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 Open Access
Wellposedness for lexicographic vector quasiequilibrium problems with lexicographic equilibrium constraints
 Rabian Wangkeeree^{1, 2}Email author,
 Thanatporn Bantaojai^{1} and
 Panu Yimmuang^{1}
https://doi.org/10.1186/s1366001506695
© Wangkeeree et al.; licensee Springer. 2015
 Received: 14 November 2014
 Accepted: 17 April 2015
 Published: 16 May 2015
Abstract
We consider the wellposedness for lexicographic vector equilibrium problems and optimization problems with lexicographic equilibrium constraints in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) wellposed at the reference point are established. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped.
Keywords
 lexicographic vector equilibrium problems
 optimization problems
 lexicographic equilibrium constraints
 wellposedness
1 Introduction
Equilibrium problems first considered by Blum and Oettli [1] have been playing an important role in optimization theory with many striking applications, particularly in transportation, mechanics, economics, etc. Equilibrium models incorporate many other important problems, such as optimization problems, variational inequalities, complementarity problems, saddle point/minimax problems, and fixed points. Equilibrium problems with scalar and vector objective functions have been widely studied. The crucial issue of solvability (the existence of solutions) has attracted the most considerable attention of researchers; see, e.g., [2, 3]. A relatively new but rapidly growing topic is the stability of solutions, including semicontinuity properties in the sense of Berge and Hausdorff; see, e.g., [4, 5] and the Hölder/Lipschitz continuity of solution mappings; see, e.g., [6–10].
On the other hand, wellposedness of optimizationrelated problems can be defined in two ways. The first and oldest is Hadamard wellposedness [11], which means existence, uniqueness, and continuous dependence of the optimal solution and optimal value from perturbed data. The second is Tikhonov wellposedness [12], which means the existence and uniqueness of the solution and convergence of each minimizing sequence to the solution. Wellposedness properties have been intensively studied and the two classical wellposedness notions have been extended and blended. Recently, the Tikhonov notion has been more interested. The major reason is its vital role in numerical methods. Any algorithm can generate only an approximating sequence of solutions. Hence, this sequence is applicable only if the problem under consideration is wellposed. For parametric problems, wellposedness is closely related to stability. Up to now, there have been many works dealing with wellposedness of optimizationrelated problems as mathematical programming [13, 14], constrained minimization [15, 16] variational inequalities [17–19], Nash equilibria [20], and equilibrium problems [21].
On the other hand, many papers appeared dealing with bilevel problems such as mathematical programming with equilibrium constraints [22], optimization problems with variational inequality constraints [20], optimization problems with Nash equilibrium constraints [20], optimization problems with equilibrium constraints [23, 24], etc. The increasing importance of these bilevel problems in mathematical applications in engineering and economics is recognized. For instance, the multileaderfollower game in economics is a bilevel problem, since each leader has to solve a Stackelberg game formulated as a mathematical program with equilibrium constraints. Recently, Anh et al. in [25] considered the bilevel equilibrium and optimization problems with equilibrium constraints. They proposed a relaxed level closedness and use it together with pseudocontinuity assumptions to establish sufficient conditions for the wellposedness and unique wellposedness.
With regard to vector equilibrium problems, most of the existing results correspond to the case when the order is induced by a closed convex cone in a vector space. Thus, they cannot be applied to lexicographic cones, which are neither closed nor open. These cones have been extensively investigated in the framework of vector optimization; see, e.g., [26–30]. For instance, Chadli et al. in [31] obtained conditions for the existence of solutions of a sequential equilibrium problem via a viscosity argument under quite strong conditions. Bianchi et al. in [32] analyzed lexicographic equilibrium problems on a topological Hausdorff vector space, and their relationship with some other vector equilibrium problems. They obtained the existence results for the tangled lexicographic problem via the study of a related sequential problem. However, for equilibrium problems, the main emphasis has been on the issue of solvability/existence. To the best of our knowledge, very recently, Anh et al. in [26] studied the wellposedness for lexicographic vector equilibrium problems in metric spaces and gave the sufficient conditions for a family of such problems to be wellposed and uniquely wellposed at the considered point. Furthermore, they derived several results on wellposedness for a class of variational inequalities.
