

2.2. Karush-Kuhn-Tucker necessary optimality conditions
for local weak efficient solutions of (CVEP1 )
To derive Karush-Kuhn-Tucker necessary optimality conditions for local weak efficient solutions of (CVEP1 ), we make the following constraint
qualifications:
(CQ1) There exist s ∈ J, v0 ∈ IT (C, x) such that
(i) yk < 0 (∀yk ∈ Dc Fk,x (x)v0 , ∀ k ∈ J, k = s);
zi < 0 (∀zi ∈ Dc gi (x)v0 ∀i ∈ I(x));
(ii) ∇hj (x), v0 = 0 (∀ j ∈ L).
(CQ2) There exists s ∈ J, v0 ∈ IT (C, x) such that for every λk ≥ 0 (∀ k ∈
J, k = s); µi ≥ 0 (∀ i ∈ I(x)), not all zero, and γj ∈ R (∀ j ∈ L), we have
0∈

λk Dc Fk,x (x)v0 +
k∈J,k=s

γj ∇hj (x), v0 .

µi Dc gi (x)v0 +
i∈I(x)

Proposition 2.1 (CQ1) implies (CQ2).

j∈L


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Karush-Kuhn-Tucker necessary optimality conditions for local weak efficient solutions of (CVEP1 ) which can be stated as follows.
Theorem 2.3 Let x be a local weak efficient solution of (CVEP1 ). Assume all hypotheses of Theorem 2.2 are fulfilled. Suppose also that the
constraint qualification (CQ2) (for some s ∈ J) holds. Then, for every
v ∈ Ker∇h(x) ∩ IT (C, x), there exist λs > 0, λk ≥ 0 (∀ k ∈ J, k = s),
µi ≥ 0 (∀ i ∈ I) satisfying
0∈

λk Dc Fk,x (x)v +

µi Dc gi (x)v,
i∈I

k∈J

µi gi (x) = 0 (∀ i ∈ I).
In what follows, we derive a strong Karush-Kuhn-Tucker necessary condition for efficiency in which all the Lagrange multipliers corresponding to
all the components of the objective are positive.
Theorem 2.4 Let x be a local weak efficient solution of (CVEP1 ). Assume all hypotheses of Theorem 2.2 are fulfilled. Suppose also that the

constraint qualification(CQ2) (for every s ∈ J) holds. Then, for every
v ∈ Ker∇h(x) ∩ IT (C, x), there exist λk > 0 (∀ k ∈ J), µi ≥ 0 (∀ i ∈ I)
satisfying
0∈

λk Dc Fk,x (x)v +

µi Dc gi (x)v,
i∈I

k∈J

µi gi (x) = 0 (∀ i ∈ I).
To close this part, we provide the following important notes.
Remark 2.4 The following assertions holds.
(i) Theorem 2.3 and Theorem 2.4 are still true if we replace the constraint
qualification (CQ2) by (CQ1).
(ii) Karush-Kuhn-Tucker necessary optimality conditions for local weak
efficient solution of (CVEP1 ) have not been yet fully discovered.
(iii) The obtained results in this parts are applied to the constrained vector
variational inequality problem (CVVI1 ) and the constrained vector
optimization problem (CVOP1 ) (see Theorems 2.6, 2.7, 2.9 and 2.10).


Chapter 3
Optimality Conditions for Vector
Equilibrium Problems in Terms of
Contingent Epiderivatives
In this paper, we study the existences of contingent epiderivatives and
the relationships between the contingent epiderivatives and the contingent

derivatives in the single-valued case. Besides, we also receive the optimality
conditions for efficient solutions of constrained and unconstrained vector
equilibrium problems.
The chapter is written on the basis of the papers [2], [3], [4] and [7] in
the list of works has been published.

3.1. Existences and relationships between contingent epiderivatives and contingent derivatives
Let ∅ = A ⊂ Y be a nonempty set and let Q ⊂ Y be a cone. Let us
recall the notions of Dinh The Luc (1989) and L. Rodríguez-Marín and M.
Sama (2007a, 2007b) which will be needed in this chapter as follows.
• The set A is said to be Q− lower (resp. -upper) bounded if there
exists y ∈ Y such that A ⊂ y + Q (resp. A ⊂ y − Q).
• A set-valued mapping F : X ⇒ Y is said to have the (LBD) (lower
bounded derivative) property at point (x, y) ∈ graphF if Dc F+ (x, y)u is
Q− lower bounded for any u ∈ L, where L is the projection of T (epi(F ), (x, y))
onto X, and graphF is the graph of F.


