 Research
 Open access
 Published:
Wellposed symmetric vector quasiequilibrium problems
Journal of Inequalities and Applications volumeÂ 2015, ArticleÂ number:Â 232 (2015)
Abstract
In this paper, we consider wellposedness of symmetric vector quasiequilibrium problems. Based on a nonlinear scalarization technique, we first establish the bounded rationality model M for symmetric vector quasiequilibrium problems, and then introduce a wellposedness concept for symmetric vector quasiequilibrium problems, which unifies its Hadamard and Tykhonov wellposedness. Finally, sufficient conditions on the wellposedness for symmetric vector quasiequilibrium problems are given.
1 Introduction
In 2003, Fu [1] introduced the symmetric vector quasiequilibrium problem (for short, SVQEP) which is a generalization of equilibrium problem proposed by Blum and Oettli [2] and gave an existence theorem for a weak Pareto solution for (SVQEP). It provides a very general model for a wide range of problems, for example, the vector optimization problem, the vector variational inequality problem, the vector complementarity problem and the vector saddle point problem. In 2006, Farajzadeh [3] considered existence theorem of the solution of (SVQEP) in the Hausdorff topological vector space. In 2008, Chen and Gong [4] studied the stability of the set of solutions for (SVQEP), proved a generic stability theorem and gave an existence theorem for essentially connected components of the set of solutions for (SVQEP). In 2012, Zhang [5] introduced the notion of a generalized LevitinPolyak wellposedness and gave sufficient conditions of the generalized LevitinPolyak wellposedness for (SVQEP). Recently, by using the same roadmap as Deng and Xiang [6], Zhang et al. [7] introduce and study wellposedness in connection with (SVQEP), which unifies its Hadamard and LevitinPolyak wellposedness.
As is well known, the notion of wellposedness can be divided into two different groups: Hadamard type and Tykhonov type [8, 9]. Roughly speaking, Hadamard type wellposedness is based on the continuous dependence of the optimal solution from the data of the considered optimization problem. Tykhonov types wellposedness such as Tikhonov and LevitinPolyak wellposedness deal with the behavior of a prescribed class of sequence of approximate solutions. Two kinds of wellposedness have been generalized to various problems related to vector optimization, e.g., vector optimization problems [10â€“15], vector variational inequality problems [10, 16], and vector equilibrium problems [5â€“7, 17]. Among many approaches for dealing with Tykhonov types wellposedness for vector optimization problems, vector variational inequality problems, and vector equilibrium problems, the nonlinear scalarization technique is of considerable interest. On the other hand, almost all the literature deals with directly specific notions of wellposedness, especially Tykhonov types of wellposedness, while some researchers have investigated a unified approaches to two different types of wellposedness. For one thing, the notion of extended wellposedness for vector optimization problems has been investigated in [12]. In some sense this notion unifies the ideas of Tykhonov and Hadamard wellposedness, allowing perturbations of the objective function and the feasible set. For another, wellposedness under perturbations (called also parametric wellposedness) for vector equilibrium problems has also been investigated in [18]. This kind of wellposedness is a blending of Hadamard and Tikhonov notions, and it gives also links to stability theory and seems well adapted to describe the behaviors of solutions under perturbations.
In this paper, we will introduce a wellposedness concept for (SVQEP), which unifies its Hadamard and Tykhonov wellposedness. The distinguishing feature of our work lies in the use of the scalarization technique and the bounded rationality model M (see [19â€“22]) to establish wellposedness results of (SVQEP). It is worthy that our research method is different from extended wellposedness and parametric wellposedness. Finally, by using the conditions of the existence theorem of the solutions to (SVQEP) (see [1]), we obtain sufficient conditions of the wellposedness for (SVQEP).
2 Preliminaries
Throughout this paper, unless otherwise specified, let C and D be a compact metric space supplied with distance \(d_{1}\), \(d_{2} \), respectively. Let \((Z,\\cdot\)\) be a Banach space and P be a nonempty, closed, convex, and pointed cone in Z with apex at the origin and \(\operatorname{int}P\neq\emptyset\).
Let \(S :C\times D\rightrightarrows C\) and \(T : C\times D\rightrightarrows D\) be two setvalued mappings and \(F , G :C \times D\rightarrow Z\) be two vectorvalued mappings. Fu [1] defined a class of symmetric vector quasiequilibrium problems (for short, SVQEP), which consist in finding \((x,y)\in C\times D\) such that
and
Now we introduce the notion of Tykhonov approximating solution sequence for (SVQEP).
Definition 2.1
A sequence \(\{(x_{n},y_{n})\}\in C\times D\) is called a Tykhonov approximating solution sequence for (SVQEP) if there exists \(\{\epsilon_{n}\}\subset\mathbb{R}_{+}^{1}\) with \(\epsilon_{n}\rightarrow 0\) such that
and
Next, we introduce a nonlinear scalarization function and their related properties. For any fixed \(e\in \operatorname{int}P\), the nonlinear scalarization function is defined by
It is well known from [23â€“25] that \(\xi_{e}\) is continuous, homogeneous, (strictly) monotone (i.e., \(\xi _{e}(z_{1})\leq \xi_{e}(z_{2})\) if \(z_{1}z_{2}\in P\) and \(\xi_{e}(z_{1}) < \xi_{e}(z_{2})\) if \(z_{2}z_{1}\in \operatorname{int}P\)) and convex. For any fixed \(e\in \operatorname{int}P\), \(z\in Z\), and \(r\in \mathbb{R}\), then \(\xi_{e}(z)\geq r\) is equivalent to \(z \notin re\operatorname{int}P\).
Remark 2.1
Note that the nonlinear scalarization function \(\xi _{e}\) is not strongly monotone (see [24]). It is for the reason that the function \(\xi_{e}\) is more useful in dealing with weakly efficient points.
Finally, we recall some useful definitions and lemmas.
Let \((X, d)\) be a metric space. Denote a family of all nonempty compact subsets of X by \(K(X)\). For any \(A,B\in K(X)\), let
denote the Hausdorff metric on \(K(X)\). It is well known that \((K(X),h)\) is complete if and only if \((X,d)\) is complete.
Definition 2.2
(see [26])
Let \(F:X \rightrightarrows Y\) be a setvalued mapping.

