Optimization A Journal of Mathematical Programming and Operations Research ISSN: 0233-1934 (Print) 1029-4945 (Online) Journal homepage: http://www.tandfonline.com/loi/gopt20 Tykhonov well-posedness for lexicographic equilibrium problems Lam Quoc Anh & Tran Quoc Duy To cite this article: Lam Quoc Anh & Tran Quoc Duy (2016): Tykhonov well-posedness for lexicographic equilibrium problems, Optimization, DOI: 10.1080/02331934.2016.1209673 To link to this article: http://dx.doi.org/10.1080/02331934.2016.1209673 Published online: 18 Jul 2016 Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=gopt20 Download by: [University of California, San Diego] Date: 19 July 2016, At: 01:32 OPTIMIZATION, 2016 http://dx.doi.org/10.1080/02331934.2016.1209673 Tykhonov well-posedness for lexicographic equilibrium problems Lam Quoc Anha and Tran Quoc Duyb,c Downloaded by [University of California, San Diego] at 01:32 19 July 2016 a Department of Mathematics, Teacher College, Cantho University, Cantho, Vietnam; b Department of Mathematics, University of Science, Vietnam National University, Hochiminh City, Vietnam; c Department of Mathematics, Cantho Technical Economic College, Cantho, Vietnam ABSTRACT ARTICLE HISTORY In this paper, we consider the vector equilibrium problems involving lexicographic cone in Banach spaces We introduce the new concepts of the Tykhonov well-posedness for such problems The corresponding concepts of the Tykhonov well-posedness in the generalized sense are also proposed and studied Some metric characterizations of well-posedness for such problems are given As an application of the main results, several results on well-posedness for the class of lexicographic variational inequalities are derived Received 15 November 2015 Accepted 27 June 2016 KEYWORDS Lexicographic order; equilibrium problems; variational inequalities; Tykhonov well-posedness AMS SUBJECT CLASSIFICATIONS 49K40; 90C31; 91B50 Introduction Well-posedness plays an important role in both theory and numerical methods for optimization theory This fact has been motivated and inspired many mathematicians to study the well-posedness for problems related to optimization In 1966, Tykhonov introduced the concept of well-posedness for unconstrained optimization problems, which has become known as Tikhonov well-posedness A minimization problem is said to be Tykhonov well-posed if it has a unique solution toward which every minimizing sequence of the problem converges (see [1]) Since then, the study of Tykhonov wellposedness and its extensions has been among the very interesting and important topics in the stability for optimization theory A generalization of Tykhonov well-posedness strengthened for this concept has been discussed for constrained optimization problems and sequence optimization problems (see, e.g [2–7]) Another generalization of the concept given for optimization problems with more than one solution requires the existence and convergence of a subsequence of each minimizing sequence towards a solution.[8] The other fundamental generalization of Tykhonov well-posedness, which was first introduced for scalar optimization problem by Zolezzi [9,10], is the well-posedness under perturbations The idea of this generalized concept is embedding the Tykhonov well-posedness and the continuous dependence of the solution on the data The study of Tykhonov well-posedness and its extensions for problems related to equilibrium problems, such as optimization problem, variational inequality, Nash equilibria and equilibrium problem, is a theme of great importance and has received increasing attention by many researchers recently The sufficient and necessary conditions and metric characterizations of the well-posedness for such problems were considered For more details, we refer the reader to [11–16] and the references therein It is well known that the class of partially ordered spaces plays an important role in vector optimization theory The vector problems related to optimization are usually based on partial orders induced by convex closed cones; i.e they base on various extensions of the Pareto order From the CONTACT Lam Quoc Anh quocanh@ctu.edu.vn © 2016 Informa UK Limited, trading as Taylor & Francis Group Downloaded by [University of California, San Diego] at 01:32 19 July 2016 L Q ANH AND T Q DUY theory of vector optimization, however, this setting leads to an optimal solution set that is usually too large (see, e.g [17–20] and the references therein) Hence, reducing the optimal solution set is the aim of many works One of the efficient approaches is to use the lexicographic cone In [21], for a fixed orthogonal base, the authors constructed a total ordering cone in Rn and showed that the lexicographic order was a unique total order in the sense that any total order on Rn was equivalent to lexicographic order Furthermore, lexicographic cone also plays a vital role in many practical problems, such as choosing products, ranking medal table in Olympic Games; see, e.g [22–25] Therefore, vector problems related to optimization involving lexicographic cone have been intensively studied recently; see, e.g for variational inequalities,[24,26] optimization problems,[22, 27] equilibrium problems [19,20,25] and the references therein As far as we know, well-posedness for the lexicographic vector equilibrium problems was discussed in only two papers [28] and [29] In these papers, this property was obtained under the lower semicontinuity of an auxiliary set-valued mapping corresponding to objective function However, this assumption is difficultly checked and hard applied to practical situations since it requires the information of a solution set of the equation In this paper, motivated and inspired by the above observations, we aim to suggest the new concepts of the Tykhonov well-posedness and its extension to the lexicographic equilibrium problems The corresponding concepts of the Tykhonov well-posedness in the generalized sense are also introduced and investigated Furthermore, we also study some metric characterizations of these properties via the Kuratowski measure of noncompactness and diameter of approximate solution sets