Motivated by the work reported above, this paper aims to consider the lexicographic vector equilibrium problems and optimization problems with lexicographic equilibrium constraints in metric spaces and establishes necessary and/or sufficient conditions for such problems to be wellposed and uniquely wellposed at the considered point assumed always that the mentioned solutions exist.
The layout of the paper is as follows. In Section 2, we propose the lexicographic vector equilibrium problems and optimization problems with lexicographic equilibrium constraints in metric spaces under our consideration and recall notions and preliminaries needed in the sequel. In Section 3, we study the wellposedness of the lexicographic vector equilibrium problems with lexicographic equilibrium constraints in metric spaces. Section 4 is devoted to the wellposedness of optimization problems with lexicographic equilibrium constraints.
2 Preliminaries
 (\(\mathrm{LQEP}_{\lambda}\)):

finding \(\bar{x} \in K_{1}(\bar{x},\lambda)\) such that$$f(\bar{x}, y, \lambda)\geq_{l} 0 ,\quad \forall y \in K_{2} (\bar{x},\lambda). $$
Remark 2.1
 (i)
When \(f:= f_{1} : X\times X\times\Lambda\rightarrow\mathbb{R}\), the (\(\mathrm{LQEP}_{\lambda}\)) collapses to the parametric quasiequilibrium problem (QEP) considered by Anh et al. [25].
 (ii)
When \(K_{i}(\bar{x},\lambda) = K(\lambda)\), for all \(i =1,2\), that is, \(K_{i}\) does not depend on \(\bar{x}\), the (\(\mathrm{LQEP}_{\lambda}\)) reduces to the lexicographic vector equilibrium problem (\(\mathrm{LEP}_{\lambda}\)) considered by Anh et al. [26].
Instead of writing \(\{(\mathrm{LQEP}_{\lambda})  \lambda\in\Lambda \}\) for the family of lexicographic vector quasiequilibrium problem, i.e., the lexicographic parametric problem, we will simply write (LQEP) in the sequel. Let \(S_{f}:\Lambda\rightarrow2^{X}\) be the solution map of (LQEP).
 (\(\mathrm{LQEP}_{\lambda,\varepsilon}\)):

find \(\bar{x} \in K_{1}(\bar{x}, \lambda)\) such that$$f(\bar{x}, y, \lambda) + \varepsilon e \geq_{l} 0, \quad \forall y \in K_{2}(\bar{x}, \lambda), $$
 (LVQEPLEC):

finding \(\bar{\mathbf{y}} \in\operatorname{gr} S_{f}\) such that$$F(\bar{\mathbf{y}}, \mathbf{y}) \geq_{l} 0, \quad \forall\mathbf {y} \in\operatorname{gr} S_{f}, $$
 (\(\mathrm{LVQEPLEC}_{\xi}\)):

find \(\bar {\mathbf{y}} \in\operatorname{gr} S_{f}\) such that$$F(\bar{\mathbf{y}}, \mathbf{y})+\xi e \geq_{l} 0, \quad \forall \mathbf {y} \in\operatorname{gr} S_{f}. $$
 (OPLEC):

finding \(\bar{\mathbf{x}} := (\bar {x},\lambda) \in\operatorname{gr}S_{f} \) such that$$ g(\bar{\mathbf{x}}) = \min\bigl\{ g(\mathbf{x})  \mathbf{x} := (x,\lambda) \in \operatorname{gr}S_{f}\bigr\} . $$
Remark 2.2
When \(f:= f_{1} : X\times X\times\Lambda\rightarrow\mathbb {R}\), the (OPLEC) collapses to the optimization problem with equilibrium constraints (OPEC) considered by Anh et al. [25].
We next give the concept of an approximating sequence, wellposedness, and unique wellposedness for (LQEP), (LVQEPLEC), and (OPLEC).
Definition 2.3
A sequence \(\{x_{n}\}\) is an approximating sequence of (LQEP) corresponding to a sequence \(\{ \lambda_{n}\}\subset\Lambda\) converging to \(\bar{\lambda}\) if there is a sequence \(\{\varepsilon_{n}\} \subset(0,\infty)\) converging to 0 such that \(x_{n} \in\tilde{S}_{f}(\lambda_{n}, \varepsilon_{n})\) for all n.