1.
F is said to be upper semicontinuous at \(x\in X\) if for any open set \(U\supset F(x)\), there is an open neighborhood \(O(x)\) of x such that \(U \supset F(x')\) for each \(x'\in O(x)\);

2.
F is said to be lower semicontinuous at x if for any open set \(U\cap F(x)\neq\emptyset\), there is an open neighborhood \(O(x)\) of x such that \(U\cap F(x')\neq\emptyset\), for each \(x'\in O(x)\);

3.
F is said to be an usco mapping if F is upper semicontinuous and \(F(x)\) is nonempty compact for each \(x\in X\);

4.
F is said to be closed if \(\operatorname{Graph}(F)\) is closed, where \(\operatorname{Graph}(F)=\{(x,y)\in X\times Y: x\in X, y\in F(x)\}\) is the graph of F.
Lemma 2.1
(see [26])
If \(F:X\rightrightarrows Y\) is closed and Y is compact, then F is upper semicontinuous at each \(x\in X\).
3 A unified approach to notions of wellposedness for (SVQEP)
Let Î› be the collection of all problem \({\lambda}=(S,T,F,G)\) such that

(a)
\(S:C\times D\rightrightarrows C\) and \(T:C\times D\rightrightarrows D\) are continuous with nonempty compact values;

(b)
\(F, G:C\times D\rightarrow Z\) are continuous;

(c)
\(\sup_{(x,y)\in C\times D}\F(x,y)\<+\infty\) and \(\sup_{(x,y)\in C\times D}\G(x,y)\<+\infty\);