of such problems The layout of the paper is as follows: In Section 2, we state the lexicographic equilibrium problems and recall some preliminary results which are needed in the succeeding sections Section is devoted to the (generalized) Tykhonov well-posedness for the lexicographic equilibrium problems In Section 4, we study sufficient conditions of the (generalized) Tykhonov well-posedness under perturbations by a sequence of approximating problems for these problems In the last section, as an application, several results on these types of well-posedness for lexicographic variational inequalities are derived from the main results Preliminaries We first recall the notion of lexicographic cone in finite-dimensional spaces and the setting of equilibrium problems involving this cone The lexicographic cone of Rn , denoted by Clex , is the collection of zero and all vectors x ∈ Rn which the first nonzero coordinate of x is positive, i.e Clex := {0} ∪ {x ∈ Rn | x1 = · · · = xk = 0, xk+1 > 0, for some k, ≤ k < n} For any x and y in Rn , the lexicographic order is defined as follows: x ≥lex y ⇐⇒ x − y ∈ Clex Since Clex ∪ ( − Clex ) = Rn , the lexicographic order is a total order Moreover, let C1 := {x ∈ Rn | x1 ≥ 0}, then int C1 Clex C1 and int Clex = int C1 and cl Clex = C1 Hence, the lexicographic cone is neither closed nor open Let E be a real Banach space and X be a nonempty closed subset of E Let E ∗ be the dual space of E The norm and the dual pair between E and E ∗ are denoted by · and ·, · , respectively Suppose that f = (f1 , f2 , , fn ) : X × X → Rn is a vector-valued function, where fi is an equilibrium function for each i ∈ In := {1, 2, , n}, i.e fi (x, x) = for all x ∈ X We consider the following lexicographic equilibrium problem: OPTIMIZATION (LEP) find x¯ ∈ X such that for all y ∈ X, f (¯x , y) ≥lex The following notions are employed in the sequel Definition 1: Let Q : X ⇒ Y be a set-valued mapping between two Banach spaces Downloaded by [University of California, San Diego] at 01:32 19 July 2016 (i) Q is said to be upper semicontinuous (usc, in short) at x0 , if for any open subset U of Y with Q(x0 ) ⊂ U, there is a neighborhood N of x0 such that Q(N) ⊂ U (ii) Q is said to be lower semicontinuous (lsc, in short) at x0 , if for any open subset U of Y with Q(x0 ) ∩ U = ∅, there is a neighborhood N of x0 such that Q(x) ∩ U = ∅, for all x ∈ N Q is said to be continuous at x0 , if it is both usc and lsc at x0 The following well-known assertions play an important role in our analysis Lemma 2.1: (see, e.g [30]) (i) If Q(x0 ) is compact, then Q is usc at x0 if and only if for any sequence {xn } converging to x0 , every sequence {yn } with yn ∈ Q(xn ) has a subsequence converging to some point in Q(x0 ) If, in addition, Q(x0 ) = {y0 } is a singleton, then such a sequence {yn } must converge to y0 (ii) Q is lsc at x0 if and only if for any sequence {xn } converging to x0 and any point y ∈ Q(x0 ), there exists a sequence {yn } with yn ∈ Q(xn ) converging to y Definition 2: (see, e.g [28]) R ∪ {+∞} is said to be Let ε be a real number An extended real-valued function g : X → (i) upper ε-level closed at x¯ ∈ X, if for any sequence {xn }, xn → x¯ , g(xn ) ≥ ε, ∀n ⇒ g(¯x ) ≥ ε ; (ii) strongly upper ε-level closed at x¯ ∈ X, if for any sequences {xn }, xn → x¯ and {μn } ⊂ [0; ∞), μn → 0, g(xn ) + μn ≥ ε, ∀n ⇒ g(¯x ) ≥ ε ; (iii) upper semicontinuous at x¯ ∈ X, if for any sequence {xn }, xn → x¯ , it holds that g(¯x ) ≥ lim sup g(xn ) n→∞ We say that F satisfies a certain property in a subset A of X if F satisfies it at each x ∈ A If A ≡ X we omit the term ‘in X’ in the statement Now we recall the concepts of the Kuratowski measure of noncompactness and the Hausdorff distance Definition 3: (see, e.g [31]) Let M be a nonempty subset of E The Kuratowski measure of noncompactness μ of the set M is defined by n μ(M) = inf ε > | M ⊂ Mi , diam Mi < ε, i = 1, , n , i=1 where diam Mi is the diameter of Mi Definition 4: Let A, B be nonempty subsets of E The Hausdorff distance between A and B is defined by H(A, B) = max H ∗ (A, B), H ∗ (B, A) , where H ∗ (A, B) = supa∈A d(a, B) with d(a, B) = inf b∈B d(a, b) 4 L Q ANH AND T Q DUY Lemma 2.2: (see, e.g [31]) The following assertions are true: (i) μ(M) = if M is compact; (ii) μ(M) ≤ μ(N) whenever M ⊂ N; (iii) If {Mn } is a sequence of closed subsets in E satisfying Mn+1 ⊂ Mn for every n ∈ N and limn→∞ μ(Mn ) = 0, then K := n∈N Mn is a nonempty compact set and limn→∞ H(Mn , K) = Downloaded by [University of California, San Diego] at 01:32 19 July 2016 Tykhonov well-posedness for lexicographic equilibrium problems In this section, we study sufficient conditions for (LEP) to be Tykhonov well-posed To start our analysis, we consider lexicographic equilibrium problems for the case n = 2, namely, f = (f1 , f2 ) : X × X → R2 , since the general case is similar Then, we can rewrite (LEP) in the following equivalent way: find x¯ ∈ X such that f1 (¯x , y) ≥ 0, f2 (¯x , z) ≥ 0, ∀y ∈ X, ∀z ∈ Z(¯x ), (1) (2) where Z : X ⇒ X is defined by Z(x) = {z ∈ X | f1 (x, z) = 0} The solution set of (LEP) is denoted by S Let e ∈ Clex \ {0} For each ε ∈ [0, +∞), we consider the following approximate problem corresponding to e: (LEPe,ε ) find x¯ ∈ X such that f (xn , y) + εn e ≥lex 0, ∀y ∈ X The solution set of this approximate problem (LEPe,ε ) is denoted by Se (ε) := {x ∈ X | f (x, y) + εe ≥lex 0, ∀y ∈ X} Definition 5: A sequence {xn } is said to be an approximating sequence for (LEP) corresponding to e, if there exists a sequence {εn } ⊂ R+ with εn → such that xn ∈ Se (εn ) for all n Definition 6: The problem (LEP) is said to be (i) generalized Tykhonov well-posed with respect to (wrt) e, if S = ∅ and for any approximating sequence {xn } for (LEP) corresponding to e, there exists a subsequence {xni } of {xn } converging to some point of S (ii) Tykhonov well-posed wrt e, if it is generalized Tykhonov well-posed wrt e and its solution set is a singleton Remark 1: (a) For each a, b > and ε > 0, let Sci (ε) := {x ∈ X | f (x, y) + εci ≥lex 0, ∀y ∈ X}, i ∈ {1, 2, 3, 4}, where c1 = (0, b), c2 = (a, −b), c3 = (a, 0), c4 = (a, b) It is easy to see that Sc1 (ε) ⊂ Sc2 (ε) ⊂ Sc3 (ε) ⊂ Sc4 (ε) OPTIMIZATION (b) For a = (a1 , a2 ) ∈ Clex \ {0} and b = (b1 , b2 ) ∈ Clex \ {0}, we define a relation ∼ on Clex \ {0} as follows: a ∼ b ⇐⇒ there exist k, l > such that a1 = kb1 and a2 = lb2 One can check that ∼ is an equivalence relation on Clex \ {0} Denote a be the equivalence class determined by a Then Clex \ {0} /∼= {e1 , e2 , e3 , e4 }, Downloaded by [University of California, San Diego] at 01:32 19 July 2016 where e1 = (0, 1), e2 = (1, −1), e3 = (1, 0), e4 = (1, 1) From Remark 1, we obtain the following inclusions: Se1 (ε) ⊂ Se2 (ε) ⊂ Se3 (ε) ⊂ Se4 (ε) The following example illustrates the above statement Example 1: Let E = R2 , X = {x = (x1 , x2 ) ∈ R2 | ≤ xk ≤ 1, k = 1, 2} and f = (f1 , f2 ) : X → R2 be defined by f1 (x, y) = x1 − y1 , f2 (x, y) = x2 − y2 For each ε > 0, direct computations give Se1 (ε) = {x = (1, x2 ) ∈ X | x2 ∈ [1 − ε, 1]}, Se2 (ε) = A, Se3 (ε) = A ∪ B, Se4 (ε) = A ∪ C, where A := {x = (1, x2 ) ∈ X | x2 ∈ [0, 1]}, B := {x = (x1 , 1) ∈ X | x1 ∈ [1 − ε, 1)}, C := {x = (x1 , x2 ) ∈ X | x1 ∈ [1 − ε, 1), x2 ∈ [1 − ε, 1]} Therefore, Se1 (ε) Se2 (ε) Se3 (ε) Se4 (ε) Proposition 3.1: Suppose that a, b ∈ Clex \ {0} and a ∼ b Then (LEP) is (generalized) Tykhonov well-posed wrt a if and only if it is (generalized, respectively) Tykhonov well-posed wrt b Proof: By the similarity we verify only the case a ∈ e2 as an example In this case, we need only to show that {xn } is an approximating sequence for (LEP) corresponding to a = (a1 , −a2 ), a1 , a2 > if and only if it is an approximating sequence for (LEP) corresponding to e2 Indeed, if {xn } is an approximating sequence for (LEP) corresponding to a Then there exists a sequence {εn } ⊂ R+ , εn → such that f (xn , y) + εn (a1 , −a2 ) ≥lex 0, ∀y ∈ X For each n, let δn = (a1 + a2 )εn Then {δn } ⊂ R+ , δn → and f1 (xn , y) + δn > f1 (xn , y) + a1 εn ≥ 0, ∀y ∈ X, which implies that f (xn , y) + δn (1, −1) ≥lex 0, ∀y ∈ X Hence, {xn } is an approximating sequence for (LEP) corresponding to e2 L Q ANH AND T Q DUY Conversely, if {xn } is an approximating sequence for (LEP) corresponding to e2 Then there exists a sequence {εn } ⊂ R+ , εn → such that f (xn , y) + εn (1, −1) ≥lex 0, ∀y ∈ X Downloaded by [University of California, San Diego] at 01:32 19 July 2016 For each n, there are ξn , ξn > such that εn = a1 ξn = a2 ξn Letting δn = ξn + ξn , one has a sequence {δn } ⊂ R+ , δn → and f (xn , y) + δn (a1 , −a2 ) ≥lex 0, ∀y ∈ X, since f1 (xn , y) + δn a1 > f1 (xn , y) + εn ≥ 0, ∀y ∈ X Thus, {xn } is an approximating sequence for (LEP) corresponding to a Motivated and inspired by the above observations, in the sequel, we choose e = e1 Then, (LEP) is (generalized) Tykhonov well-posed wrt e if it is (generalized) Tykhonov well-posed wrt c, for all c ∈ Clex \ {0} The following two examples illustrate that the converse is not true Example 2: Let E = R, X = [0, +∞) and f = (f1 , f2 ) : X → R2 be defined by f (x, y) = (0, y − x) It is clear that S = {0} For each ε > 0, f (x, y) + εe ≥lex 0, ∀y ∈ X if and only if x ≤ ε; i.e Se (ε) = [0, ε] Hence, (LEP) is Tykhonov well-posed wrt e However, for all c = (c1 , c2 ) ∈ intClex , one has Sc (ε) = X, so (LEP) is not Tykhonov well-posed wrt c Indeed, let xn = (n+1)/n Then, {xn } is an approximating sequence for (LEP) corresponding to c since f (xn , y)+εc = (εc1 , xn −y +εc2 ) >lex 0, ∀y ∈ X But xn → ∈ / S Example 3: Let E be an m-dimensional Euclidean space Rm , X = {x = (x1 , x2 , , xm ) ∈ Rm | ≤ xk ≤ 1, k ∈ Im } and f = (f1 , f2 ) : X → R2 be defined by f1 (x, y) = x1 − y1 , f2 (x, y) = x2 − y2 + x−y By simple computations, we have S = {x ∈ X | x = (1, 1, x3 , xm ), xk ∈ [0, 1], k ∈ {3, , m}} For each ε > 0, one has Se (ε) = {x ∈ X | x = (1, x2 , x3 , , xm ), x2 ∈ [1 − ε, 1], xk ∈ [0, 1], k ∈ {3, , m}} Hence, (LEP) is generalized Tykhonov well-posed wrt e However, for each c = (c1 , c2 ) ∈ intClex , (LEP) is not generalized Tykhonov well-posed wrt c Indeed, let x (n) = (1, 1/n, 1, , 1) Then, for all ε > 0, one has f (x (n) , y) + εc = − y1 + εc1 , 1 − y2 + x (n) − y + εc2 n >lex 0, ∀y ∈ X, since − y1 + εc1 > 0, ∀y1 ∈ [0, 1] Hence, {x (n) } is an approximating sequence for (LEP) / S corresponding to c But x (n) → (1, 0, 1, , 1) ∈ To simplify the presentation, in the sequel, if (LEP) is (generalized) Tykhonov well-posed with respect to e, we omit the term ‘with respect to e’ in the statement For each ε > 0, the approximate solution set of (LEP) corresponding to e is denoted by: S(ε) := Se (ε) = {x ∈ X | f1 (x, y) ≥ 0, ∀y ∈ X and f2 (x, z) + ε ≥ 0, ∀z ∈ Z(x)} The following theorem provides sufficient conditions of the Tykhonov well-posedness for (LEP) Theorem 3.2: Assume that X is compact and (i) f1 is continuous; the Fréchet derivative D2 f1 of f1 with respect to the second argument exists and D2 f1 (x, y) is surjective for all x, y ∈ X, x = y; (ii) f2 is strongly upper 0-level closed OPTIMIZATION Then, (LEP) is generalized Tykhonov well-posed Furthermore, it is Tykhonov well-posed if S is a singleton Proof: Let {xn } be an arbitrary approximating sequence for (LEP) Then, there exists a sequence {εn } ⊂ R+ with εn → such that f1 (xn , y) ≥ 0, ∀y ∈ X, (3) f2 (xn , z) + εn ≥ 0, ∀z ∈ Z(xn ) (4) Downloaded by [University of California, San Diego] at 01:32 19 July 2016 By the compactness of X, there is a subsequence (still denoted by {xn }) converging to some x¯ in X Combining the continuity of f1 and (3), we conclude that f1 (¯x , y) ≥ 0, for all y ∈ X To complete the first conclusion of the theorem, we only need to show that x¯ ∈ S Suppose, on the contrary, that x¯ ∈ / S, then there is a point z¯ ∈ Z(¯x ) \ {¯x }, such that f2 (¯x , z¯ ) < (5) We prove that for each neighborhood V of z¯ , V ⊂ X, there exist a neighborhood U of x¯ and a mapping s : U → V such that s(x) ∈ Z(x) for all x ∈ U Let m = D2 f1 (¯x , z¯ )−1 Since D2 f1 is surjective, Theorem 5A.1 in [32] implies that m = Let α be a positive real number such that Bα (¯z ) ⊂ V , where Bα (¯z ) is the closed ball with center at z¯ and radius α Since f1 is Fréchet differentiable with respect to the second argument, one can choose a real number β, with < β ≤ α satisfying f1 (¯x , z) − f1 (¯x , z¯ ) − D2 f1 (¯x , z¯ ), z − z¯ ≤ z − z¯ , for all z ∈ Bβ (¯z ), 2m which implies that f1 (¯x , z) − D2 f1 (¯x , z¯ ), z − z¯ ≤ β , for all z ∈ Bβ (¯z ), 2m as f1 (¯x , z¯ ) = From the continuity of f1 , there is γ ∈ (0, β] such that f1 (¯x , z) − f1 (x, z) ≤ β , for all x ∈ Bγ (¯x ), and z ∈ Bβ (¯z ) 2m For each x ∈ Bγ (¯x ), we construct the function ξx : Bβ (¯z ) → X defined by ξx (z) = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), z − f1 (x, z) Since f1 is continuous, ξx is continuous in Bβ (¯z ) Moreover, for any z ∈ Bβ (¯z ), one has ξx (z) − z¯ = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), z − f1 (x, z) − z¯ = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), z − f1 (x, z) − D2 f1 (¯x , z¯ ), z¯ = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), z − z¯ − f1 (x, z) D2 f1 (¯x , z¯ ), z − z¯ − f1 (¯x , z) + f1 (¯x , z) − f1 (x, z) β β ≤m = β + 2m 2m ≤m This means that ξx maps Bβ (¯z ) into itself Since Bβ (¯z ) is compact and convex, the Brouwer’s fixedpoint theorem implies that, for each x ∈ Bγ (¯x ), there exists a point denoted by s(x) ∈ Bβ (¯z ) ⊂ V such that ξx (s(x)) = s(x), i.e s(x) = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), s(x) − f1 (x, s(x)) , L Q ANH AND T Q DUY or equivalently, f1 (x, s(x)) = Thus, s(x) ∈ Z(x) This argument ensures the existence of a sequence {zn } with zn ∈ Z(xn ), zn → z¯ It follows from (4) that f2 (xn , zn ) + εn ≥ 0, for all n Since f2 is strongly upper 0-level closed at (¯x , z¯ ), we have f2 (¯x , z¯ ) ≥ 0, which contradicts (5) Therefore, x¯ ∈ S, i.e (LEP) is generalized Tykhonov well-posed The second conclusion of the theorem follows directly from Definition 6, and hence the proof is complete Going back to Example 3, we immediately check that all assumptions in Theorem 3.2 are satisfied Hence, (LEP) is generalized Tykhonov well-posed The following examples show that the assumptions of Theorem 3.2 are essential Example 4: (Compactness of X cannot be dispensed) Let X = E = R, and Downloaded by [University of California, San Diego] at 01:32 19 July 2016 f (x, y) = ((x − y)2 , x − y) Obviously, assumptions (i) and (ii) of Theorem 3.2 are satisfied and S = R However, (LEP) is not generalized Tykhonov well-posed since the approximating sequence {xn }, xn = n, for (LEP) has no any convergent subsequence The reason is that X is not compact Example 5: (Surjectivity of D2 f1 is essential) Let E = R, X = [0, 1] and f (x, y) = ((1 − x)(x − y)2 , y − x) Then, assumption (ii) is satisfied by the continuity of f2 By direct computations, we see that S = [0, 1) Let xn = (n − 1)/n We can easily verify that {xn } is an approximating sequence for (LEP) but {xn } converges to ∈ / S Hence, (LEP) is not generalized Tykhonov well-posed The reason is that assumption (i) is violated Indeed, for all y ∈ X, D2 f1 (1, y) = Example 6: (Assumption (ii) is essential) Let E = R2 , X = [0, 1] × [0, 1] and f (x, y) = (f1 (x, y), f2 (x, y)), where x = (x1 , x2 ), y = (y1 , y2 ) ∈ X, f1 (x, y) = x1 − y1 , and f2 (x, y) = It is clear that f1 is continuous and D2 f1 (x, y) = y1 − x1 , y2 − x2 , if x2 > 12 , otherwise −1 Thus, assumption (i) is fulfilled By direct computations, we have S = (1, x2 ) | x2 ∈ {0} ∪ 12 , 1 / S Hence, (LEP) and S(ε) = (1, x2 ) | x2 ∈ [0, ε] ∪ 12 , Let xn = (1, n+1 2n ), then xn → (1, ) ∈ is not generalized Tykhonov well-posed The reason is that the strong upper 0-level closedness of f2 is violated Indeed, taking xn = (1, (n + 1)/2n), yn = (1, 1/2n) and εn = 1/n, we have xn → x = (1, 1/2), yn → y = (1, 0), εn → Then, f2 (xn , yn ) + εn > 0, but f2 (x, y) = −1/2 < Using the diameter of the approximate solution sets, we obtain a metric characterization of the Tykhonov well-posedness for (LEP) Theorem 3.3: Assume that (i) f1 is continuous; the Fréchet derivative of f1 with respect to the second argument exists and D2 f1 (x, y) is surjective for all x, y ∈ X, x = y; (ii) f2 is strongly upper 0-level closed Then, (LEP) is Tykhonov well-posed if and only if S(ε) = ∅, ∀ε > and lim diam (S(ε)) = ε→0 Downloaded by [University of California, San Diego] at 01:32 19 July 2016 OPTIMIZATION Proof: Suppose that (LEP) is Tykhonov well-posed Then, (LEP) has a unique solution x¯ , and hence S(ε) = ∅ for all ε > If limε→0 diam (S(ε)) = 0, then there exist r > 0, εn > with εn → and xn , un ∈ S(εn ) such that r d(xn , un ) ≥ , ∀n ∈ N (6) Then, {xn } and {un } are approximating sequences for (LEP) By the Tykhonov well-posedness of (LEP), they must converge to the unique solution x¯ of (LEP) Thus, limn→∞ d(xn , un ) = 0, which contradicts (6) Conversely, suppose that {xn } is an approximating sequence for (LEP) Then, there exists a sequence {εn }, εn → such that xn ∈ S(εn ) for all n By taking a subsequence, if necessary, we can assume that {εn } is non-increasing, and hence S(εn ) ⊂ S(εm ) whenever m ≤ n Since limn→∞ diam (S(εn )) = 0, {xn } is a Cauchy sequence and converges to x¯ , x¯ ∈ X Using the same arguments as for Theorem 3.2, we conclude that x¯ ∈ S To complete the proof, we show that (LEP) has a unique solution Suppose, by contradiction, that S contains another point u with u = x¯ It is obvious that x¯ and u belong to S(ε) for any ε > It follows that < d(¯x , u) ≤ diam (S(ε)), which is impossible Using the Kuratowski measure of noncompactness of approximate solution sets, we establish a metric characterization of the generalized Tykhonov well-posedness for (LEP) Theorem 3.4: Assume that (i) f1 is continuous; the Fréchet derivative of f1 with respect to the second argument exists and D2 f1 (x, y) is surjective for all x, y ∈ X, x = y; (ii) f2 is upper δ-level closed for all δ < Then, (LEP) is generalized Tykhonov well-posed if and only if S(ε) = ∅, ∀ε > and lim μ(S(ε)) = ε→0 Proof: Suppose that (LEP) is generalized Tykhonov well-posed We first prove that S is compact Indeed, let {xn } be an arbitrary sequence in S Obviously, it is also an approximating sequence for (LEP), and hence it has a subsequence converging to some point in S Thus, S is compact Take any ε > and S ⊂ ∪ni=1 Mi with diam Mi ≤ ε for all i = 1, , n Set Ni := {x ∈ X | d(x, Mi ) ≤ H(S(ε), S)} We show that S(ε) ⊂ ∪ni=1 Ni Let x ∈ S(ε) Then, d(x, S) ≤ H(S(ε), S) Since