Definition 2.4
 (i)
\(F(\mathbf{x}_{n},\mathbf{y}) + \varepsilon_{n}e \geq_{l} 0\), for all \(\mathbf{y}:= (y, \lambda) \in S_{f}(\lambda) \times\Lambda\);
 (ii)
\(\{ x_{n}\}\) is an approximating sequence for (LQEP) corresponding to \(\{ \lambda_{n} \}\).
Definition 2.5
 (i)
\(g(\mathbf{x}_{n}) \leq g(\mathbf{y}) + \varepsilon_{n}\), for all \(\mathbf{y}:= (y, \lambda) \in S_{f}(\lambda) \times\Lambda\);
 (ii)
\(\{x_{n}\}\) is an approximating sequence for (LQEP) corresponding to \(\{\lambda_{n}\}\).
Definition 2.6
 (i)
it has solutions;
 (ii)
for any approximating sequence \(\{ \mathbf{x}_{n} \}:= \{(x_{n}, \lambda_{n})\}\) for (LVQEPLEC), where \(\lambda_{n} \rightarrow\bar {\lambda}\), has a subsequence converging to a solution.
Definition 2.7
 (i)
it has a unique solution \(\bar{\mathbf{x}} := (\bar{x}, \bar {\lambda})\);
 (ii)
every approximating sequence \(\{ \mathbf{x}_{n} \}:= \{(x_{n}, \lambda_{n})\}\) for (LVQEPLEC) or (OPLEC), where \(\lambda_{n} \rightarrow \bar{\lambda}\), converges to \(\bar{\mathbf{x}}\).
Now we recall the continuitylike properties which will be used for our analysis.
Definition 2.8
[34]
 (i)
Q is upper semicontinuous (usc) at \(\bar{x}\) if, for any open set \(U\supseteq Q(\bar{x})\), there is a neighborhood N of \(\bar{x}\) such that \(Q(N)\subseteq U\).
 (ii)
Q is lower semicontinuous (lsc) at \(\bar{x}\) if, for any open subset U of Y with \(Q(\bar{x})\cap U\neq\emptyset\), there is a neighborhood N of \(\bar{x}\) such that \(Q(x)\cap U \neq \emptyset\) for all \(x \in N\).
 (iii)
Q is closed at \(\bar{x}\) if, for any sequences \(\{x_{k}\} \) and \(\{y_{k}\}\) with \(x_{k}\rightarrow\bar{x}\) and \(y_{k} \rightarrow\bar{y}\) and \(y_{k} \in Q(x_{k})\), we have \(\bar{y} \in Q(\bar{x})\).
Lemma 2.9
 (i)
If Q is usc at \(\bar{x}\) and \(Q(\bar{x})\) is compact, then for any sequence \(\{x_{n}\}\) converging to \(\bar{x}\), every sequence \(\{y_{n}\}\) with \(y_{n} \in Q(x_{n})\) has a subsequence converging to some point in \(Q(\bar{x})\). If, in addition, \(Q(\bar{x}) = \{\bar{y}\}\) is a singleton, then such a sequence \(\{y_{n}\}\) must converge to \(\bar{y}\).
 (ii)
Q is lsc at \(\bar{x}\) if and only if, for any sequence \(\{ x_{n}\}\) with \(x_{n}\rightarrow\bar{x}\) and any point \(y \in Q(\bar{x})\), there is a sequence \(\{y_{n}\}\) with \(y_{n} \in Q(x_{n})\) converging to y.
