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

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. Research optimality conditions for weak, Henig, global and superefficient solutions in vector equilibrium problems with steady. 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.

MINISTRY OF EDUCATION AND VIETNAM ACADEMY

GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY

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

Supervisors 1: Assoc Prof Dr Do Van Luu

Supervisors 2: Dr Nguyen Cong Dieu

Gradu-The thesis can be found at:

- Library of Graduate University of Science and Technology

- Vietnam National Library

The vector equilibrium problem plays an important role in nonlinearanalysis and has attracted extensive attention in recent years because ofits 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 fromthe scalar equilibrium problem which was first introduced by Blum-Oettli(1994) and the optimality condition for its efficient solutions is a main sub-ject 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 second-order optimality conditions for vector equilibrium problems in terms ofcontingent derivatives and epiderivatives in which the conditions of orderone using stable functions and two using arbitrary functions

The contingent derivative plays a central role in analysis and appliedanalysis, and it will be used to establish the optimality conditions Aubin(1981) first introduced a concept of a contingent derivative for set-valuedmapping and their applications to express the optimality conditions invector optimization problems like Aubin-Ekeland (1984), Corley (1988)and Luc (1991) Jahn-Rauh (1997) provided a concept of a contingentepiderivative for set-valued mapping and obtained the respectively opti-mality conditions Chen-Jahn (1998) proposed a concept of a general con-tingent epiderivative for set-valued mapping and the result is applied tothe set-valued vector equilibrium problems In the case of single-valuedoptimization problems, we don’t need to move from set-valued results intosingle-valued results which establishing the new results are sharper

1

Based on the concept of Aubin (1981), Jiménez-Novo (2008) have provedthe good calculus rules of contingent derivatives with steady, stable, Hadamarddifferentiable, Fréchet differentiable functions as well as their applicationsfor establishing optimality conditions in unconstrained vector equilibriumproblems The author also derived the necessary and sufficient optimal-ity conditions for multiobjective optimization problems involving equalityand inequality constraints with stable functions via contingent derivatives.One limitation in the results of Jiménez-Novo (2008) is not consideredthe Fritz John and Kuhn-Tucker necessary optimality conditions for lo-cal weakly efficient solutions of constrained vector equilibrium problemincluding 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 exis-tences, uniqueness and some properties of contingent epiderivatives andhypoderivatives, the relationships between contingent epiderivatives/ hy-poderivatives and contingent derivatives with both stable functions andset-valued mappings in case the finite-dimensional image spaces One lim-itation in the results of Rodríguez-Marín and Sama (2007a, 2007b) is notconsidered the existences of contingent epiderivatives and hypoderivativesfor arbitrary single-valued functions with Banach image spaces On opti-mality conditions, Jiménez-Novo and Sama (2009) only derived the suf-ficient and necessary optimality conditions for strict local minimums oforder one via the contingent epiderivatives and hypoderivatives with sta-ble objective functions in multiobjective optimization problems In case thesufficient and necessary optimality conditions for weakly efficient, Henigefficient, global efficient and superefficient solutions of vector equilibriumproblems in terms of contingent epiderivatives and hypoderivatives withstable functions are not considered by Jiménez-Novo and Sama (2009) andother authors Our thesis has studied the existence results of contingentepiderivatives and hypoderivatives with arbitrary single-valued functions

in Banach spaces, the relationships between them and contingent tives, and obtaining the sufficient and necessary optimality conditions forefficient solutions of vector equilibrium problems via the contingent epi-derivatives with steady functions in Banach spaces, and providing, in ad-

dition, a sufficient optimality condition for weakly efficient solution of constrained vector equilibrium problem with stable functions as a basis forextending the results to research the second order optimality conditions

un-In a recent decade, the second-order optimality conditions for vectorequilibrium problems and its special cases via contingent derivatives andepiderivatives has been intensively studied by many authors like Jahn-Khan-Zeilinger (2005), Durea (2008), Li-Zhu-Teo (2012), Khan-Tammer(2013), etc We see that the existence results of second order contingentepiderivatives and hypoderivatives with arbitrary single-valued functions inBanach spaces are not considerd, and the sufficient optimality conditionsfor weakly efficient solutions via second-order composed contingent epi-derivatives only studied to the unconstrained optimization problem Ourdissertation has researched the existence results for second-order generalcontingent epiderivatives and hypoderivatives with arbitrary single-valuedfunctions as well as constructed the sufficient, sufficient and necessary op-timality conditions for efficient solutions of constrained vector equilibriumproblems in terms of contingent epiderivatives in Banach spaces

The main purpose of this thesis is to study the first- and second-orderoptimality 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 vec-tor equilibrium problem involving set, inequality and equality constraintswith stable functions via contingent derivatives in finite-dimensional spaces.2) Research optimality conditions for weak, Henig, global and super-efficient solutions in vector equilibrium problems with steady, Hadamarddifferentiable, Fréchet differentiable functions in terms of contingent epi-derivatives in Banach spaces

3) Research second order optimality conditions for weak, Henig, global,super-efficient solutions in vector equilibrium problems with arbitrary func-tions in terms of contingent epiderivatives in Banach spaces

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

Chapter 1 introduces some concepts from efficient solutions to (CVEP),contingent cones, contingent sets, contingent derivatives, epiderivativesand hypoderivatives Besides, it provides the concept of stable, steady,Hadamard differentiable and Fréchet differentiable functions and severalcontingent derivatives related fomulars Finally, the concept of ideal andPareto efficient points with respect to a cone is also derived as well