(d)
there exists \((x,y)\in C\times D\) meets (1), (2), and (3).
For any \(\lambda_{1}=(S_{1},T_{1},F_{1},G_{1})\), \(\lambda_{2}=(S_{2},T_{2},F_{2},G_{2})\in{ \varLambda }\), we define
where \(h_{1}\), \(h_{2}\) are Hausdorff metrics on \(K(C)\) and \(K(D)\), respectively. By PropositionÂ 3.1 in [4], it is easy to prove that \(({\varLambda },\rho)\) is a complete metric space.
Let \(X^{*}=C\times D\), \(x^{*}=(x,y)\), and \(d=\max\{d_{1},d_{2}\}\). The bounded rationality model \(M=\{{ \varLambda },X^{*},f,{\varPhi }\}\) for (SVQEP) corresponding to \(\lambda\in{ \varLambda }\) is defined as follows:

(i)
\(({ \varLambda },\rho)\) is a metric space and \((X^{*},d)\) is a compact metric space;

(ii)
the feasible set of Î» is defined by
$$f(\lambda):=\bigl\{ x^{*}\in X^{*}:x^{*}\in(S\times T) \bigl(x^{*}\bigr)\bigr\} ; $$ 
(iii)
the solution set of Î» is defined by
$$E(\lambda):=\bigl\{ x^{*}\in X^{*}: x^{*} \mbox{ meets (\ref{eq:1}), (\ref{eq:2}) and (\ref{eq:3})}\bigr\} ; $$ 
(iv)
the rationality function of Î» is defined by
$$\begin{aligned}& {\varPhi }\bigl(\lambda,x^{*}\bigr) \\& \quad :=\operatorname{Max} \Bigl\{ \sup_{u\in S(x,y)}\bigl\{  \xi_{e}\bigl(F(u,y)F(x,y)\bigr)\bigr\} ,\sup_{v\in T(x,y)} \bigl\{ \xi_{e}\bigl(G(x,v)G(x,y)\bigr)\bigr\} \Bigr\} . \end{aligned}$$
Remark 3.1
In (iv), a nonlinear scalarization function \(\xi_{e}\) is applied to reduce (SVQEP) to a scalar optimization problem since (SVQEP) does not possess linearity and convexity. Referring to [25], if one needs to solve exactly one representation to catch all the solution of (SQVEP), then the nonlinear scalarization technique is feasible.
Example 3.1
Let \(C=D=[0,2]\), \(P=\mathbb{R}_{+}\), and \(e=1\). For any \((x,y)\in C\times D\), assume that
and, for any \((u,v)\in C\times D\),
Then it is easy to see that

1.
If \(x^{*}\in f(\lambda)\), then \(x\in[0,1]\) and \(y\in[0,1]\). Obviously, \({\varPhi }(\lambda,x^{*})\geq0\).

2.
For any \(\lambda\in \varLambda \), one has
$$E(\lambda)=\bigl\{ (x,y):x=1,y=1\bigr\} \neq\emptyset. $$ 
3.
It is easy to check that \((x,y)\in E(\lambda)\) if and only if \({\varPhi }(\lambda,x^{*})=0\). Moreover, taking \(x=0\), \(y=\frac{1}{2}\), then \((x,y)=(0,\frac{1}{2})\notin E(\lambda)\) and \({\varPhi }(\lambda,x^{*})=1\neq0\).
Lemma 3.1

1.
\(\forall\lambda\in \varLambda \), \(E(\lambda)\neq\emptyset\) and \(\forall x^{*}\in f(\lambda)\), \({\varPhi }(\lambda,x^{*})\geq0\).

2.
\(\forall\lambda\in \varLambda \), \({\varPhi }(\lambda,x^{*})\leq \epsilon\) if and only if \(x^{*}\) meets (5) and (6).

