Optimality conditions for vector equilibrium problems in terms of contingent derivatives

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), Jim†nez-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.

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MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING 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 Gradu- ate 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 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 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 opti- mality conditions. Chen-Jahn (1998) proposed a concept of a general con- tingent 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. 1 2Based 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 optimal- ity 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 lo- cal 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 exis- tences, uniqueness and some properties of contingent epiderivatives and hypoderivatives, the relationships between contingent epiderivatives/ hy- poderivatives and contingent derivatives with both stable functions and set-valued mappings in case the finite-dimensional image spaces. One lim- itation 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 opti- mality conditions, Jiménez-Novo and Sama (2009) only derived the suf- ficient and necessary optimality conditions for strict local minimums of order one via the contingent epiderivatives and hypoderivatives with sta- ble 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 deriva- tives, and obtaining the sufficient and necessary optimality conditions for efficient solutions of vector equilibrium problems via the contingent epi- derivatives with steady functions in Banach spaces, and providing, in ad- 3dition, a sufficient optimality condition for weakly efficient solution of un- constrained 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 Jahn- Khan-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 epi- derivatives 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 op- timality 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 vec- tor 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 super- efficient solutions in vector equilibrium problems with steady, Hadamard differentiable, 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 of the thesis consists of four chapters and the main results of the dissertation are contained in Chapters 2,3,4. 4Chapter 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 vec- tor 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 inves- tigating 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 func- tions 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 epideriva- 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 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 Eco- nomics 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 con- tingent 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 5 6subset 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) 6∈ −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, W = Rl and the cones Q = Rm+ , S = Rr+, 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 VB = {y ∈ Y : | 〈y∗, y〉 | < r 2 }. The notion VB will be used throughout this dissertation. It is evident that inf{〈y∗, y〉 : y ∈ B + VB} ≥ r 2 , 7and 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 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 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} ⊂ 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) = 〈Tx, y − x〉 , where T is a mapping fromX 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}. 8 9Let 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 neigh- bourhood 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 solu- tion 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 ∈ 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 〈λ, y〉+ 〈µ, z〉 ≥ 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 0 ∈ ∑ k∈J λkDcFk,x(x)v + ∑ i∈I µiDcgi(x)v + ∑ j∈L γj 〈∇hj(x), v〉 , (2.1) µigi(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) 6= (0, 0) such that 0 ∈ ∑ k∈J λkDcFk,x(x)v + ∑ i∈I µiDcgi(x)v, (2.3) µigi(x) = 0 (∀ i ∈ I). 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 ∈ 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 0 ∈ ∑ k∈J λkDcFk,x(x)v + ∑ i∈I µiDcgi(x)v + ∑ j∈L γj 〈∇hj(x), v〉 , (2.4) µigi(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) 6= (0, 0) such that 0 ∈ ∑ k∈J λkDcFk,x(x)v + ∑ i∈I µiDcgi(x)v, µigi(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, furth
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