Chapter 2 studies the Fritz John and Karush-Kuhn-Tucker necessaryoptimality conditions for local weak efficient solution of constrained vec-tor equilibrium problems with stable functions via contingent derivatives

in finite-dimensional spaces and presents some its applications to vectorinequality variational problems, vector optimization problems Besides, wehave proposed two constraint qualifications (CQ1) and (CQ2) for inves-tigating Karush-Kuhn-Tucker and strong Karush-Kuhn-Tucker necessaryoptimality conditions Many examples to illustrate the results are derived.Chapter 3 studies the existences of contingent epiderivatives as well asthe necessary and sufficient optimality condition for weak, Henig, global,super-efficient solutions in vector equilibrium problems with stable func-tions via contingent epiderivatives in two cases the initial and final spacesare 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 tives and second order sufficient optimality conditions for weakly efficient,Henig efficient, global efficient and superefficient solutions in vector equi-librium problems with constraints with arbitrary functions via contingentepiderivatives in Banach spaces The last part of this chapter makes anassumption 4.1 as a basis for studying second order optimality conditions.The result of the thesis is presented in:

epideriva-• The 4th National Conference on Applied Mathematics, National nomics University, Hanoi 23-25/12/2015;

Eco-• 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 thepresentation of research results achieved in the next chapters and exactly:Section 1.1 deals with several concepts such as: tangent sets, stablefunctions, contingent derivatives, epiderivatives and hypoderivatives

• In section 1.1.1 presents the concepts of contingent cone, adjacentcone, interior tangent cone, sequential interior tangent cone, normal cone,second order contingent set, second order adjacent set, second order interiortangent set and some its properties

• In section 1.1.2 presents the definitions of first and second order tingent derivatives

con-• In section 1.1.3 presents the definitions of Hadamard derivative, stablefunction, 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; theconcepts of first and second order contingent epiderivatives along withsome results on its existences

Section 1.2 deals with general vector equilibrium problem and some itsspecial 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, globalefficient 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

5

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 beconstraints 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

Vector x is called a weakly efficient solution of problem (CVEP) If thereexists a neighborhood U of x such that (1.1) holds for every y ∈ K ∩ Uthen x is called a local weakly efficient solution of problem (CVEP) If theproblem (CVEP) with a set constraint (in short, (VEP)), and called theunconstrained vector equilibrium problem If X = Rn, Y = Rm, Z = Rr,

W = Rl and the cones Q = Rm+, S = Rr+, then the problem (CVEP) issaid 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 dualcone of Q ⊂ Y, which means that

Making use of the seperation theorem of disjoint convex sets {0} and B,

it yields that there exists y∗ ∈ Y∗ \ {0} satisfying

and for any convex neighborhood U of zero with U ⊂ VB, it holds that

B + U is a convex set and 0 6∈ cl(B + U ) Thus, cone(B + U ) is a pointedconvex cone satisfying Q \ {0} ⊂ int cone(U + B)

Based on the preceding illustrations, Gong (2008, 2010) has constructedthe concept for globally efficient, Henig efficient and super-efficient solu-tions 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} ⊂ intHsuch 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 hh, xi instead of the value of h ∈ L(X, Y ) at x ∈ X The vectorvariational inequality problem with constraints is denoted by (CVVI) andgiven as F (x, y) = hT x, y − xi , where T is a mapping from X into L(X, Y )

In this case, the concept of efficient solutions of (CVEP) is similar as theconcept 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 alocal weak minimum and a strict local minimum of order m (m ∈ N) aswell as the optimality condition for strict local minimum of order one viacontingent derivatives of multiobjective optimization problems is derived

• In section 1.2.3 introduces vector variational inequality problem andsome 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 necessaryoptimality conditions for local weakly efficient solutions of (CVEP1) andsome its applications to the vector variational inequality problem (CVVI1),the vector optimization problem (CVOP1), the transportion - productionproblem 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:

neigh-Fritz John necessary optimality conditions for local weak efficient tion of (CVEP1) which can be stated as follows.

solu-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 ∈ Dcg(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) 6= (0, 0) such that

hλ, yi + hµ, zi ≥ 0,

µigi(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 ∈ Dcg(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

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

µi ≥ 0 (∀ i ∈ I) with (λ, µ) 6= (0, 0) such that

Remark 2.1 Theorem 2.2 is applied to establish the necessary optimalityconditions 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 multiobjectiveoptimization problems with set constraint while the author Jiménez andNovo (2008) have not been yet fully discovered The author only studiedthe optimality conditions for weak efficient solutions of problem (CVEP1)involving equality and inequality constraints In addition, if C ≡ Rn thenTheorem 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, gare steady x Suppose, furthermore, that for every v ∈ Ker∇h(x), thereexists z ∈ Dcg(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

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 Hadamarddifferentiable and steady at x Suppose, furthermore, that for every v ∈Ker∇h(x) ∩ IT (C, x), dgi(x; v) < 0 (∀ i ∈ I(x)) Then,

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

µi ≥ 0 (∀ i ∈ I) with (λ, µ) 6= (0, 0) such that

2.2 Karush-Kuhn-Tucker necessary optimality conditions for local weak efficient solutions of (CVEP1)

To derive Karush-Kuhn-Tucker necessary optimality conditions for cal weak efficient solutions of (CVEP1), we make the following constraintqualifications:

lo-(CQ1) There exist s ∈ J, v0 ∈ IT (C, x) such that

(i) yk < 0 (∀yk ∈ DcFk,x(x)v0, ∀ k ∈ J, k 6= s);

zi < 0 (∀zi ∈ Dcgi(x)v0 ∀i ∈ I(x));

(ii) h∇hj(x), v0i = 0 (∀ j ∈ L)

(CQ2) There exists s ∈ J, v0 ∈ IT (C, x) such that for every λk ≥ 0 (∀ k ∈

J, k 6= s); µi ≥ 0 (∀ i ∈ I(x)), not all zero, and γj ∈ R (∀ j ∈ L), we have