14

• The notation of efficient points:
IM in(A|Q) = {y ∈ A : A ⊂ y + Q},
M in(A|Q) = {y ∈ A : A ∩ (y − Q) ⊂ y + Q ∩ (−Q)},
IM ax(A|Q) = {y ∈ A : A ⊂ y − Q},
M ax(A|Q) = {y ∈ A : A ∩ (y + Q) ⊂ y + Q ∩ (−Q)},
infQ A = IM ax

y ∈ Y : A ⊂ y + Q |Q ,

supQ A = IM in

y ∈ Y : A ⊂ y − Q |Q .

Hereafter, we derive an existence result for contingent epiderivative in
case the set-valued mapping f+ has (LBD) property at (x, y) ∈ graph f+ .
Proposition 3.1 Let f : X → Y and x ∈ X. Assume that f+ : X ⇒ Y
has (LBD) property or Dc f+ (x, f (x))u is Q− lower bounded for every
u ∈ L, where L is the projection of T (epif, (x, f (x))) onto X. The following
statements are equivalent:
(i) Df (x) exists.
(ii) infQ Dc f+ (x, f (x))u ∈ Dc f+ (x, f (x))u ∀ u ∈ L.
The existence of contingent hypoderivative can be stated as follows.
Proposition 3.2 Let f : X → Y and x ∈ X. Assume that Dc f+ (x, f (x))u
is Q− upper bounded for every u ∈ L and Q is pointed. The following
statements are equivalent:
(i) Df (x) exists.
(ii) supQ Dc f+ (x, f (x))u ∈ Dc f+ (x, f (x))u ∀ u ∈ L.
Using the concept of Pareto minimum points, a relationship between contingent epiderivatives and contingent derivatives can be stated as follows.
Proposition 3.3 Let f : X → Y and x ∈ X. Suppose that Q has a
compact base B and Df+ (x, f (x)) exists. Then, for any u ∈ X, we have
M in Df+ (x, f (x))u + Q|Q ⊂ Dc f (x)u.

(3.1)

Df+ (x, f (x))u ∈ Dc f (x)u.

(3.2)

In addition,

15

We next have the representation formular of contingent epiderivative,
which can be formulated as follows.
Proposition 3.4 Let f : X → Y and x ∈ X. Suppose that cone Q has a
compact base B and u ∈ dom Dc f+ (x, f (x)). Then, if Df (x)u exists then
Df (x)u = IM in Dc f (x)u|Q = IM in Dc f+ (x, f (x))u|Q

(3.3)

= M in Dc f (x)u|Q = M in Dc f+ (x, f (x))u|Q .
Remark 3.1 Proposition 3.4 has solved a case involving the contingent
epiderivatives of a single-valued mapping with Banach image space, while
Jiménez-Novo and Sama (2009) only received the results with stable functions in finite-dimensional spaces.

3.2. Optimality conditions for efficient solutions of (VEP)
3.2.1. Banach space case
Necessary and sufficient optimality conditions for efficient solutions of
problem (VEP) can be stated as follows.
Lemma 3.1 Let x ∈ K and assume that
(i) Q has a compact base B;
(ii) DFx (x)u exists for every u ∈ dom Dc Fx+ (x, Fx (x));
(iii) Fx (K) ⊂ Dc Fx (x)u + Q for every u ∈ dom Dc Fx+ (x, Fx (x)).
Then, for every u ∈ dom Dc Fx+ (x, Fx (x)), the following inequality holds
ξ, DFx (x)u ≤ ξ, Fx (y)

∀ y ∈ K, ∀ ξ ∈ Q+ .

Theorem 3.1 Let x ∈ K with Fx (x) = 0. Under the assumptions of

Lemma 3.1 and assume, in addition, the function Fx is steady at x. Then,
vector x is a weakly efficient solution of (VEP) if and only if for every
u ∈ A(K, x) ∩ domDc Fx+ (x, Fx (x)), there exists ξ ∈ Q+ \ {0} such that
0 ≤ ξ, DFx (x)u ≤ ξ, Fx (y) , ∀ y ∈ K.

