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 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 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- 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 Chapter 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 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 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 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 Some Knowledge of Preparing Chapter 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 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 ≥ ∀ y ∈ Q} We denote the quasi-interior of Q+ by Q , i.e Q = {y ∗ ∈ Y ∗ : y ∗ , y > ∀ y ∈ Q \ {0}} Let B be a base of cone Q Set Q∆ (B) = {y ∗ ∈ Q : ∃ t > 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 ∗ , = Let us consider an open absolutely convex neighborhood VB of zero in Y be of the form r VB = {y ∈ Y : | y ∗ , y | < } The notion VB will be used throughout this dissertation It is evident that r inf { y ∗ , y : y ∈ B + VB } ≥ , and for any convex neighborhood U of zero with U ⊂ VB , it holds that B + U is a convex set and ∈ 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 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 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 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 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) ≤ (∀ i ∈ I), hj (x) = (∀ j ∈ L)} Let us denote by Ker∇h(x) = {v ∈ X : ∇h(x), v = 0}, I(x) = {i ∈ I : gi (x) = 0} 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 < (∀ i ∈ I(x)) Then, (i) For every v ∈ Rn , there exist λk ≥ (∀ k ∈ J), µi ≥ (∀ 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) = (∀ i ∈ I) (2.5) (ii) For every v ∈ Ker∇h(x), there exist λk ≥ (∀ k ∈ J), µi ≥ (∀ 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) = (∀ 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) < (∀ i ∈ I(x)) Then, 11 (i) For every v ∈ IT (C, x), there exist λk ≥ (∀ k ∈ J), µi ≥ (∀ 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) = (∀ i ∈ I) (2.7) (ii) For every v ∈ Ker∇h(x) ∩ IT (C, x), there exist λk ≥ (∀ k ∈ J), µi ≥ (∀ 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) = (∀ 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 < (∀yk ∈ Dc Fk,x (x)v0 , ∀ k ∈ J, k = s); zi < (∀zi ∈ Dc gi (x)v0 ∀i ∈ I(x)); (ii) ∇hj (x), v0 = (∀ j ∈ L) (CQ2) There exists s ∈ J, v0 ∈ IT (C, x) such that for every λk ≥ (∀ k ∈ J, k = s); µi ≥ (∀ 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 12 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 ≥ (∀ k ∈ J, k = s), µi ≥ (∀ i ∈ I) satisfying 0∈ λk Dc Fk,x (x)v + µi Dc gi (x)v, i∈I k∈J µi gi (x) = (∀ 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 > (∀ k ∈ J), µi ≥ (∀ i ∈ I) satisfying 0∈ λk Dc Fk,x (x)v + µi Dc gi (x)v, i∈I k∈J µi gi (x) = (∀ 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 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) = 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 ≤ 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) = 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 ≤ 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) = 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 = 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) = 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 ≥ (3.6) Theorem 3.5 (Kuhn-Tucker necessary condition) Let x ∈ K with Fx (x) = 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 ≥ ∀ (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 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: (i) D k(x, u, v)x exists (ii) IM in Dc2 k(x, u, v)x|Q = ∅ In addition, 19 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: (i) D k(x, u, v)x exists (ii) IM in Dc2 k+ (x, k(x), u, v)x|Q = ∅ In addition, 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 (epi k, (x, k(x)), (u, v)) onto X Then the following statements are equivalent: (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 (epi k, (x, k(x)), (u, v)) onto X Then the following statements are equivalent: (i) D 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) = 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 (i) D (Fx , g)(x, u, v, w)x exists; (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 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) = 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 (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 = satisfying λ, ax + η, bx ≥ (iii) For every y ∈ K, there exist e1 ∈ Q, e2 ∈ IT (S, −w) such that 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 (A) (ax , bx ) := D (Fx , g)(x, u, v, w)x exists for every x ∈ Mx (u, v, w); (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 ≥ ∀ 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 ≥ ∀ 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 24 LIST OF WORKS HAS BEEN PUBLISHED Do Van Luu, Tran Van Su, Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints, RAIRO - Oper Res., 2017, (online) https://doi.org/10.1051/ro/2017042 (SCI-E) Tran Van Su, Optimality conditions for vector equilibrium problems in terms of contingent epiderivatives, Numer Funct Anal Optim., 2016, 37, 640-665 (SCI-E) Tran Van Su, New optimality condition for unconstrained vector equilibrium problem in terms of contingent derivatives in Banach spaces, 4OR- Q J Oper Res., 2017, (online) https://doi.org/10.1007/s10288017-0360-4 (SCI-E) Tran Van Su, A new optimality condition for weakly efficient solutions of convex vector equilibrium problems with constraints, J Nonlinear Funct Anal., 2017, 7, 1-14 (Scopus) Tran Van Su, Optimality conditions for weak efficient solution of vector equilibrium problem with constraints, J Nonlinear Funct Anal., 2016, 4, 1-16 (Scopus) Tran Van Su, Second-order optimality conditions for vector equilibrium problems, J Nonlinear Funct Anal., 2015, 6, 1-31 (Scopus) Tran Van Su, Fritz John type optimality conditions for weak efficient solutions of vector equilibrium problems with constrains in terms of contingent epiderivatives, Appl Math Sci., 2015, 126, 6249-6261 (Scopus) Do Van Luu, Tran Van Su, Nguyen Cong Dieu, Second-order efficiency conditions for vector equilibrium problem with constraints via contingent epiderivatives, Ann Oper Res., (submitted) (SCI) ... 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. .. 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. .. 2.9 and 2.10) Chapter Optimality Conditions for Vector Equilibrium Problems in Terms of Contingent Epiderivatives In this paper, we study the existences of contingent epiderivatives and the relationships