 Research Article
 Open Access
 Published:
LevitinPolyak WellPosedness in Vector Quasivariational Inequality Problems with Functional Constraints
Fixed Point Theory and Applications volume 2010, Article number: 984074 (2010)
Abstract
We introduce several types of LevtinPolyak wellposedness for a vector quasivariational inequality with functional constraints. Necessary and/or sufficient conditions are derived for them.
1. Introduction
It is well known that, under certain conditions, a Nash equilibrium problem can be formulated and solved as a variational inequality problem. A generalized Nash game is a Nash game in which each player's strategy depends on other players' strategies [1]. The connection between generalized Nash games and quasivariational inequalities was first recognized by Bensoussan [2]. Recently, some researchers [1, 3, 4] found that mathematical models of many real world problems, including some engineering problems, can be formulated as certain kinds of variational inequality problems, including quasivariational inequality problems. However, as noted in [5], compared with variational inequality problems, the study on quasivariational inequality problems is still in its infancy, in particular only a few algorithms have been proposed to solve variational inequalities numerically.
Vector variational inequality problems were introduced by Giannessi [6] and are related to vector network equilibrium problems [7]. Since then, various types of vector variational inequalities were introduced and studied (see, e.g., [8, 9] and the references therein).
In this paper, we will consider vector quasivariational inequality problems with functional constraints, which are described below.
Let be a normed space and a metric space. Let be nonempty and closed sets. Let be a locally convex space and be a nontrivial closed and convex cone with nonempty interior . Define the following order in , for any ,
Let be the space of all the linear continuous operators from to . Let and be two functions. We denote by the function value , where . Let be a strict setvalued map (i.e., ).
Let
The vector quasivariational inequality problem with functional and abstract set constraints considered in this paper is:
Denote by the solution set of (VQVI).
Wellposedness for unconstrained and constrained optimization problems was first studied by Tikhonov [10] and Levitin and Polyak [11]. The issue being considered is that for each approximating solution sequence, there exists a subsequence that converges to a solution of the problem.
In Tikhonov's well posedness, the approximating solution is always feasible. However, it should be noted that many algorithms in optimization and variational inequalities, such as penaltytype methods and augmented Lagrangian methods, terminate when the constraint is approximately satisfied. These methods may generate sequences that may not be necessarily feasible [12].
Up to now, various extensions of these well posednesses have been developed and well studied (see, e.g., [13–18]). Studies on well posedness of optimization problems have been extended to vector optimization problems (see e.g., [19–24]). The study of LevitinPolyak well posedness for scalar convex optimization problems with functional constraints originates from [25]. Recently, this research was extended to nonconvex optimization problems with abstract and functional constraints [12] and nonconvex vector optimization problems with both abstract and functional constraints [26]. Wellposedness of variational inequality problems, mixed variational inequality problems, and equilibrium problems without functional constraints was investigated in the literature (see, e.g., [27–30]). Wellposedness in variational inequality problems with both abstract and functional constraints was investigated in [31]. Wellposedness of (generalized) quasivariational inequality and mixed quasivariationallike inequalities has been studied in the literature [32–35]. The study of well posedness for (generalized) vector variational inequality, vector quasiequilibria and vector equilibrium problems can be found in [36–39] and the references therein.
In this paper, we will introduce and study several types of LevitinPolyak (LP in short) well posednesses and generalized LP well posednesses for vector quasivariational inequalities with functional constraints. The paper is organized as follows. In Section 2, four types of LP well posednesses and generalized LP well posednesses for vector quasivariational inequality problems will be defined. In Section 3, we will derive various criteria and characterizations for the various (generalized) LP well posednesses of constrained vector quasivariational inequalities.
2. Definitions and Preliminaries
Let , be two normed spaces. A setvalued map from to is
(i)closed, on , if for any sequence with and with , one has ;
(ii)lower semicontinuous (l.s.c. in short) at , if , , and imply that there exists a sequence satisfying such that for sufficiently large. If is l.s.c. at each point of , we say that is l.s.c on .
Let be a metric space, , and . In the sequel, we denote by the distance function from point to set . For a topological vector space , we denote by its dual space. For any cone , we will denote the (positive) polar cone of by
Let be fixed. Denote
Throughout this paper, we always assume that the feasible set is nonempty and the function is continuous on .
Definition 2.1.

(i)
A sequence is called a type I LevtinPolyak (LP in short) approximating solution sequence if there exists with such that
(2.3)
(ii) is called a type II LP approximating solution sequence if there exist with and with such that (2.3)–(2.5) hold and
(iii) is called a generalized type I LP approximating solution sequence if there exists with such that
and (2.4), (2.5) hold.