(3.4)

In particular, if K is convex then vector x is a weak efficient solution of
(VEP) if and only if for every u ∈ T (K, x) ∩ dom Dc Fx+ (x, Fx (x)), there
exists ξ ∈ Q+ \ {0} such that (3.4) is satisfied.


16

Remark 3.2 Theorem 3.1 has solved a case of the weakly efficient solution
of vector equilibrium problem without constraints in terms of contingent
epiderivatives, while Jiménez-Novo and Sama (2009) have not been yet
fully discovered, and they only obtained the first-order sufficient and necessary optimality conditions for strict local minimum of a multiobjective
optimization problem without constraints.
Theorem 3.2 Let x ∈ K with Fx (x) = 0 and assume that all the conditions (i), (ii) and (iii) in Lemma 3.1 be fulfilled. Suppose, furthermore,
that the function Fx is steady at x. Then, vector x ∈ K is a Henig efficient (resp., global efficient, superefficient) solutions of (VEP) if and only
if for every u ∈ A(K, x) ∩ dom Dc Fx+ (x, Fx (x)), there exists ξ ∈ Q∆ (B)
(resp., Q , int(Q+ )) such that
0 ≤ ξ, DFx (x)u ≤ ξ, Fx (y) ,

∀ y ∈ K.

Remark 3.3 The results obtained in Theorem 3.2 are fully new and we
have not seen any similar research before for above efficient solutions that
used the tool of contingent epiderivatives.

3.2.2. Finite-dimensional case
In the case Fx (.) is stable at x, a sufficient optimality condition for weak
efficient solution of (VEP) can be stated as follows.
Theorem 3.3 Let dimY < +∞ and let Q ⊂ Y be a closed convex cone
with intQ = ∅. Let x ∈ K and assume that Fx : X → Y be stable at x
with Fx (x) = 0. Suppose, furthermore, that for each u ∈ A(K, x) satisfying
Dc Fx (x)u ∩ (−intQ) = ∅,

(3.5)

and for every y ∈ K, there exists e ∈ Q such that
DFx (x)u ∈ IM in((Fx (.) ± Q)(y) − e | Q).
Then, vector x ∈ K is a weak efficient solution of (VEP).
Remark 3.4 If we replace the condition in (3.5) by an other condition like
DFx (x)u ∈ −intQ, then the results obtained in Theorem 3.3 are still valid.
Theorem 3.3 is a new result about the sufficient optimality condition for
weak efficient solution of (VEP) with stable functions at optimal point.


17

3.3. Optimality conditions for efficient solutions of (CVEP)
Let us consider problem (CVEP) in which X = Rn , Y = Rm , Z = Rr ,
W = Rl , the cones Q ⊂ Rm and S ⊂ Rr with nonempty interiors have
compact bases B and B , respectively. Then, h = (h1 , h2 , . . . , hl ) : Rn → Rl
with hk : Rn → R for every k = 1 . . . l. The constraint qualification of
Kurcyusz-Robinson-Zowe type is denoted by (KRZ) and given as
z ∈ Z : (y, z) ∈ cone D(Fx , g)(x) Ker∇h(x) ∩ IT (C, x)
+ cone S + g(x) = Z.

Fritz John and Kuhn-Tucker necessary optimality conditions for efficient solutions of (CVEP) can be stated as follows.
Theorem 3.4 (Fritz John necessary condition) Let x ∈ K with Fx (x) = 0.
Assume that Fx , g are steady at x; h is continuous in a neighbourhood of
x and Fréchet differentiable at x with ∇h1 (x), . . . , ∇hl (x) linearly independent. Then, if x ∈ K is a weak efficient solution of (CVEP) then for
every u ∈ Ker∇h(x) ∩ IT (C, x) and (v1 , v2 ) = D(Fx , g)(x)u, there exist
(λ, η) ∈ Rm × Rr with (λ, η) = (0, 0) such that
λ ∈Q+ , η ∈ N (−S, g(x)),
λ, v1 + η, v2 ≥ 0.