S ⊂ ∪ni=1 Mi , we conclude that d(x, ∪ni=1 Mi ) ≤ H(S(ε), S) So, there is k0 ∈ {1, 2, , n} such that d(x, Mk0 ) ≤ H(S(ε), S), i.e x ∈ Nk0 Therefore, S(ε) ⊂ ∪ni=1 Ni Notice further that diam Ni = diam Mi + 2H(S(ε), S) ≤ ε + 2H(S(ε), S) Since S is compact, we conclude that μ(S) = Therefore, μ(S(ε)) ≤ μ(S) + 2H(S(ε), S) = 2H(S(ε), S) Now we show that H(S(ε), S) → as ε → Since S ⊂ S(ε), we get H ∗ (S, S(ε)) = 0, and hence we only need to prove that H ∗ (S(ε), S) → as ε → Assume, by contradiction, that there exist a real number r > 0, a sequence {εn }, εn → 0, and xn ∈ S(εn ) such that d(xn , S) ≥ r, for all n Since {xn } is an approximating sequence for (LEP), it has a subsequence {xnk } converging to some point 10 L Q ANH AND T Q DUY Downloaded by [University of California, San Diego] at 01:32 19 July 2016 x ∈ S Thus, for nk sufficiently large, we have d(xnk , x) < r, which is a contradiction Therefore, μ(S(ε)) → as ε → Conversely, suppose that S(ε) = ∅, for all ε > and μ(S(ε)) → as ε → We first prove that for each ε > 0, S(ε) is closed Let {xn } ⊂ S(ε) and xn → x¯ Then, for each n, one has f1 (xn , y) ≥ 0, ∀y ∈ X, (7) f2 (xn , z) + ε ≥ 0, ∀z ∈ Z(xn ) (8) Combining the continuity of f1 and (7), we conclude that f1 (¯x , y) ≥ for all y ∈ X Suppose that there is z¯ ∈ Z(¯x ) \ {¯x } such that f2 (¯x , z¯ ) + ε < Using the techniques given in the proof of Theorem 3.2, there is a sequence {zn }, zn ∈ Z(xn ) such that zn → z¯ By assumption (ii), we have f2 (xn , zn ) < −ε for n sufficiently large, which contradicts (8) Hence, x¯ ∈ S(ε), i.e S(ε) is closed It is worth noting that S = ∩ε>0 S(ε) Applying Lemma 2.2 and taking into account that limε→0 μ(S(ε)) = 0, we conclude that S is compact and H(S(ε), S) → as ε → Let {xn } be an approximating sequence for (LEP) Then, there is εn > 0, εn → such that xn ∈ S(εn ) Since d(xn , S) ≤ H(S(εn ), S) and H(S(ε), S) → 0, there exists a sequence {¯xn } ⊂ S such that d(xn , x¯ n ) → as n → ∞ Inasmuch as S is compact, there is a subsequence {¯xnk } of {¯xn } converging to some x¯ ∈ S Thus, {xn } has the corresponding subsequence {xnk } converging to x¯ This brings the proof to its end Remark 2: The assertions of Theorem 3.4 are still valid if the Kuratowski measure is replaced by the Hausdorff or Istrˇatescu measure For further information about such noncompact measures including their relations, we refer the reader to [31] Sequential Tykhonov well-posedness for lexicographic equilibrium problems In this section, we study well-posedness for lexicographic equilibrium problems under perturbations by a sequence of approximating problems which are expressed in terms of perturbing constraints Cn ⊂ X and f (n) : X → R2 Then, our problems are embedded into the following family (LEP(Cn ,f (n) ) ) find x¯ ∈ Cn such that f (n) (¯x , y) ≥lex 0, ∀y ∈ Cn Let C := {C ⊂ X | C is nonempty closed}, F := {f = (f1 , f2 ) : X → R2 | f is a function} and M := {(C, f ) ∈ C × F | there exists x¯ ∈ C such that f (¯x , y) ≥lex 0, ∀y ∈ C} Instead of writing {(LEPϕ ) | ϕ ∈ C × F} for the family of lexicographic equilibrium problems, i.e the sequence of approximating problems, we will simply write (LEP) in the sequel Now we propose the following basic assumptions: (A1 ) f1 is continuous; the Fréchet derivative of f1 with respect to the second argument exists and D2 f1 (x, y) is surjective for all x, y ∈ X, x = y; (A2 ) f2 is upper semicontinuous We consider the following subset of F: F d = {(f1 , f2 ) ∈ F | f1 satisfies (A1 ) and f2 satisfies (A2 )} For each ϕm = (Cm , f (m) ), ϕn = (Cn , f (n) ) in C × F, we define the distance between ϕm and ϕn as follows: d(ϕm , ϕn ) := H(Cm , Cn ) + sup d(f (m) (x, y), f (n) (x, y)), (x,y)∈X OPTIMIZATION 11 where d((f1 , f2 ), (g1 , g2 )) = sup {|f1 (x, y) − g1 (x, y)| + |f2 (x, y) − g2 (x, y)|} (x,y)∈X Then, (C × F, d) is a pseudo-quasi-metric space We refer the reader to [33] for further information about this space For each ϕ ∈ M, we denote the solution set of (LEPϕ ) by S(ϕ) Then ϕ → S(ϕ) is a set-valued mapping from M into X Definition 7: For a given ϕ ∈ C × F, let {ϕn } ⊂ C × F be a sequence converging to ϕ A sequence {xn }, xn ∈ Cn , is said to be an approximating sequence for (LEPϕ ) corresponding to {ϕn }, if there exists {εn } ⊂ R+ , with εn → such that Downloaded by [University of California, San Diego] at 01:32 19 July 2016 f (n) (xn , y) + εn e ≥lex 0, ∀y ∈ Cn , where e = (0, 1) In what follows, for each ϕ = (C, f ) ∈ C × F and ε, δ ∈ [0, +∞), let S(ϕ, ε) := {x ∈ C | f1 (x, y) ≥ 0, ∀y ∈ C, and f2 (x, z) + ε ≥ 0, ∀z ∈ Z(x)}, (ϕ, δ, ε) = S( , ε), ∈Bδ (ϕ) where Z(x) = {z ∈ X | f1 (x, z) = 0} Definition 8: (LEP) is said to be generalized Tykhonov well-posed under perturbations by a sequence of approximating problems (sequentially generalized Tykhonov well-posed, in short) at ϕ, if (i) the solution set S(ϕ) is nonempty; (ii) for any sequence {ϕn } converging to ϕ, every approximating sequence for (LEPϕ ) corresponding to {ϕn } must exist a subsequence converging to an element in S(ϕ) Definition 9: (LEP) is said to be sequentially Tykhonov well-posed at ϕ, if (i) there exists a unique solution x¯ to (LEPϕ ); (ii) for any sequence {ϕn } converging to ϕ, every approximating sequence for (LEPϕ ) corresponding to {ϕn } will converge to x¯ One says that (LEP) is sequentially generalized Tykhonov well-posed (sequentially Tykhonov well-posed, respectively) in a subset A of C × F, if it is sequentially generalized Tykhonov well-posed (sequentially Tykhonov well-posed, respectively) at each element of A Theorem 4.1: Assume that X is compact Then, (LEP) is sequentially generalized Tykhonov wellposed in M ∩ (C × F d ) Furthermore, it is sequentially Tykhonov well-posed if its solution set is a singleton Proof: Let ϕ = (C, f ) ∈ M ∩ (C × F d ) We check that S is usc at (ϕ, 0) For reductio ad absurdum, suppose that there exist an open superset N of S(ϕ, 0) and a sequence {(ϕn , εn )} converging to (ϕ, 0) such that for each n, there is xn ∈ S(ϕn , εn ) \ N Then, one has (n) f1 (xn , y) ≥ 0, (n) f2 (xn , z) + εn ∀y ∈ Cn , ≥ 0, ∀z ∈ Zn (xn ) (9) (10) Since X is compact, we can assume that xn → x¯ for some x¯ ∈ X Thus, for all b > 0, there