3.
\(x^{*}\in E(\lambda)\) if and only if \({\varPhi }(\lambda,x^{*})= 0\).
Proof
1. By the definition of Î›, \(\forall \lambda\in \varLambda \), \(E(\lambda)\neq\emptyset\). If \(x^{*}=(x,y)\in f(\lambda )\), then \(x\in S(x,y)\), \(y\in T(x,y)\) and
2. If \(x^{*}\) meets (5) and (6), then
and
Thus, we have
Conversely, if \({\varPhi }(\lambda,x^{*})\leq\epsilon\), then we get
and
It follows that
and
Hence, \(x^{*}\) meets (5) and (6).
3. Using the above results, this result can be obtained.â€ƒâ–¡
Let \(x^{*}_{n}=(x_{n},y_{n})\) and \(x^{*}_{n_{k}}=(x_{n_{k}},y_{n_{k}})\). By LemmaÂ 3.1, we get some new representations on (1)(6) as follows:
Therefore, the set of solutions for the problem \(\lambda\in{ \varLambda }\) and \(\lambda_{n}\in{ \varLambda }\) (\(n=1,2,3,\ldots\)) is defined as
The Tykhonov approximating solution set for the problem \(\lambda\in {\varLambda }\) and \(\lambda_{n}\in{ \varLambda }\) (\(n=1,2,3,\ldots\)) is defined as
Tykhonov wellposedness for (SVQEP) corresponding to the problem Î» is given as follows.
Definition 3.1

1.
If \(\forall x^{*}_{n}\in E(\lambda,\epsilon _{n})\), \(\epsilon_{n}>0\) with \(\epsilon_{n}\rightarrow0\), there must exist a subsequence \(\{x^{*}_{n_{k}}\}\subset\{x^{*}_{n}\}\) such that \(x^{*}_{n_{k}}\rightarrow x^{*}\in E(\lambda)\), then the problem Î» is said to be generalized Tykhonov wellposed (for short GTwp).

2.
If \(E(\lambda)=\{x^{*}\}\) (a singleton), \(\forall x^{*}_{n}\in E(\lambda,\epsilon_{n})\), \(\epsilon_{n}>0\) with \(\epsilon_{n}\rightarrow 0\), we must have \(x^{*}_{n}\rightarrow x^{*}\), then the problem Î» is said to be Tykhonov wellposed (for short Twp).
Referring to [9], Hadamard wellposedness for (SVQEP) corresponding to the problem Î» is defined as follows.
Definition 3.2

1.
If \(\forall\lambda_{n}\in \varLambda \), \(\lambda _{n}\rightarrow \lambda\), \(\forall x^{*}_{n}\in E(\lambda_{n})\), there must exist a subsequence \(\{x^{*}_{n_{k}}\}\subset\{x^{*}_{n}\}\) such that \(x^{*}_{n_{k}}\rightarrow x^{*}\in E(\lambda)\), then the problem Î» is said to be generalized Hadamard wellposed (for short GHwp).

2.
If \(E(\lambda)=\{x^{*}\}\) (a singleton), \(\forall\lambda_{n}\in \varLambda \), \(\lambda_{n}\rightarrow\lambda\), \(\forall x^{*}_{n}\in E(\lambda_{n})\), we must have \(x^{*}_{n}\rightarrow x^{*}\), then the problem Î» is said to be Hadamard wellposed (for short Hwp).
Finally, we establish a wellposedness concept for (SVQEP) corresponding to the problem Î», which unifies its Hadamard and Tykhonov wellposedness.
Definition 3.3

1.
If \(\forall\lambda_{n}\in \varLambda \), \(\lambda _{n}\rightarrow \lambda\), \(\forall x^{*}_{n}\in E(\lambda_{n},\epsilon_{n})\), \(\epsilon_{n}>0\) with \(\epsilon_{n}\rightarrow0\), there must exist a subsequence \(\{x^{*}_{n_{k}}\}\subset\{x^{*}_{n}\}\) such that \(x^{*}_{n_{k}}\rightarrow x^{*}\in E(\lambda)\), then the problem Î» is said to be generalized wellposed (for short Gwp).