(iv)
is called a generalized type II LP approximating solution sequence if there exist with and with such that (2.4)–(2.7) hold.
Definition 2.2.
(VQVI) is said to be type I (resp., type II, generalized type I, generalized type II) LP well posed if the solution set of (VQVI) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence , there exist a subsequence of and such that .
Remark 2.3.

(i)
It is easily seen that if , , then type I (resp., type II, generalized type I, generalized type II) LP well posedness of (VQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well posedness of (QVI) defined in [34].

(ii)
It is clear that any (generalized) type II LP approximating solution sequence is a (generalized) type I LP approximating solution sequence. Thus, (generalized) type I LP well posedness implies (generalized) type II LP well posedness.
(iii)Each type of LP well posedness of (VQVI) implies that its solution set is compact.
To see that the various LP well posednesses of (VQVI) are adaptations of the corresponding LP well posednesses in minimizing problems by using the Auslender gap function, we consider the following general constrained optimization problem:
where is nonempty and is proper. The feasible set of (P) is , where . The optimal set and optimal value of (P) are denoted by , respectively. Note that if , where
then . In this paper, we always assume that . We note that LP well posedness for the special case, where is finite valued and l.s.c., is closed, has been studied in [12].
Definition 2.4.

(i)
A sequence is called a type I LP minimizing sequence for (P) if
(2.9)
(ii) is called a type II LP minimizing sequence for (P) if
and (2.10) hold.

(iii)
is called a generalized type I LP minimizing sequence for (P) if
(2.12)
and (2.9) hold.

(iv)
is called a generalized type II LP minimizing sequence for (P) if (2.11) and (2.12) hold.
Definition 2.5.

(P)
is said to be type I (resp., type II, generalized type I, generalized type II) LP well posed if the solution set of (P) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP minimizing sequence , there exist a subsequence of and such that .
The Auslender gap function for (VQVI) is
From Lemma in [40], we know that is weak* compact. This fact combined with that when implies that
Recall the following nonlinear scalarization function (see, e.g., [9]):
It is known that is a continuous, (strictly) monotone (i.e., for any , , implies that and implies that ), subadditive, and convex function. Moreover, for any , it holds that . Furthermore, following the proof of [9, Proposition ], we can prove that
Let be defined by
First we have the following lemma.
Lemma 2.6.
Let be defined by (2.14), then
(i), for all ,
(ii) and if and only if .
Proof.

(i)
Let , then . We let in (2.14) be equal to , then .

(ii)
Assume that . Suppose to the contrary that , then, there exists such that
(2.18)
Thus,
It follows that
Hence, , contradicting the assumption, so . Conversely, assume that , then we have
As a result, for any , there exists such that
It follows that . This fact combined with (i) implies that .
In the rest of this paper, we set in (P) equal to . Note that if the setvalued map is closed on , then is closed. By Lemma 2.6, if and only if minimizes (defined by (2.26)) over with .
The following lemma establishes some relationship between LP approximating solution sequence and LP minimizing sequence.
Lemma 2.7.
Let the function be defined by (2.14) as follows:
(i) is a sequence such that there exists with satisfying (2.4)(2.5) if and only if and (2.9) holds with .
(ii) is a sequence such that there exist with and with satisfying (2.4)–(2.6) if and only if and (2.11) holds with .
Proof.

(i)
Let be any sequence, if there exists with satisfying (2.4)(2.5), then we can easily verify that
(2.23)
It follows that (2.9) holds with .
For the converse, let and (2.9) hold. We can see that and (2.4) hold. Furthermore, by (2.9), we have that there exists
such that
That is,
Now, we will show that (2.5) holds, otherwise there exists such that
As a result, for any , Since is a weak* compact set, we have
which contradicts (2.26).

(ii)
Let be any sequence, we can check that
(2.29)
holds if and only if there exists with and with such that (2.6) (with replaced by ) holds. From the proof of (i), we know that
and hold if and only if such that there exists with satisfying (2.4)(2.5) (with replaced by ). Finally, we set and the conclusion follows.
The next proposition establishes relationships between the various LP well posednesses of (VQVI) and those of (P) with defined by (2.14).
Proposition 2.8.
Assume that , then
(i)(VQVI) is generalized type I (resp., generalized type II) LP well posed if and only if (P) is generalized type I (resp., generalized type II) LP well posed with defined by (2.14).
(ii)If (VQVI) is type I (resp., type II) LP well posed, (P) is type I (resp., type II) LP well posed with defined by (2.14).
Proof.
By Lemma 2.6, if , is a solution of (VQVI) if and only if is an optimal solution of (P) with and defined by (2.14).