is a natural number n1 ∈ N such that b d(xn , x¯ ) ≤ , ∀n ≥ n1 12 L Q ANH AND T Q DUY Since Cn → C, there exists n2 ∈ N such that H(C, Cn ) ≤ b , ∀n ≥ n2 , d(xn , xn ) ≤ b , ∀n ≥ n2 (11) and hence, there is xn ∈ C such that Then, for all n ≥ max{n1 , n2 }, we get Downloaded by [University of California, San Diego] at 01:32 19 July 2016 d(xn , x¯ ) ≤ d(xn , xn ) + d(xn , x¯ ) ≤ b, i.e xn → x¯ as n → ∞ Since C is closed, we have x¯ ∈ C Now we show that x¯ ∈ S(ϕ, 0) = S(ϕ) The proof is divided into three steps Step We check that f1 (¯x , y) ≥ 0, for all y ∈ C Indeed, for each y ∈ C, by the closedness of C, there is a sequence {yn } ⊂ C such that yn → y, and so there is n3 ∈ N such that d(yn , y) ≤ b , ∀n ≥ n3 Applying (11), we conclude that there exists yn ∈ Cn such that d(yn , yn ) ≤ b2 , ∀n ≥ n2 Hence, for all n ≥ max{n2 , n3 }, one has d(yn , y) ≤ d(yn , yn ) + d(yn , y) ≤ b, i.e yn converges to y Suppose that there is y¯ ∈ C and θ < such that f1 (¯x , y¯ ) < continuous, there is a subsequence {(xnk , ynk )} such that f1 (xnk , ynk ) < θ Since f1 is θ On the other hand, (nk ) ≤ d(f1 (nk ) (xnk , ynk ), f1 (xnk , ynk )) ≤ sup d(f1 (x,y)∈X (x, y), f1 (x, y)) (n ) (n ) Since sup(x,y)∈X d(f1 k (x, y), f1 (x, y)) → as nk → ∞, we derive the fact that d(f1 k (xnk , ynk ), f1 (xnk , ynk )) converges to (n ) It follows from (9) that there is n4 ∈ N such that f1 k (xnk , ynk ) ≥ θ4 , ∀nk ≥ n4 Then, (nk ) d(f1 n (xnk , ynk ), f1 (xnk , ynk )) ≥ d f1 k (xnk , ynk ), −∞, θ ≥ |θ | , which is a contradiction Therefore, f1 (¯x , y) ≥ 0, for all y ∈ C Step For each z¯ ∈ Z(¯x ) \ {¯x }, we show that there exists a sequence {zn }, zn ∈ Zn (xn ) with zn → z¯ We first prove that for each neighborhood V of z¯ , V ⊂ X, there exist a neighborhood U of x¯ and a mapping sn : U → V such that sn (x) ∈ Zn (x) for all x ∈ U Let m = D2 f1 (¯x , z¯ )−1 Combining the fact that f ∈ F d and Theorem 5A.1 in [32], we conclude that m = Let α be a positive real number such that Bα (¯z ) ⊂ V As f1 satisfies assumption (A1 ), there is a real number β ∈ (0, α] such that f1 (¯x , z) − f1 (¯x , z¯ ) − D2 f1 (¯x , z¯ ), z − z¯ ≤ z − z¯ , for all z ∈ Bβ (¯z ), 2m OPTIMIZATION 13 and consequently, as f1 (¯x , z¯ ) = 0, f1 (¯x , z) − D2 f1 (¯x , z¯ ), z − z¯ ≤ (n) Since f1 is continuous in Bα (¯x ) × Bα (¯z ) and f1 that for all x ∈ Bγ (¯x ), z ∈ Bβ (¯z ), one has β , for all z ∈ Bβ (¯z ) 2m → f1 , there exists a positive number γ ≤ β such f1 (x, z) − f1 (¯x , z) ≤ Downloaded by [University of California, San Diego] at 01:32 19 July 2016 and f1(n) (x, z) − f1 (x, z) ≤ β , 4m (12) β , for n sufficiently large 4m (13) Applying (12) and (13), we have (n) (n) f1 (x, z) − f1 (¯x , z) ≤ f1 (x, z) − f1 (x, z) + f1 (x, z) − f1 (¯x , z) ≤ (n) For each x ∈ Bγ (¯x ), we consider the function ξx ξx(n) (z) = D2 f1 (¯x , z¯ )−1 β 2m : Bβ (¯z ) → X defined by (n) D2 f1 (¯x , z¯ ), z − f1 (x, z) It is clear that ξx(n) is continuous in Bβ (¯z ) and furthermore, for each z ∈ Bβ (¯z ), one has ξx(n) (z) − z¯ = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), z − f1(n) (x, z) − z¯ = D2 f1 (¯x , z¯ )−1 D2 f1 (¯x , z¯ ), z − z¯ − f1 (x, z) ≤m ≤m (n) (n) D2 f1 (¯x , z¯ ), z − z¯ − f1 (¯x , z) + f1 (¯x , z) − f1 (x, z) β β + 2m 2m = β Thanks to the Brouwer’s fixed-point theorem, for all x ∈ Bγ (¯x ), there exists a point denoted by (n) sn (x) ∈ Bβ (¯z ) ⊂ V such that ξx (sn (x)) = sn (x) Thus, sn (x) = D2 f1 (¯x , z¯ )−1 (n) D2 f1 (¯x , z¯ ), sn (x) − f1 (x, sn (x)) , which is equivalent to f1(n) (x, sn (x)) = 0, i.e sn (x) ∈ Zn (x) Step We prove that f2 (¯x , z) ≥ for all z ∈ Z(¯x ) Indeed, for each z¯ ∈ Z(¯x ), we proceed in the same way as in Step to ensure that there exists a sequence {zn }, zn ∈ Zn (xn ) satisfying zn → z¯ Taking (n) (n) into account (10), we obtain f2 (xn , zn ) + εn ≥ So, f2 (xn , zn ) + f2 (xn , zn ) − f2 (xn , zn ) + εn ≥ 0, i.e f2 (xn , zn ) ≥ −|f2(n) (xn , zn ) − f2 (xn , zn )| − εn (n) Since f2 (xn , zn ) − f2 (xn , zn ) → 0, and f2 is upper semicontinuous, we conclude that f2 (¯x , z¯ ) ≥ Therefore, x¯ ∈ S(ϕ), which is again a contradiction, since xn ∈ / N for all n Hence, S is usc at (ϕ, 0) To complete the proof, we check that S(ϕ, 0) = S(ϕ) is compact Let {xn } ⊂ S(ϕ), xn → x¯ , be an arbitrary sequence Using the same argument as above, we can conclude that x¯ ∈ S(ϕ) Thus, S(ϕ) is closed, and hence S(ϕ) is compact due to the compactness of X The following example is given as an illustration for Theorem 4.1 14 L Q ANH AND T Q DUY Example 7: Let C = X = {x = (x1 , x2 , , xm ) ∈ Rm | ≤ xk ≤ 1, k ∈ Im } and f = (f1 , f2 ) : X → R2 be defined by Downloaded by [University of California, San Diego] at 01:32 19 July 2016 f1 (x, y) = (x1 − y1 ) x − y , f2 (x, y) = (x2 − y2 ) exp ( x − y ) Then, it is easy to see that the assumptions of Theorem 4.1 are satisfied By Direct computations, we have S(ϕ) = {x ∈ X | x = (1, 1, x3 , , xm )}, where ϕ = (C, f ) Let ϕn = (Cn , f (n) ) and x (n) ∈ S(ϕn , εn ) be arbitrary, where ϕn → ϕ, and εn ∈ [0, +∞), εn → Since x (n) ∈ Cn and Cn → C, there exists x¯ ∈ C such that x (n) → x¯ Using the same arguments as in the proof of Theorem 4.1, we have x¯ ∈ S(ϕ) Hence, (LEP) is sequentially generalized Tykhonov well-posed at ϕ The next examples prove that assumptions of Theorem 4.1 are essential Example 8: (Compactness of X is essential) Let C = X = R and f (x, y) = exp (x)(x − y)2 , exp (y)(x − y) Then, all assumptions of Theorem 4.1, except for the compactness of X, are satisfied It is clear that S(ϕ) = R, where ϕ = (C, f ) Let Cn = R and f (n) (x, y) = (x − y)2 + x n n , (x − y) + y n n Then, ϕn = (Cn , f (n) ) converges to ϕ as n → ∞ We have S(ϕn ) = R Let {xn }, xn = n, be an approximating sequence for (LEPϕ ) corresponding to ϕn = (Cn , f (n) ), ϕn → ϕ But then, {xn } has no any convergent subsequence Hence, (LEP) is not sequentially generalized Tykhonov well-posed at ϕ = (C, f ) Example 9: (Surjectivity of D2 f1 is essential) Let X = [−2, 2], C = [0, 1] and f (x, y) = x(x − y)2 , x − y Then, X is compact and f2 is continuous We see that S(ϕ) = (0, 1], where ϕ = (C, f ) Let Cn = − n1 , + n1 and f (n) (x, y) = x+ 1 (x − y)2 , n exp n n − (x − y) It is clear that ϕn = (Cn , f (n) ) converges to ϕ as n tends to ∞ Let xn = n1 One can check {xn } is an / S(ϕ) Hence, approximating sequence for (LEPϕ ) corresponding to {ϕn } but {xn } converges to ∈ (LEP) is not sequentially generalized Tykhonov well-posed The reason is that D2 f1 (0, y) = 0, for all y ∈ X Example 10: (Upper semicontinuity of f2 is essential) Let C = X = [0, 2] × [0, 2] and f (x, y) = (f1 (x, y), f2 (x, y)), where x = (x1 , x2 ), y = (y1 , y2 ) ∈ X and f1 (x, y) = y2 − x2 , f2 (x, y) = x1 − y1 , 0, if x1 ≤ 1, if x1 > Thus, assumption (A1 ) is fulfilled For ϕ = (C, f ), one can easily check that S(ϕ) = {(x1 , 0) | x1 ∈ (1, 2]} Let Cn = [0, 2] × [0, 2], and (n) (n) f (n) (x, y) = f1 (x, y), f2 (x, y) , where It is easy to see that f1 is continuous and D2 f1 (x, y) = (n) f1 (x, y) = x1 − y1 n + y2 − x2 , OPTIMIZATION and Downloaded by [University of California, San Diego] at 01:32 19 July 2016 (n) f2 (x, y) = ⎧ ⎨(x1 − y1 ) exp ⎩(x2 − y2 ) exp x1 −y1 , n x2 −y2 n 15 if x1 ≤ 1, −1 , if x1 > Obviously, ϕn = (Cn , f (n) ) converges to ϕ as n → ∞ Let x (n) = ( n+1 n , 0) It is not hard to verify that {x (n) } is an approximating sequence for (LEPϕ ) corresponding to {ϕn } But x (n) → (1, 0) ∈ / S(ϕ) Therefore, (LEP) is not sequentially generalized Tykhonov well-posed The reason is that f2 is not (n) = (2, 1), we have x (n) → x = (1, 1), y (n) → y = (2, 1) and usc Indeed, taking x (n) = ( n+1 n , 1), y f2 (x (n) , y (n) ) = 0, but f2 (x, y) = −1 < The following theorem gives us a metric characterization of the sequential Tykhonov wellposedness for (LEP) in terms of the behavior of the approximate solution sets Theorem 4.2: (LEP) is sequentially Tykhonov well-posed at ϕ¯ ∈ C × F d if and only if the following condition holds (ϕ, ¯ δ, ε) = ∅, ∀δ, ε > and diam (ϕ, ¯ δ, ε) → as (δ, ε) → (0, 0) (14) Proof: Suppose that (LEP) is sequentially Tykhonov well-posed at ϕ¯ ∈ C × F d Then, (LEPϕ¯ ) has a unique solution x¯ , and hence (ϕ, ¯ δ, ε) = ∅ as x¯ ∈ (ϕ, ¯ δ, ε) for all δ, ε > If diam (ϕ, ¯ δ, ε) as (δ, ε) → (0, 0), then there are r > 0, and sequences of positive real numbers {δn } and {εn }, ¯ δn , εn ), for each n ∈ N, such that (δn , εn ) → (0, 0), and sequences {xn }, {ˆxn } with xn , xˆ n ∈ (ϕ, d(xn , xˆ n ) > r (15) Thus, there are ϕn = (Cn , f (n) ), ϕˆ n = (Cˆ n , fˆ (n) ) both in Bδn (ϕ) ¯ such that xn ∈ Cn , xˆ n ∈ Cˆ n , and f (n) (xn , y) + εn e ≥lex 0, ∀y ∈ Cn , fˆ (n) (ˆxn , yˆ ) + εn e ≥lex 0, ∀ˆy ∈ Cˆ n Since {ϕn } and {ϕˆ n } converge to ϕ¯ as n → ∞, the sequences {xn } and {ˆxn } are approximating sequences for (LEPϕ¯ ) corresponding to {ϕn } and {ϕˆ n }, respectively By the sequentially Tykhonov well-posedness of (LEP) at ϕ, ¯ they must converge to the unique solution x¯ of (LEPϕ¯ ) Therefore, limn→∞ d(xn , xˆ n ) = 0, which contradicts (15) Conversely, suppose that condition (14) holds for ϕ¯ ∈ C × F d Let ϕn = (Cn , f (n) ) → ϕ¯ and {xn } be an approximating sequence for (LEPϕ¯ ) corresponding to {ϕn } Then, there exists εn > 0, εn → 0, ¯ δn , εn ) with δn = d(ϕn , ϕ) → 0, such that xn ∈ S(ϕn , εn ) for all n ∈ N It is clear that xn ∈ (ϕ, and hence δn → as n → ∞ We take into account (14) to ensure that {xn } is a Cauchy sequence, and so it converges to some x¯ ∈ X Applying the same argument given in the proof of Theorem 4.1, we conclude that x¯ ∈ S(ϕ) To finish the proof, we need to verify that (LEPϕ¯ ) has a unique solution ¯ δ, ε), for all Indeed, if S(ϕ) ¯ had two distinct solutions x¯ and x¯ , then they would belong to (ϕ, δ, ε > Hence, ¯ δ, ε), < d(¯x1 , x¯ ) ≤ diam (ϕ, which contradicts (14) Using the Kuratowski measure of noncompactness of the approximate solution sets, we obtain characterized conditions of the sequentially generalized Tykhonov well-posedness for (LEP) Theorem 4.3: (a) If (LEP) is sequentially generalized Tykhonov well-posed at ϕ¯ ∈ C × F, then (ϕ, ¯ δ, ε) = ∅, ∀δ, ε > and μ( (ϕ, ¯ δ, ε)) → as (δ, ε) → (0, 0) (16) 16 (b) L Q ANH AND T Q DUY Assume that C ×F is compact or finite dimensional and ϕ¯ ∈ C ×F d Then, (LEP) is sequentially generalized Tykhonov well-posed at ϕ¯ if condition (16) holds Proof: (a) Suppose that (LEP) is sequentially generalized Tykhonov well-posed at ϕ ¯ Then S(ϕ) ¯ is a nonempty compact set Since S(ϕ) ¯ ⊂ (ϕ, ¯ δ, ε) for each δ, ε > 0, we conclude that (ϕ, ¯ δ, ε) = ∅ and H( (ϕ, ¯ δ, ε), S(ϕ)) ¯ = H ∗ ( (ϕ, ¯ δ, ε), S(ϕ)) ¯ Using the same techniques as in the proof of Theorem 3.4, we also obtain, μ( (ϕ, ¯ δ, ε)) ≤ 2H( (ϕ, ¯ δ, ε), S(ϕ)) ¯ + μ(S(ϕ)) ¯ = 2H ∗ ( (ϕ, ¯ δ, ε), S(ϕ)) ¯ Downloaded by [University of California, San Diego] at 01:32 19 July 2016 Now we prove that H ∗ ( (ϕ, ¯ δ, ε), S(ϕ)) ¯ → as (δ, ε) → (0, 0) If H ∗ ( (ϕ, ¯ δ, ε), S(ϕ)) ¯ → as ¯ δn , εn ) such that (δ, ε) → (0, 0), then there are ρ > 0, (δn , εn ) → (0, 0) and xn ∈ (ϕ, d(xn , S(ϕ)) ¯ ≥ ρ, ∀n ∈ N (17) Since xn ∈ (ϕ, ¯ δn , εn ), {xn } is an approximating sequence for (LEPϕ¯ ) The sequentially generalized Tykhonov well-posedness of (LEP) at ϕ¯ gives us the existence of a subsequence of {xn } converging to some point of S(ϕ); ¯ this contradicts (17) ¯ δ, ε) with xn → x Then, (b) We first show that (ϕ, ¯ δ, ε) is closed for all δ, ε > Let xn ∈ (ϕ, for each n ∈ N, there is ϕn ∈ Bϕ¯ (δ), ϕn = (Cn , f (n) ), such that xn ∈ S(ϕn , ε), i.e (n) f1 (xn , y) ≥ 0, (n) f2 (xn , z) + ε ≥ 0, ∀y ∈ Cn , (18) ∀z ∈ Zn (xn ) (19) Since C × F is compact or finite dimensional, Bδ (ϕ) ¯ is compact Hence, we can assume that the ¯ With the similar techniques as in the proof of sequence {ϕn } converges to some point ϕ ∈ Bδ (ϕ) Theorem 4.1, we get x ∈ C and f1 (x, y) ≥ for all y ∈ C We verify that f2 (x, z) + ε ≥ for all z ∈ Z(x) Let z ∈ Z(x) be arbitrary Using the same argument as in the proof of Theorem 4.1, there (n) is a sequence {zn }, zn ∈ Zn (xn ) satisfying zn → z It follows from (19) that f2 (xn , zn ) + ε ≥ 0; this is equivalent to (n) f2 (xn , zn ) + f2 (xn , zn ) − f2 (xn , zn ) + ε ≥ (n) Since f2 is upper semicontinuous and f2 (xn , zn ) − f2 (xn , zn ) → 0, we arrive at f2 (x, z) + ε ≥ Since ϕ ∈ Bδ (ϕ), ¯ we have x ∈ (ϕ, ¯ δ, ε) Hence, (ϕ, ¯ δ, ε) is closed Since S(ϕ) ¯ = ∩δ,ε>0 (ϕ, ¯ δ, ε) and μ( (ϕ, ¯ δ, ε)) → 0, Lemma 2.2 implies the fact that S(ϕ) ¯ is compact and