2.
If \(E(\lambda)=\{x^{*}\}\) (a singleton), \(\forall\lambda_{n}\in \varLambda \), \(\lambda_{n}\rightarrow\lambda\), \(\forall x^{*}_{n}\in E(\lambda_{n},\epsilon_{n})\), \(\epsilon_{n}>0\) with \(\epsilon_{n}\rightarrow0\), we must have \(x^{*}_{n}\rightarrow x^{*}\), then the problem Î» is said to be wellposed (for short wp).
4 Sufficient conditions for wellposedness of (SVQEP)
Assume that the bounded rationality model \(M=\{\varLambda ,X^{*},f,\varPhi \}\) for (SVQEP) is given. In order to show sufficient conditions for wellposedness of (SVQEP), we first give the following lemmas.
Lemma 4.1
\(f:{\varLambda }\rightrightarrows X^{*}\) is an usco mapping.
Proof
Since \(X^{*}\) is a compact metric space, by LemmaÂ 2.1, it suffices to show that \(\operatorname{Graph}(f)\) is closed. That is to say, \(\forall\lambda_{n}\in{ \varLambda }\), \(\lambda_{n}\rightarrow\lambda\in \varLambda \), \(\forall x_{n}^{*}\in f(\lambda_{n})\), \(x_{n}^{*}\rightarrow x^{*}\), we need to show that \(x^{*}\in f(\lambda)\).
Let \(h_{1}(S_{n}(x_{n},y_{n}),S(x_{n},y_{n}))\leq\epsilon_{n}\) and \(h_{2}(T_{n}(x_{n},y_{n}),T(x_{n},y_{n}))\leq\epsilon_{n}\). For each \(n=1,2,3,\ldots\)â€‰, since \((x_{n},y_{n})\in f(\lambda_{n})\), then there exists \((x_{n},y_{n})\in X^{*}\) such that \((x_{n},y_{n})\in (S_{n}\times T_{n})(x_{n},y_{n})\). So there exists \(x_{n}'\in S(x_{n},y_{n})\) such that \(d_{1}(x_{n},x_{n}')\leq\epsilon_{n}\). By
we get \(x_{n}'\rightarrow x\). Note that setvalue mapping S is continuous on \(X^{*}\), then we get
By compactness of \(S(x,y)\), we have \(x\in S(x,y)\). Similarly, we can prove that \(y\in T(x,y)\). Hence, \((x,y)\in(S\times T)(x,y)\). It shows that \(x^{*}\in f(\lambda)\).â€ƒâ–¡
Lemma 4.2
Suppose that \(f:\varLambda \rightrightarrows X^{*}\) is a usco mapping. Then, for any \(\lambda_{n}\rightarrow\lambda\) and any \(x_{n}^{*}\in f(\lambda_{n})\), there is a subsequence \(\{x^{*}_{n_{k}}\}\subset\{ x^{*}_{n}\}\) such that \(x_{n_{k}}^{*}\rightarrow x^{*}\in f(\lambda)\).
Lemma 4.3
Î¦ is lower semicontinuous at \((\lambda,x^{*})\).
Proof
We only need to show that \(\forall\epsilon>0\), \(\forall\lambda_{n}=(S_{n},T_{n},F_{n},G_{n})\in {\varLambda }\), \(\lambda_{n}\rightarrow\lambda=(S,T,F,G) \in {\varLambda }\), \(\forall x_{n}^{*}\in X^{*}\), \(x^{*}_{n}\rightarrow x^{*}\in X^{*}\), there exists a positive integer N such that, \(\forall n\geq N\),
Let
and
By the definition of the least upper bound, there exists \(u_{0}\in S(x,y)\) such that
Note that \(\sup_{(x,y)\in C\times D}h_{1}(S_{n}(x,y),S(x,y))\rightarrow0\) and \(h_{1}(S(x_{n},y_{n}),S(x,y))\rightarrow0\), we have
By (9), there exists \(u_{n}\in S_{n}(x_{n},y_{n})\) such that \(d_{1}(u_{n}, u_{0})\rightarrow0\). Since F is continuous on \(C\times D\) and \(\sup_{(x,y)\in C\times D}\F_{n}(x,y)F(x,y)\\rightarrow0\), letting \(n\rightarrow \infty\), we have
Using continuity of \(\xi_{e}\) and (10), we have
By (11), there exists a positive integer \(N_{1}\) such that, for any \(n\geq N_{1}\),
From (8) and (12), for any \(n\geq N_{1}\), we get
Similarly, we can prove that there exists a positive integer \(N_{2}\) such that, for any \(n\geq N_{2}\),
Let \(N=\max\{N_{1},N_{2}\}\), \(\forall n\geq N\), by (13) and (14), we get (7), that is,
â€ƒâ–¡
Finally, we give sufficient conditions for Gwp and wp of (SVQEP) corresponding to \(\lambda\in{ \varLambda }\).
Theorem 4.1