(3.6)

Theorem 3.5 (Kuhn-Tucker necessary condition) Let x ∈ K with Fx (x) =
0. Assume that Fx , g and h satisfy the conditions of Theorem 3.4, the set
M := D(Fx , g)(x)(Ker∇h(x) ∩ IT (C, x)) is convex and the constraint
qualification of (KRZ) holds. Then, if x is a weak efficient solution (resp.,
Henig efficient solution, global efficient solution, superefficient solution) of
(CVEP) then there exist (λ, η) ∈ Rm × Rr \ {(0, 0)} such that
λ ∈ Q+ \ {0} (t.ứ., Q∆ (B), Q , int(Q+ )),

(3.7)

η ∈ N (−S, g(x)),

(3.8)

λ, v1 + η, v2 ≥ 0 ∀ (v1 , v2 ) ∈ M.

(3.9)

Remark 3.4 The obtained results for the necessary optimality conditions of efficient solutions to the constrained vector equilibrium problems

including equality, inequality and set constraints in terms of contingent
epiderivatives in Theorems 3.4 and 3.5 are new.


Chapter 4
Second-Order Optimality
Conditions for Vector Equilibrium
Problems in Terms of Contingent
Epiderivatives
In this chapter, we first study the existences of second-order contingent
epiderivatives with arbitrary single-valued mappings in Banach spaces. We
second present the second-order optimality conditions for efficient solutions
of constrained vector equilibrium problems (CVEP) in terms of contingent
epiderivatives with arbitrary objective functions in Banach spaces.
This paper is written on the basis of the papers [6] and [8] in the list of
works has been published.

4.1. Existences and relationships between second-order
contingent epiderivatives and contingent derivatives
Based on a cone with a compact base, we have the characterzations of
second-order contingent epiderivatives as follows.
Proposition 4.1 Let k : X → Y, x ∈ X and (u, v) ∈ X × Y. Assume
that Q has a compact base B and x ∈ dom Dc2 k+ (x, k(x), u, v). Then the
following conditions are equivalent:
2

(i) D k(x, u, v)x exists.
(ii) IM in Dc2 k(x, u, v)x|Q = ∅. In addition,

19
2

D k(x, u, v)x = IM in Dc2 k(x, u, v)x|Q .
In case the cone Q is pointed, making use of the set-valued mapping
k+ = k + Q instead of k we obtain the following result.
Proposition 4.2 Let k : X → Y, x ∈ X and (u, v) ∈ X × Y. Assume
that Q is pointed and x ∈ dom Dc2 k+ (x, k(x), u, v). Then the following
conditions are equivalent:
2

(i) D k(x, u, v)x exists.
(ii) IM in Dc2 k+ (x, k(x), u, v)x|Q = ∅. In addition,
2

D k(x, u, v)x = IM in Dc2 k+ (x, k(x), u, v)x|Q .
Remark 4.1 The results obtained in Propositions 4.1 and 4.2 are still
valid for D2 k(x, u, v)x if we replace IMin by IMax. Besides, Propositions
4.1 and 4.2 are extensions of Theorem 2.8 (see Jiménez et al. (2009)).
Making use of the concept of Q− lower boundedness, an other characterzation for the existence of second-order contingent epiderivatives can be
stated as follows.
Proposition 4.3 Let k : X → Y, x ∈ X and (u, v) ∈ X × Y. Suppose, in
addition, that Q is pointed and Dc2 k+ (x, k(x), u, v)x is Q− lower bounded
for every x ∈ L, where L is the projection of T 2 (epi k, (x, k(x)), (u, v)) onto
X. Then the following statements are equivalent:
2

(i) D k(x, u, v)x exists for every x ∈ L.
(ii) infQ Dc2 k+ (x, k(x), u, v)x ∈ Dc2 k+ (x, k(x), u, v)x for every x ∈ L.
A dual form of Proposition 4.3 is the following result.

Proposition 4.4 Let k : X → Y, x ∈ X and (u, v) ∈ X × Y. Suppose, in
addition, that Q is pointed and Dc2 k+ (x, k(x), u, v)x is Q− upper bounded
for every x ∈ L, where L is the projection of T 2 (epi k, (x, k(x)), (u, v)) onto
X. Then the following statements are equivalent:
(i) D 2 k(x, u, v)x exists for all x ∈ L.
(ii) supQ Dc2 k+ (x, k(x), u, v)x ∈ Dc2 k+ (x, k(x), u, v)x for every x ∈ L.