H( (ϕ, ¯ δ, ε), S(ϕ)) ¯ → as (δ, ε) → (0, 0) Let {xn } be an approximating sequence for (LEPϕ¯ ) corresponding to {ϕn }, where ϕn = (Cn , f (n) ) → ϕ¯ = (C, f ) Then, there exists a sequence of positive real numbers {εn } with εn → such that ¯ δn , εn ) with δn = d(ϕ, ¯ ϕn ) Then, xn ∈ S(ϕn , εn ) for all n ∈ N Consequently, xn ∈ (ϕ, d(xn , S(ϕ)) ¯ ≤ H( (ϕ, ¯ δn , εn ), S(ϕ)) ¯ → as n → ∞ Hence, there is x¯ n ∈ S(ϕ) ¯ such that d(xn , x¯ n ) → as n → ∞ The compactness of S(ϕ) ¯ ensures that ¯ Thus, the corresponding subsequence {xnk } {¯xn } has a subsequence {¯xnk } converging to x¯ ∈ S(ϕ) ¯ The of {xn } converges to x¯ Therefore, (LEP) is sequentially generalized Tykhonov well-posed at ϕ proof is complete OPTIMIZATION 17 Applications to lexicographic variational inequalities In this section, let E, E ∗ and X be the same as in Section and Ti : E → E ∗ , i = 1, 2, be a mapping We consider the following lexicographic variational inequality: (LVI) find x¯ ∈ X such that T1 (¯x ), y − x¯ , T2 (¯x ), y − x¯ ≥lex 0, ∀y ∈ X We denote S be the set of solutions for (LVI) For a number ε > 0, we use denotation S for the approximate solution set of (LVI), i.e Downloaded by [University of California, San Diego] at 01:32 19 July 2016 S(ε) := x ∈ X | T1 (¯x ), y − x¯ ≥ 0, ∀y ∈ X; T2 (x), z − x + ε ≥ 0, ∀z ∈ Z(x) , where Z(x) = {z ∈ X | T1 (x), y − x = 0} In the following, we use the notions defined in Definitions and by replacing the term ‘LEP’ with ‘LVI’ The following corollaries are derived from the corresponding results in Section Corollary 5.1: Assume that X is compact and (i) T1 is continuous and surjective; (ii) T2 is continuous Then, (LVI) is generalized Tykhonov well-posed Moreover, it is Tykhonov well-posed if S is a singleton Proof: To apply Theorem 3.2, let fi (x, y) := Ti (x), y − x for i = 1, It is clear that f1 , f2 are continuous and D2 f1 (x, y) = T1 (x), so assumptions (i) and (ii) of Theorem 3.2 are fulfilled Applying Theorem 3.2, we obtain the conclusions of Corollary 5.1 Using the techniques given in the proof for Corollary 5.1, we also establish the characterizations of the Tykhonov well-posedness for (LVI) as follows Corollary 5.2: Assume that (i) T1 is continuous and surjective; (ii) T2 is continuous Then, (LVI) is Tykhonov well-posed if and only if S(ε) = ∅, ∀ε > and lim diam (S(ε)) = ε→0 Corollary 5.3: Assume that (i) T1 is continuous and surjective; (ii) T2 is continuous Then, (LVI) is generalized Tykhonov well-posed if and only if S(ε) = ∅, ∀ε > and lim μ(S(ε)) = ε→0 For winding up this section, we study the Tykhonov well-posedness for lexicographic variational inequalities under perturbations by a sequence of approximating problems, which are expressed in (n) terms of perturbing constraints Cn ⊂ X, and Ti : X → X ∗ , i = 1, Then, they are embedded into the following family LVI(C ,(T (n) ,T (n) )) find x¯ ∈ Cn such that n (n) (n) T1 (¯x ), y − x¯ , T2 (¯x ), y − x¯ ≥lex ∀y ∈ Cn 18 L Q ANH AND T Q DUY Let C = {C ⊂ X | C is nonempty closed }, A = {(T1 , T2 ) | Ti : X → X ∗ , i = 1, 2} and M := {(C, (T1 , T2 )) | ∃¯x ∈ C s.t T1 (¯x ), y − x¯ , T2 (¯x ), y − x¯ ≥lex ∀y ∈ C} For each ϕ = (C, (T1 , T2 )), follows: Downloaded by [University of California, San Diego] at 01:32 19 July 2016 d(ϕ, ) := H(C, C ) + = (C , (B1 , B2 )) in C × A the distance between ϕ and is defined as sup (x,y)∈X×X (T1 − B1 )(x), y − x + (T2 − B2 )(x), y − x Then, (C × A, d) is a pseudo-quasi metric space We denote (LVI) := {(LVIϕ ) | ϕ ∈ C × A} For each ϕ ∈ C × A, the solution set of (LVIϕ ) is denoted by S(ϕ) For ε, δ > 0, set S(ϕ, ε) := {x ∈ C | T1 (x), y − x ≥ 0, ∀y ∈ C; T2 (x), z − x + ε ≥ 0, ∀z ∈ Z(x)}, S( , ε) (ϕ, δ, ε) := ∈Bδ (ϕ) We consider the following subset of A Ad := {(T1 , T2 ) | T1 is continuous and surjective and T2 is continuous} In the sequel, we use again Definitions and by replacing the term ‘LEP’ with ‘LVI’ We also obtain the following results derived directly from the corresponding results in Section by setting fi (x, y) := Ti (x), y − x for i = 1, Corollary 5.4: Assume that X is compact Then, (LVI) is sequentially generalized Tykhonov wellposed in M ∩ (C × Ad ) Furthermore, it is sequentially Tykhonov well-posed if its solution set is singleton Corollary 5.5: (LVI) is sequentially Tykhonov well-posed at ϕ¯ ∈ C × Ad if and only if the following condition holds (ϕ, ¯ δ, ε) = ∅, ∀δ, ε > and diam (ϕ, ¯ δ, ε) → as (δ, ε) → (0, 0) Corollary 5.6: (a) If (LVI) is sequentially generalized Tykhonov well-posed at ϕ¯ ∈ C × A, then (ϕ, ¯ δ, ε) = ∅, ∀δ, ε > and μ( (ϕ, ¯ δ, ε)) → as (δ, ε) → (0, 0) (b) (20) Assume that C ×A is compact or finite dimensional and ϕ¯ ∈ C ×Ad Then, (LVI) is sequentially generalized Tykhonov well-posed at ϕ, ¯ if condition (20) holds Applying the results obtained in Sections and to lexicographic variational inequalities, we establish sufficient conditions for the considered problems to be well-posed We not employ the assumption related to the lower semicontinuity of the solution map Z required in [28], and hence our assumptions are easier to verify and sharpen the earlier results Disclosure statement No potential conflict of interest was reported by the authors OPTIMIZATION 19 Funding This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) [grant number 101.01-2014.44] Downloaded by [University of California, San Diego] at 01:32 19 July 2016 References [1] Tikhonov AN On the stability of the functional optimization problem USSR Comput Math Math Phys 1966;6:28–33 [2] Levitin ES, Polyak BT On the 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Tykhonov well-posedness for lexicographic equilibrium problems In this section, we study well-posedness for lexicographic equilibrium problems under perturbations by a sequence of approximating problems. .. numerical methods for optimization theory This fact has been motivated and inspired many mathematicians to study the well-posedness for problems related to optimization In 1966, Tykhonov introduced... optimization problems and vector variational inequalities J Info Optim Sci 2006;27:259–270 [12] Fang YP, Hu R, Huang NJ Well-posedness for equilibrium problems and for optimization problems with equilibrium