1.
Every \(\lambda\in{ \varLambda }\) is Gwp.

2.
Let \(\lambda\in{ \varLambda }\) and suppose furthermore \(E(\lambda)=\{x^{*}\}\) (a singleton), then Î» is wp.
Proof
1. \(\forall\lambda_{n}\in {\varLambda }\), \(\lambda_{n}\rightarrow\lambda\), \(\forall x_{n}^{*}\in E(\lambda_{n},\epsilon_{n})\), \(\epsilon_{n}>0\) with \(\epsilon_{n}\rightarrow0\), then we have \(x_{n}^{*}\in f(\lambda_{n})\) and \({\varPhi }(\lambda_{n},x_{n}^{*})\leq\epsilon_{n}\). First, by LemmaÂ 4.1 and LemmaÂ 4.2, if \(\lambda_{n}\rightarrow\lambda\), then there exists \(\{x_{n_{k}}^{*}\}\subset\{x_{n}^{*}\}\) such that \(x_{n_{k}}^{*}\rightarrow x^{*}\in f(\lambda)\). Secondly, by \({\varPhi }(\lambda_{n},x_{n}^{*})\leq\epsilon_{n}\) and LemmaÂ 4.3, we have
which implies that \({\varPhi }(\lambda,x^{*})=0\). It shows that the problem Î» is Gwp.
2. By way of contradiction. If the sequence \(\{x_{n}^{*}\}\) does not converge \(x^{*}\), then there exist an open neighborhood O at \(x^{*}\) and a subsequence \(\{x_{n_{k}}^{*}\}\) of \(\{x_{n}^{*}\}\) such that \(x_{n_{k}}^{*}\notin O\). Since \(E(\lambda)=\{x^{*}\}\) (a singleton), using the above proof, we get \(x_{n_{k}}^{*}\rightarrow x^{*}\). This is a contradiction to \(x_{n_{k}}^{*}\notin O\).â€ƒâ–¡
Example 4.1
Let \(C=D=[0,2]\times[0,2]\), \(Z=\mathbb{R}^{2}\), \(P=\mathbb{R}_{+}^{2}\), and \(e=(1,1)\). For any \((x,y)\in X\times Y\), assume that
and for any \((u,v)\in C\times D\),
Then it is easy to see that, for \(\lambda=(S,T,F,G)\in \varLambda \),
and
Moreover, by TheoremÂ 4.1, the problem Î» must be Gwp.
Finally, by DefinitionÂ 3.1, DefinitionÂ 3.2, DefinitionÂ 3.3, and TheoremÂ 4.1, it is easy to check the following.
Corollary 4.1