Definition 2.10
 (i)g is upper εlevel closed at \(\bar{x} \in X\) if, for any sequence \(\{x_{n}\}\) satisfying\(g(\bar{x})\geq\varepsilon\).$$x_{n}\rightarrow\bar{x} \quad \text{and} \quad g(x_{n})\geq \varepsilon\quad \text{for all } n, $$
 (ii)g is strongly upper εlevel closed at \(\bar {x} \in X\) if, for any sequences \(\{x_{n}\}\) in X and \(\{r_{n}\}\subset [0,\infty)\) satisfying\(g(\bar{x}) \geq\varepsilon\).$$x_{n}\rightarrow\bar{x},\qquad r_{n} \rightarrow0 \quad \text{and}\quad g(x_{n}) + r_{n} \geq\varepsilon \quad \text{for all } n, $$
Definition 2.11
 (i)f is called upper pseudocontinuous at \(x_{0} \in X\) iff for any point x and sequence \(\{x_{n}\} \) in X such that\(\limsup_{n\rightarrow\infty} f(x_{n}) < f(x)\).$$f(x_{0}) < f(x)\quad \text{and}\quad x_{n} \rightarrow x_{0}, $$
 (ii)f is called lower pseudocontinuous at \(x_{0} \in X\) iff for any point x and sequence \(\{x_{n}\} \) in X such that\(f(x) < \liminf_{n\rightarrow\infty} f(x_{n}) \).$$f(x) < f(x_{0})\quad \text{and}\quad x_{n} \rightarrow x_{0}, $$
 (iii)
f is termed pseudocontinuous at \(x_{0} \in X\) iff it is both lower and upper pseudocontinuous at this point.
Remark 2.12
The class of the upper pseudocontinuous functions strictly contains that of the usc functions; see [16].
3 Lexicographic vector equilibrium problems with lexicographic equilibrium constraints (LVQEPLEC)
In this section, we shall establish necessary and/or sufficient conditions for (LVQEPLEC) to be (uniquely) wellposed at the reference point \(\bar{\lambda} \in \Lambda\). To simplify the presentation, in the sequel, the results will be formulated for the case \(n=2\).
 (\(\mathrm{LQEP}_{\lambda,\varepsilon}\)):

find \(\bar{x} \in K_{1}(\bar{x},\lambda)\) such that$$ \left \{ \begin{array}{l} f_{1} (\bar{x}, y, \lambda)\geq0, \quad \forall y \in K_{2}(\bar{x},\lambda); \\ f_{2}(\bar{x},z,\lambda)+\varepsilon\geq0 ,\quad \forall z\in Z_{f}(\bar{x},\lambda). \end{array} \right . $$
 (\(\mathrm{LVQEPLEC}_{\xi}\)):

find \(\bar{\mathbf{y}} \in\operatorname{gr}S_{f}\) such that$$ \left \{ \begin{array}{l} F_{1} (\bar{\mathbf{y}}, \mathbf{y})\geq0, \quad \forall \mathbf{y} \in\operatorname{gr}S_{f}; \\ F_{2}(\bar{\mathbf{y}},\mathbf{y}')+\varepsilon\geq0, \quad \forall\mathbf{y}' \in Z_{F}(\bar{\mathbf{y}}). \end{array} \right . $$
Lemma 3.1
Let \(\{x_{n}\}\) converging to \(\bar{x} \in Z_{1,f}(\bar{\lambda})\) be an approximating sequence of (\(\mathrm{LQEP}_{\bar{\lambda}}\)) corresponding to a sequence \(\lambda _{n}\rightarrow\bar{\lambda}\) and assume that \(Z_{f}\) is lsc at \((\bar{x},\bar{\lambda})\) and \(f_{2}\) is strongly upper 0level closed on \(\{\bar{x}\} \times Z_{f}(\bar{x},\bar{\lambda})\times\{\bar {\lambda}\}\). Then \(\bar{x} \in S_{f}(\bar{\lambda})\).
Proof
Theorem 3.2
 (i)
in \(X \times\Lambda\), \(K_{1}\) is closed and \(K_{2}\) is lsc;
 (ii)
\(Z_{f}\) is lsc on \(Z_{1,f}(\bar{\lambda}) \times\{\bar{\lambda }\}\);
 (iii)
\(f_{1}\) is upper 0level closed on \(K_{1}(\bar{x},\bar{\lambda}) \times K_{2}(\bar{x}, \bar{\lambda}) \times\{\bar{\lambda}\}\);
 (iv)
\(f_{2}\) is strongly upper 0level closed on \(K_{1}(\bar{x},\bar {\lambda}) \times K_{2}(\bar{x}, \bar{\lambda}) \times\{\bar {\lambda}\}\);
 (v)
\(F_{1}(\cdot,\mathbf{y})\) is upper 0level closed at \((\bar {x},\bar{\lambda})\), for all \(\mathbf{y} \in X \times\Lambda\);
 (vi)
\(F_{2}(\cdot,\mathbf{y})\) is strongly upper 0level closed at \((\bar{x}, \bar{\lambda})\), for all \(\mathbf{y} \in X \times \Lambda\).