20

Remark 4.2 The results obtained in Propositions 4.3 and 4.4 are true
extensions of Propositions 3.1 and 3.2, respectively.

4.2. Second-order sufficient optimality conditions for efficient solutions of (CVEP)
Let us consider problem (CVEP) with set and cone constraints, which
means that the feasible set be of the following form
K = {x ∈ C : g(x) ∈ −S}.
For each x, u ∈ X and (v, w) ∈ Y × Z, we set
Mx (u, v, w) := dom Dc2 (Fx+ , g+ ) x, (Fx , g)(x), u, (v, w) .
We next derive the second-order sufficient optimality condition for weakly
efficient solution of (CVEP).
Theorem 4.1 Let x ∈ K with Fx (x) = 0 and the cones Q, S have interiors nonempty. Assume, in addition, that there exist u ∈ IT (C, x) and
(v, w) ∈ Dc (Fx+ , g+ )(x, (Fx , g)(x))u ∩ (−Q) × (−S) such that for every
x ∈ Mx (u, v, w), we have
2

(i) D (Fx , g)(x, u, v, w)x exists;
2

(ii) D (Fx , g)(x, u, v, w)x ∈ (−intQ) × IT (−S, w);

(iii) For every y ∈ K, there exist e1 ∈ Q, e2 ∈ IT (S, −w) such that
2

D (Fx , g)(x, u, v, w)x ∈ IM in (Fx , g)(y) − (e1 , e2 ) + P | Q × S .
Where, P ⊂ Y × Z such that either P = Q × S, or P = −(Q × S).
Then, vector x is a weakly efficient solution of (CVEP).
Remark 4.3 Theorem 4.1 is extended from the first-order sufficient optimality condition in terms of contingent epiderivatives in Theorem 3.3.
Furthermore, this theorem has solved the second-order sufficient optimality condition for vector optimization problem with cone constraint, while
Li-Zhu and Teo (2012) have not been yet fully discovered.
Similarly as in Theorem 4.1, a second-order sufficient optimality condition for Henig, global and super-efficient solutions of (CVEP) can be
stated as follows.


21

Theorem 4.2 Let x ∈ K with Fx (x) = 0 and the cones Q, S have
nonempty interiors. Suppose, in addition, that the cone Q has a base B
and there exist u ∈ IT (C, x), (v, w) ∈ Dc (Fx+ , g+ )(x, (Fx , g)(x))u∩(−Q)×
(−S) such that for every x ∈ Mx (u, v, w) satisfying
2

(i) D (Fx , g)(x, u, v, w)x exists.
(ii) There exists λ ∈ Q∆ (B) (resp., Q , int(Q+ ) if in addition B compact),
η ∈ S + with η, w = 0 satisfying λ, ax + η, bx ≥ 0.
(iii) For every y ∈ K, there exist e1 ∈ Q, e2 ∈ IT (S, −w) such that
2

D (Fx , g)(x, u, v, w)x ∈ IM in (Fx , g)(y) − (e1 , e2 ) + P | Q × S .
Where, P ⊂ Y × Z such that, either P = Q × S, or P = −(Q × S).
Then, x is a Henig efficient (resp., global efficient, superefficient) solution to the (CVEP).

Remark 4.4 The obtained result for second-order sufficient optimality
condition of Henig efficient, global efficient and superefficient solutions of
(CVEP) via contingent epiderivatives has not been yet fully discovered.

4.3. Second-order necessary and sufficient optimality conditions for efficient solutions of (CVEP)
Let us consider problem (CVEP) in which the feasible set K is of the
following form
K = {x ∈ C : g(x) ∈ −S, h(x) = 0}.
In order to derive second-order necessary and sufficient optimality conditions for efficient solutions of (CVEP), we make an assumption as follows.
Assumption 4.1 For each x ∈ K, there exist u ∈ IT (C, x)∩IT (h−1 (0), x)
and (v, w) ∈ Dc (Fx , g)(x, (Fx , g)(x))u ∩ (−Q) × (−S) satisfying
2

(A) (ax , bx ) := D (Fx , g)(x, u, v, w)x exists for every x ∈ Mx (u, v, w);
2

(B) (Fx , g)(K) ⊂ D (Fx , g)(x, u, v, w)x + Q × S for every x ∈ Mx (u, v, w);
(C) The following constraint qualification holds
z ∈ Z : (y, z) ∈ cone (Fx , g)(K)

+ cone(S + w) = Z.