1.
Every \(\lambda\in{ \varLambda }\) must be GTwp and GHwp.

2.
Let \(\lambda\in{ \varLambda }\), if \(E(\lambda)=\{x^{*}\}\) (a singleton), then Î» must be Twp and Hwp.
Remark 4.1
In TheoremÂ 4.1 and CorollaryÂ 4.1, \(\lambda\in \varLambda \) means that the problem \(\lambda=(S,T,F,G)\) holds for all conditions (a), (b), (c), and (d).
References
Fu, JY: Symmetric vector quasiequilibrium problems. J. Math. Anal. Appl. 285, 708713 (2003)
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123145 (1994)
Farajzadeh, AP: On the symmetric vector quasiequilibrium problems. J. Math. Anal. Appl. 322, 10991110 (2006)
Chen, JC, Gong, XH: The stability of set of solutions for symmetric vector quasiequilibrium problems. J. Optim. Theory Appl. 136, 359374 (2008)
Zhang, WY: Wellposedness for convex symmetric vector quasiequilibrium problems. J. Math. Anal. Appl. 387, 909915 (2012)
Deng, XC, Xiang, SW: Wellposed generalized vector equilibrium problems. J. Inequal. Appl. 2014, Article ID 127 (2014)
Zhang, WB, Huang, NJ, Oâ€™Regan, D: Generalized wellposedness for symmetric vector quasiequilibrium problems. J.Â Appl. Math. 2015, Article ID 108357 (2015)
Dontchev, AL, Zolezzi, T: WellPosed Optimization Problems. Lecture Notes in Mathematics, vol.Â 1543. Springer, Berlin (1993)
Yu, J, Yang, H, Yu, C: Wellposed Ky Fanâ€™s point, quasivariational inequality and Nash equilibrium problems. Nonlinear Anal. 66, 777790 (2007)
Crespi, GP, Guerraggio, A, Rocca, M: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132, 213226 (2007)
Durea, M: Scalarization for pointwise wellposed vectorial problems. Math. Methods Oper. Res. 66, 409418 (2007)
Huang, XX: Extended wellposedness properties of vector optimization problems. J. Optim. Theory Appl. 106, 165182 (2000)
Huang, XX, Yang, XQ: LevitinPolyak wellposedness of constrained vector optimization problems. J. Glob. Optim. 37, 287304 (2007)
Li, Z, Xia, FQ: Scalarization method for LevitinPolyak wellposedness of vectorial optimization problems. Math. Methods Oper. Res. 76, 361375 (2012)
Miglierina, E, Molho, E, Rocca, M: Wellposedness and scalarization in vector optimization. J. Optim. Theory Appl. 126, 391409 (2005)
Zui, X, Zhu, DL, Huang, XX: LevitinPolyak wellposedness in generalized vector variational inequality problem with functional constraints. Math. Methods Oper. Res. 67, 505524 (2008)
Li, SJ, Li, MH: LevitinPolyak wellposedness of vector equilibrium problems. Math. Methods Oper. Res. 69, 125140 (2009)
Kimura, K, Liou, YC, Wu, SY: Wellposedness for parametric vector equilibrium problems with applications. J. Ind. Manag. Optim. 4, 313327 (2008)
Anderlini, L, Canning, D: Structural stability implies robustness to bounded rationality. J. Econ. Theory 101, 395422 (2001)
Yu, C, Yu, J: On structural stability and robustness to bounded rationality. Nonlinear Anal. TMA 65, 583592 (2006)
Yu, C, Yu, J: Bounded rationality in multiobjective games. Nonlinear Anal. TMA 67, 930937 (2007)
Yu, J, Yang, H, Yu, C: Structural stability and robustness to bounded rationality for noncompact cases. J. Glob. Optim. 44, 149157 (2009)
Gerth, C, Weidner, P: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67, 297320 (1990)
Chen, GY, Huang, XX, Yang, XQ: Vector Optimization: SetValued and Variational Analysis. Springer, Berlin (2005)
Luc, DT: Theory of Vector Optimization. Springer, Berlin (1989)
Aliprantis, CD, Border, KC: Infinite Dimensional Analysis. Springer, Berlin (1999)
Acknowledgements
The authors would like to thank the editor and the reviewers for their helpful comments and suggestions, which have improved the presentation of the paper. This work is supported by NSFC (Grant No. 11161008) and Natural Science Foundation Guizhou Province, P.R.Â China (Grant Nos. 20132235, 20122289).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authorsâ€™ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Deng, X., Xiang, S. Wellposed symmetric vector quasiequilibrium problems. J Inequal Appl 2015, 232 (2015). https://doi.org/10.1186/s1366001507496
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366001507496