(i)
Similar to the proof of Lemma 2.7, it is also routine to check that a sequence is a generalized type I (resp., generalized type II) LP approximating solution sequence if and only if it is a generalized type I (resp., generalized type II) LP minimizing sequence of (P). So (VQVI) is generalized type I (resp., generalized type II) LP well posed if and only if (P) is generalized type I (resp., generalized type II) LP well posed with defined by (2.26).

(ii)
Since , for any . This fact together with Lemma 2.7 implies that a type I (resp., type II) LP minimizing sequence of (P) is a type I (resp., type II) LP approximating solution sequence. So type I (resp., type II) LP well posedness of (VQVI) implies type I (resp., type II) LP well posedness of (P) with defined by (2.26).
To end this section, we note that all the results in [12] for the well posedness hold for (P) so long as is closed, is l.s.c. on , and .
3. Criteria and Characterizations for Various LP WellPosedness of (VQVI)
In this section, we give necessary and/or sufficient conditions for the various types of (generalized) LP well posednesses defined in Section 2.
Consider the following statement:
The next proposition can be straightforwardly proved.
Proposition 3.1.
If (VQVI) is type I (resp., type II, generalized type I, generalized type II) LP well posed, then (3.1) holds. Conversely, if (3.1) holds and is compact, then (VQVI) is type I (resp., type II, generalized type I, generalized type II) LP well posed.
Now, we consider a realvalued function defined for sufficiently small such that
With the help of Lemma 2.7, analogously to [35, Theorems , and ], we can prove the following two theorems.
Theorem 3.2.
If (VQVI) is type II LP well posed, the setvalued map is closed valued, then there exists a function c satisfying (3.2) such that
where is defined by (2.14). Conversely, suppose that is nonempty and compact, and (3.3) holds for some satisfying (3.2), then (VQVI) is type II LP well posed.
Theorem 3.3.
If (VQVI) is type II LP well posed in the generalized sense, the setvalued mapping is closed, then there exists a function satisfying (3.2) such that
where is defined by (2.14). Conversely, suppose that is nonempty and compact, and (3.4) holds for some satisfying (3.2), then (VQVI) is generalized type II LP well posed.
Next we give FuriVignoli type characterizations [41] for the (generalized) type I LP well posednesses of (VQVI).
Let be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset of is defined as
where diam is the diameter of defined by
For any , define
Lemma 3.4.
Let be defined by (2.14) and . Let
then one has and .
Proof.
First, we prove the former result. For any satisfying
we have and . We will show that , for all . Otherwise, there exists such that . By the weak* compactness of , we have , which leads to and gives rise to a contradiction. Furthermore, we observe that . This fact combined with implies that .
Now, we prove the equivalence between and . Firstly, we can establish the same inclusion for and analogously to the proof stated above. Then if satisfies and
It is routine to check that . From (3.11), we know that for each , there exists such that . As a result, we can see that . Thus, we prove the conclusion.
The next lemma can be proved analogously to ([25, Theorem ]).
Lemma 3.5.
Let be a Banach space. Suppose that is l.s.c. on and bounded below on . Assume that the optimal solution set of (P) is nonempty and compact, then, (P) is (generalized) type I LP well posed if and only if
To continue our study, we make some assumptions below.
Assumption.

(i)
is a Banach space.

(ii)
The setvalued map is closed, and lower semicontinuous on .