Proof
Step III: We have to prove that \(\tilde{S}_{f}(\bar {\lambda}, 0)\) is compact by checking its closedness. Take an arbitrary sequence \(\{ x_{n}\}\) in \(S(\bar{\lambda})= \tilde{S}_{f}(\bar{\lambda}, 0)\) converging to \(\bar{x}\). Setting \(\lambda_{n} := \bar{\lambda}\) for all n, we have \(\lambda_{n} \rightarrow\bar{\lambda}\) and \(x_{n}\in Z_{1,f}(\lambda_{n})\) for all n. This together with the closedness of \(Z_{1,f}\) at \(\bar{\lambda}\) implies that \(\bar{x} \in Z_{1,f}(\bar {\lambda})\). Note that \(\{x_{n}\}\) is, of course, an approximating sequence of (\(\mathrm{LQEP}_{\bar{\lambda}}\)) corresponding to \(\{\lambda_{n}\} \). Then Lemma 3.1 again implies that \(\bar{x} \in S_{f}(\bar {\lambda})= \tilde{S}_{f}(\bar{\lambda}, 0)\), and hence \(S_{f}(\bar {\lambda})\) is compact; that is, \(\tilde{S}_{f}(\bar{\lambda}, 0)\) is compact.
Furthermore, suppose that \(S_{f} : \Lambda\rightarrow X\) is singlevalued and (LVQEPLEC) admits a unique solution \(\bar{\mathbf{x}}\). We have to show that (LVQEPLEC) is uniquely wellposed. Let \(\{\mathbf {x}_{n}\}\) be an approximating sequence for (LVQEPLEC). By the same argument as in the preceding part, there is a subsequence converging to \(\bar{\mathbf{x}}\). If \(\{ \mathbf{x}_{n}\}\) did not converge to \(\bar{\mathbf{x}}\), there would be an open set U containing \(\bar{\mathbf{x}}\) such that some subsequence was outside U. By the above argument, this subsequence has a subsequence convergent to \(\bar {\mathbf{x}}\), an impossibility. □
The following examples show that none of the assumptions in Theorem 3.2 can be dropped.
Example 3.3
(The compactness of X cannot be dropped)
Example 3.4
(The closedness of \(K_{1}\) is essential)
Therefore, (LVQEPLEC) is not wellposed. Indeed, let \(x_{n}=\frac {1}{n}\), \(\lambda_{n}=\frac{1}{n}\) for all \(n \in\mathbb{N}\). Then \(\mathbf{x}_{n}:=(x_{n}, \lambda_{n})\) is a solution of (LVQEPLEC) and \(\{\mathbf{x}_{n}\}\) converges to \(\mathbf {x}:=(0,0)\). But x does not belong to the solution set of (LVQEPLEC).
Example 3.5
(The lower semicontinuity of \(K_{2}\) cannot be dispensed)
Example 3.6
(The lower semicontinuity of \(Z_{f}\) cannot be dropped)
Example 3.7
(Upper 0level closedness of \(f_{1}\))
Example 3.8
(Strong upper 0level closedness of \(f_{2}\))
Example 3.9
(Upper 0level closedness of \(F_{1}\))
Example 3.10
(Strong upper 0level closedness of \(F_{2}\))
Theorem 3.11
 (i)
If (LVQEPLEC) is uniquely wellposed at \(\bar{\lambda}\), then \(\operatorname{diam} \Gamma(\xi,\varepsilon) \downarrow0\) as \((\xi,\varepsilon)\downarrow(0,0)\).