22

Theorem 4.3 Let x be a feasible point of (CVEP). Assume that intQ = ∅
and Assumption 4.1 is fulfilled. Then, x is a weakly efficient solution of
(CVEP) if and only if for every x ∈ Mx (u, v, w), there exist (λ, η) ∈
(Y ∗ × Z ∗ ) \ {(0, 0)} such that
λ ∈ Q+ \ {0}, η ∈ S + với

∼

η, w = 0;

∼

(4.1)

∼

λ, Fx (x) + η, g(x) ≥ λ, ax + η, bx ≥ 0 ∀ x ∈ K.

(4.2)

Remark 4.5 Theorem 4.3 is the extension result from the first-order sufficient and necessary optimality condition of unconstrained vector equilibrium problem (VEP) in terms of contingent epiderivatives and based on
a basis from the obtained results of Theorem 3.1. This a new result for
the second-order sufficient and necessary optimality condition for weakly
efficient solution of (CVEP) via contingent epiderivatives with arbitrary
functions in Banach spaces.
In the case of Henig, global and super-efficient solutions, a second-order
necessary and sufficient optimality condition can be illustrated as follows.
Theorem 4.4 Let x be a feasible point of (CVEP). Assume that Assumption 4.1 is fulfilled and the cone Q has a base B. Then, x is a
Henig efficient solution (resp., global efficient solution, superefficient solution) of (CVEP) if and only if for every x ∈ Mx (u, v, w), there exist
(λ, η) ∈ (Y ∗ × Z ∗ ) \ {(0, 0)} satisfying
λ ∈ Q∆ (B) (resp., Q , int(Q+ ));
η ∈ S + with
∼

∼

(4.3)

η, w = 0;

(4.4)
∼

λ, Fx (x) + η, g(x) ≥ λ, ax + η, bx ≥ 0 ∀ x ∈ K.

(4.5)

Remark 4.6 This is a new result about the second-order necessary and sufficient optimality condition for efficient solution types of problem (CVEP)
in terms of contingent epiderivatives with arbitrary functions in Banach
spaces and based on the result obtained of Theorem 3.2.


23

GENERAL CONCLUSIONS
The thesis has achieved the following results.
1) Established the existence results and representation fomulars of firstand second-order contingent epiderivatives and hypoderivatives with a
single-valued map in Banach spaces and given some the relationships between first- and second-order contingent epiderivatives and hypoderivatives
and first- and second-order contingent derivatives, respectively.
2) Constructed the (strong Karush-Kuhn-Tucker) Karush-Kuhn-Tucker
and Fritz John necessary optimality conditions for local weak efficient solutions of (CVEP1 ) with the class of steady, stable, Hadamard directional
differentiable and Fréchet differentiable functions in terms of contingent
derivatives and its applications to the constrained vector variational inequality problems (CVVI1 ), the constrained vector optimization problems
(CVOP1 ) and obtained the necessary optimality conditions for local weak
efficient solutions to the models of the transportion-production problem
and Nash-Cournot equilibria problem.

3) Constructed the Fritz John and Kuhn-Tucker types first- and second order necessary and sufficient optimality conditions via contingent
epiderivatives for vector equilibrium problems (VEP) and (CVEP) with
a class of steady, stable, Hadamard directional differentiable and Fréchet
differentiable functions to the first-order conditions and with a class of
arbitrary functions to the second-order conditions.
Recommendations for future research:
• Research the optimality conditions for efficient solution of vector equilibrium problems with applications.
• Research the first- and second-order optimality conditions in terms
of contingent derivatives for constrained vector equilibrium problems
(CVEP) with applications.
• Research the fundamental formulas of second-order contingent derivatives and epiderivatives in Banach spaces.
• Research and build some reality models by using the tools of firstand second-order contingent derivatives and epiderivatives.