(iii)
The map is continuous on .
We have the following lemma concerning the l.s.c. of defined by (2.14).
Lemma 3.6.
Let function be defined by (2.14) and Assumption 1 hold, then is l.s.c. function from to . Further assume that the solution set of (VQVI) is nonempty, then .
Proof.
First we show that , for all . Suppose to the contrary that there exists such that , then,
That is,
Namely,
which is impossible since is a finite function on . Second, we show that is l.s.c. on . Note that the function
is continuous on by the continuity of on and the continuity of . We also note that . Let . Suppose that the sequence satisfies
and . For any , by the lower semicontinuity of and continuity of , we have a sequence with converging to such that
It follows that . Hence, is l.s.c. on . Furthermore, if , by Lemma 2.6, we see that .
Theorem 3.7.
Let Assumption 1 hold and let the solution set of (QVVI) be nonempty and compact, then, (VQVI) is generalized type I LP well posed if and only if
Proof.
Note that the function defined by (2.14) is nonnegative on . By the lower semicontinuity of and Lemma 3.6, is l.s.c. on . Moreover, is closed, since is closed on . By Proposition 2.8, Lemmas 3.4 and 3.5, the conclusion follows.
Although the type I (type II) LP well posedness of (VQVI) is not equivalent to the type I (type II) LP well posedness of (P), we can still establish the same characterization for type I (type II) LP well posedness of (VQVI) as Theorem 3.7. We need the next lemma.
Lemma 3.8.
Let Assumption 1 hold, then defined by (3.8) is closed.
Proof.
Let and . We show that . It is obvious that . Since and , by the closedness of , we have . Moreover, since
hold and is l.s.c., for any , we can find that with such that
Hence, is closed.
Theorem 3.9.
Let Assumption 1 hold and let be defined by (2.14). Assume that the solution set of (QVVI) is nonempty and compact, then (VQVI) is type I LP well posed if and only if
Proof.
The proof is similar to that of Theorem in [35] and thus omitted.
Example 3.10.

(i)
Let , , , , and . maps into an identical mapping, that is to say , for any . The set valued mapping is defined as follows, given for some , then
(3.23)
with , of course is closed and l.s.c. Now, we can show that, when , , which is bounded. Thus, , by applying Theorem 3.9, we know that (VQVI) is type I LP well posed.
(i)Suppose that is a setvalued mapping from to , for fixed , implies that
with , obviously is still closed and l.s.c. If we replace by in (i), then with , which is unbounded. Therefore, and the (VQVI) is not LP well posed in sense of type I. Actually, the solution set of this problem is and thus unbounded.
Definition 3.11.

(i)
Let be a topological space, and let be nonempty. Suppose that is an extended realvalued function. is said to be level compact on if, for any , the subset is compact.