 (ii)
Conversely, suppose that X and Λ are complete, assumptions (i)(vi) in Theorem 3.2 hold and \(\operatorname{diam} \Gamma(\xi,\varepsilon) \downarrow0\) as \(\xi\downarrow0\) and \(\varepsilon\downarrow0\). Then (LVQEPLEC) is uniquely wellposed at \(\bar{\lambda}\).
Proof
To weaken the assumption of unique wellposedness in Theorem 3.11, we are going to use the notions of measures of noncompactness in a metric space X. We recall that a subset A of a metric space X is εdiscrete iff \(d(x, y) \geq\varepsilon\) for all \(x,y \in A\) with \(x \neq y\).
Definition 3.12
 (i)The Kuratowski measure of M is$$ \mu(M)=\inf \Biggl\{ \varepsilon>0 \Big M\subseteq\bigcup _{k=1}^{n} M_{k} \text{ and } \operatorname{diam} M_{k} \leq\varepsilon, k=1,\ldots ,n, \exists n \in \mathbb{N} \Biggr\} . $$
 (ii)The Hausdorff measure of M is$$ \eta(M)=\inf \Biggl\{ \varepsilon>0 \Big M\subseteq\bigcup _{k=1}^{n} B(x_{k},\varepsilon), x_{k} \in X \text{ for some } n \in\mathbb {N} \Biggr\} . $$
 (iii)The Istrǎtescu measure of M is$$ \iota(M)=\inf \{ \varepsilon>0  M \text{ have no infinite } \varepsilon \text{discrete subset} \}. $$
 (i)
\(\gamma(M) = +\infty\) if and only if the set M is unbounded;
 (ii)
\(\gamma(M) = \gamma(\operatorname{cl}M)\);
 (iii)
from \(\gamma(M) = 0\) it follows that M is a totally bounded set;
 (iv)if X is a complete space and if \(\{A_{n}\}\) is a sequence of closed subsets of X such that \(A_{n+1} \subseteq A_{n}\) for each \(n \in \mathbb{N}\) and \(\lim_{n \rightarrow+\infty}\gamma(A_{n}) = 0\), then \(K := \bigcap_{n \in\mathbb{N}}A_{n}\) is a nonempty compact set andwhere H is the Hausdorff metric;$$\lim_{n \rightarrow+\infty}H(A_{n},K) = 0, $$
 (v)
from \(M \subseteq N\) it follows that \(\gamma(M) \leq\gamma(N)\).
Theorem 3.13
 (i)
If (LVQEPLEC) is wellposed at \(\bar{\lambda}\), then \(\gamma (\Gamma(\xi,\varepsilon))\downarrow0\) as \(\xi\downarrow0\) and \(\varepsilon\downarrow0\).
 (ii)Conversely, suppose that \(\gamma(\Gamma(\xi,\varepsilon ))\downarrow0\) as \(\xi\downarrow0\) and \(\varepsilon\downarrow0\), and the following conditions hold:
 (a)
X and Λ are complete;
 (b)
\(K_{1}\) is closed and \(K_{2}\) is lsc at \((\bar{x},\bar{\lambda})\);
 (c)
\(Z_{f}\) is lsc on \((X \times\Lambda) \cap\operatorname{gr}Z_{1,f}\);
 (d)
\(f_{1}\) is upper 0level closed on \(K_{1}(\bar{x},\bar{\lambda }) \times K_{2}(\bar{x},\bar{\lambda})\times\{\bar{\lambda} \}\);
 (e)
\(f_{2}\) is upper alevel closed on \(K_{1}(\bar{x},\bar{\lambda }) \times K_{2}(\bar{x},\bar{\lambda})\times\{\bar{\lambda}\}\) and \(a<0\);
 (f)
\(F_{1}(\cdot, \mathbf{y})\) is upper 0level closed at \((\bar{x}, \bar{\lambda})\), for all \(\mathbf{y} \in X\times\Lambda\);
 (g)
\(F_{2}(\cdot, \mathbf{y})\) is upper blevel closed at \((\bar {x},\bar{\lambda})\), for all \(\mathbf{y} \in X\times\Lambda\) and \(b <0\).
 (a)
Proof
By (3.4) the proof is similar for the three mentioned measures of noncompactness. We discuss only the case \(\gamma=\mu\), the Kuratowski measure.
The following examples show that all assumptions of Theorem 3.13(ii) are essential.
Example 3.14
(The closedness of \(K_{1}\) is essential)
Let X, Λ, \(K_{1}\), \(K_{2}\), f, and F be as in Example 3.4. It is easy to check that \(Z_{f}\) is lsc and X is complete and \(K_{2}\) is lsc in \(X \times\Lambda\). Assumptions (ii)(c)(ii)(f) are fulfilled since f and F are continuous in \(X\times X \times\Lambda \) and \((X\times\Lambda) \times(X,\Lambda)\), respectively. Moreover, \(\Gamma(\xi, \varepsilon) \subseteq[1, 1] \times[0, 1]\), and hence \(\gamma(\Gamma(\xi,\varepsilon)) \leq\gamma([1, 1] \times [0, 1]) = 0\). It is easy to see that the solution set of (LVQEPLEC) coincides with \(\operatorname{gr} S_{f}\). But \(S_{f}(\lambda)=(0,1] \) for all \(\lambda\in[0,1]\), i.e., \(\operatorname{gr} S_{f}=\{ (x, \lambda) x \in(0,1], \lambda\in [0,1]\}\). With the same arguments as in Example 3.4, (LVQEPLEC) is not wellposed. The reason is that \(K_{1}\) is not closed at \((0, 0)\).
Example 3.15
Example 3.16
(The lower semicontinuity of \(Z_{f}\) cannot be dropped)
Example 3.17
(Upper 0level closedness of \(f_{1}\))
Finally, we show that assumption (iii) is not satisfied. Indeed, take \(\{x_{n}\}\) and \(\{\lambda_{n}\}\) as above and \(y_{n} = 1\), we have \((x_{n}, y_{n}, \lambda_{n}) \rightarrow(0,1,0)\) and \(f_{1}(x_{n},y_{n},\lambda_{n}) = 1 > 0\) for all n, while \(f_{1}(0,1,0) = 1 < 0\).
Example 3.18
(Strong upper 0level closedness of \(f_{2}\))
Finally, we show that assumption (iv) is not satisfied. Indeed, take sequences \(x_{n} = 0\), \(y_{n} = 1\), \(\lambda_{n} = \frac{1}{n} \), and \(\varepsilon_{n} = \frac{1}{n}\), we have \(\{ (x_{n}, y_{n}, \lambda_{n}, \varepsilon_{n})\}\) and \(f_{2}(x_{n}, y_{n}, \lambda_{n})+\varepsilon_{n} > 0\) for all n, while \(f_{2}(0,1,0) = 1 < 0\).
Example 3.19
(Upper 0level closedness of \(F_{1}\))
Let X, Λ, \(K_{1}\), \(K_{2}\), f, and F be as in Example 3.9. Then assumptions (i)(vi) and (vi) are satisfied. Moreover, \(\Gamma (\xi,\varepsilon)\subseteq[0,1] \times[0, 1]\), and hence \(\gamma(\Gamma(\xi,\varepsilon)) = 0\). We have \(\operatorname{gr} S_{f}:= [0,1] \), \(\lambda\in[0,1]\). The solution set of (LVQEPLEC) is \((1,0)\cup\{ (x,\lambda) x=0,1, \lambda\in(0,1]\}\). We can conclude that all the assumptions of Theorem 3.2 except (vii) are satisfied. Therefore, (LVQEPLEC) is not wellposed. Indeed, let \(x_{n}=0\), \(\lambda_{n}=\frac{1}{n}\). We see that \(\mathbf {x}_{n} =(x_{n}, \lambda_{n})\) is a solution of (LVQEPLEC) and \(\mathbf{x}_{n}\) converges to \(\mathbf {x}=(0,0)\). But x does not belong to the solution set of (LVQEPLEC).
Example 3.20
(Strong upper 0level closedness of \(F_{2}\))
4 Optimization problem with lexicographic equilibrium constraints (OPLEC)
We prove first a sufficient condition for the wellposedness in topological settings.
Theorem 4.1
 (i)
in \(X \times\Lambda\), \(K_{1}\) is closed and \(K_{2}\) is lsc;
 (ii)
\(Z_{f}\) is lsc on \(Z_{1,f}(\bar{\lambda}) \times\{\bar{\lambda }\} \);
 (iii)
\(f_{1}\) is upper 0level closed on \(K_{1}(\bar{x},\bar{\lambda}) \times K_{2}(\bar{x}, \bar{\lambda}) \times\{\bar{\lambda}\}\);
 (iv)
\(f_{2}\) is strongly upper 0level closed at \(K_{1}(\bar{x},\bar {\lambda}) \times K_{2}(\bar{x}, \bar{\lambda}) \times\{\lambda\}\);
 (v)
g is lower pseudocontinuous in \((x,\bar{\lambda})\).
Proof
Theorem 4.2
 (i)
If (OPLEC) is uniquely wellposed, then \(\operatorname{diam}M(\xi, \varepsilon) \downarrow0\) as \((\xi, \varepsilon) \downarrow(0, 0)\).
 (ii)Conversely, assume that \(\operatorname{diam}M(\xi, \varepsilon) \downarrow0\) as \((\xi, \varepsilon) \downarrow (0, 0)\), and that the following conditions hold:
 (a)
X and Λ are complete;
 (b)
\(K_{1}\) is closed and \(K_{2}\) is lsc at \((\bar{x}, \bar{\lambda})\);
 (c)
\(Z_{f}\) is lsc on \(Z_{1,f}(\bar{\lambda}) \times\bar{\lambda}\);
 (d)
\(f_{1}\) is upper 0level closed at \(K_{1}(\bar{x},\bar{\lambda}) \times K_{2}(\bar{x}, \bar{\lambda}) \times\{\bar{\lambda}\}\);
 (e)
\(f_{2}\) is strongly upper 0level closed on \(K_{1}(\bar{x},\bar {\lambda}) \times K_{2}(\bar{x}, \bar{\lambda}) \times\{\bar {\lambda}\}\);
 (f)
g be lower pseudocontinuous at \((\bar{x}, \bar{\lambda})\).
 (a)
Proof
For the wellposedness of (OPLEC) in terms of measures of noncompactness we have the following result. Let us consider only the case of the Hausdorff measure η; we get the corresponding results for the case μ and ι.
Theorem 4.3
 (i)
If (OPLEC) is wellposed at \(\bar{\lambda}\), then \(\eta(M(\xi ,\varepsilon)) \downarrow0\) as \((\xi, \varepsilon) \downarrow(0, 0)\).
 (ii)Conversely, suppose that \(\eta(M(\xi, \varepsilon)) \downarrow 0\) as \((\xi, \varepsilon) \downarrow(0, 0)\), and the following conditions hold:
 (a)
X and Λ are complete;
 (b)
\(K_{1}\) is closed and \(K_{2}\) is lsc on \(X \times\Lambda\);
 (c)
\(Z_{f}\) is lsc on \(Z_{1,f}(\bar{\lambda}) \times\bar{\lambda}\);
 (d)
\(f_{1}\) is upper bupper level closed in \(K_{1}(X,\Lambda) \times K_{2}(X,\Lambda) \times\Lambda\), for all \(b <0\);
 (e)
\(f_{2}\) is strongly upper bupper level closed in \(K_{1}(X,\Lambda ) \times K_{2}(X,\Lambda) \times\Lambda\), for all \(b <0\);
 (f)
g is lsc in \(X \times\Lambda\).
 (a)
Proof
5 Conclusions
In this paper, we obtain the wellposedness for lexicographic vector equilibrium problems and optimization problems with lexicographic equilibrium constraints in metric spaces. Sufficient conditions for a family of such problems to be (uniquely) wellposed at the reference point are established. Numerous examples are provided to explain that all the assumptions we impose are very relaxed and cannot be dropped. The results presented in this paper extend and improve some known results.
Declarations
Acknowledgements
This work was partially supported by the Thailand Research Fund, Grant No. PHD/0035/2553 and Naresuan University. The authors would like to thank the referees for their remarks and suggestions, which helped to improve the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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