(ii)
Let be a finite dimensional normed space, and let be nonempty. A function is said to be level bounded on if is bounded or
(3.25)
The following proposition presents some sufficient conditions for type I LP well posedness of (VQVI)
Proposition 3.12.
Let Assumption 1 hold. Further assume that one of the following conditions holds.
(i)There exists such that is compact, where
(ii)the function defined by (2.14) is level compact on ,
(iii) is finite dimensional and
where is defined by (2.14).
(iv)There exists such that is levelcompact on defined by (3.26). Then, (VQVI) is type I LP well posed.
Proof.
First, we show that each one of (i), (ii), and (iii) implies (iv). Clearly, either of (i) and (ii) implies (iv). Now, we show that (iii) implies (iv). We notice that the set is closed by the closedness of . Then, we need only to show that for any , the set
is bounded since is a finite dimensional space and the function defined by (2.14) is l.s.c. on and, thus, is closed. Suppose to the contrary that there exist and such that and . From , we have
Thus,
which contradicts condition (3.27).
Now, we show that if (iv) holds, then (VQVI) is type I LP well posed. Let be a type I LP approximating solution sequence of (VQVI). Then, there exist with and such that
From (3.32) and (3.33), we can assume without loss of generality that . By Lemma 2.7, we can assume without loss of generality that
where is defined by (2.14). By the level compactness of on , there exist a subsequence of of and such that . From this fact and (3.32), we have . Since is closed and (3.33) holds, we also have . That is,
Furthermore, by Lemmas 2.7 and 3.6, we have
We know that by Lemma 2.6, so . This fact combined with (3.35) and Lemma 2.6 implies that .
Similarly, we can prove the next proposition.
Proposition 3.13.
Let Assumption 1 hold. Further assume that one of the following conditions holds.
(i)There exists such that is compact, where
(ii)the function defined by (2.14) is level compact on ,
(iii) is finite dimensional and
(iv)There exists such that is level compact on defined by (3.37). Then, (VQVI) is generalized type I LP well posed.
Remark 3.14.
If is finite dimensional, then the "levelcompactness" condition in Propositions 3.12 and 3.13 can be replaced by the "levelboundedness" condition.
Now, we consider the case when is a normed space, is a closed and convex cone with nonempty interior and let .
Let and denote
The next proposition follows immediately from Proposition 2.8(i), Lemma 3.6, and [12, Proposition (iv)].
Proposition 3.15.
Let be a normed space, let be a closed and convex cone with nonempty interior and . Let the setvalued map be closed and l.s.c on . Assume that the solution set of (VQVI) is nonempty. Further assume that there exists such that the function defined by (2.14) is level compact on , then (VQVI) is generalized type I LP well posed.
Remark 3.16.
If is finite dimensional, then the levelcompactness condition of can be replaced by the level boundedness of .
References
Facchinei F, Pang JS: FiniteDimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research. Volume 1–2. Springer, New York, NY, USA; 2003:xxxiv+624+I69.
Bensoussan A: Points de nash dans le cas de fonctionnelles quadratiques et jeux différentiels linéaires à personnes. SIAM Journal on Control and Optimization 1974, 12: 460–499. 10.1137/0312037
Pang JS, Fukushima M: Quasivariational inequalities, generalized Nash equilibria, and multileaderfollower games. Computational Management Science 2005,2(1):21–56. 10.1007/s1028700400100
Baiocchi C, Capelo A: Variational and Quasivariational Inequalities, A WileyInterscience Publication. John Wiley & Sons, New York, NY, USA; 1984:ix+452.
Fukushima M: A class of gap functions for quasivariational inequality problems. Journal of Industrial and Management Optimization 2007,3(2):165–171.
Giannessi F: Theorems of alternative, quadratic programs and complementarity problems. In Variational Inequalities and Complementarity Problems. Edited by: Cottle RW, Giannessi F, Lions JL. John Wiley & Sons, New York, NY, USA; 1980:151–186.
Chen GY, Yen ND: On the variational inequality model for network equilibrium. In Internal Report. Department of Mathematics, University of Pisa, Pisa, Italy; 1993.
Ceng LC, Chen GY, Huang XX, Yao JC: Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications. Taiwanese Journal of Mathematics 2008,12(1):151–172.
Chen GY, Huang XX, Yang XQ: Vector Optimization, SetValued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems. Volume 541. Springe, Berlin, Germany; 2005:x+306.
Tikhonov AN: On the stability of the functional optimization problem. USSR Computational Mathematics and Mathematical Physics 1966,6(4):28–33. 10.1016/00415553(66)900036
Levitin ES, Polyak BT: Convergence of minimizing sequences in conditional extremum problems. Soviet Mathematics Doklady 1966, 7: 764–767.
Huang XX, Yang XQ: Generalized LevitinPolyak wellposedness in constrained optimization. SIAM Journal on Optimization 2006,17(1):243–258. 10.1137/040614943
Beer G, Lucchetti R: The epidistance topology: continuity and stability results with applications to convex optimization problems. Mathematics of Operations Research 1992,17(3):715–726. 10.1287/moor.17.3.715
Dontchev AL, Zolezzi T: WellPosed Optimization Problems, Lecture Notes in Mathematics. Volume 1543. Springer, Berlin, Germany; 1993:xii+421.
Revalski JP: Hadamard and strong wellposedness for convex programs. SIAM Journal on Optimization 1997,7(2):519–526. 10.1137/S1052623495286776
Lucchetti R, Revalski J (Eds): Recent Developments in WellPosed Variational Problems, Mathematics and its Applications. Volume 331. Kluwer Academic, Dodrecht, The Netherlands; 1995:viii+266.
Zolezzi T: Wellposedness criteria in optimization with application to the calculus of variations. Nonlinear Analysis: Theory, Methods & Applications 1995,25(5):437–453. 10.1016/0362546X(94)001425
Zolezzi T: Extended wellposedness of optimization problems. Journal of Optimization Theory and Applications 1996,91(1):257–266. 10.1007/BF02192292
Bednarczuk E: Wellposedness of vector optimization problems. In Recent Advances and Historical Development of Vector Optimization Problems, Lecture Notes in Economics and Mathematical Systems. Volume 294. Edited by: Jahn J, Krabs W. Springer, Berlin, Germany; 1987:51–61.
Deng S: Coercivity properties and wellposedness in vector optimization. RAIRO Operations Research 2003,37(3):195–208. 10.1051/ro:2003021
Huang XX: Extended wellposedness properties of vector optimization problems. Journal of Optimization Theory and Applications 2000,106(1):165–182. 10.1023/A:1004615325743
Huang XX: Extended and strongly extended wellposedness of setvalued optimization problems. Mathematical Methods of Operations Research 2001,53(1):101–116. 10.1007/s001860000100
Loridan P: Wellposedness in vector optimization. In Recent Developments in WellPosed Variational Problems, Mathematics and Its Applications. Volume 331. Kluwer Academic, Dodrecht, The Netherlands; 1995:171–192.
Lucchetti R: WellPosedness Towards Vector Optimization, Lecture Notes in Economics and Mathematical System. Volume 294. Springer, New York, NY, USA; 1987.
Konsulova AS, Revalski JP: Constrained convex optimization problemswellposedness and stability. Numerical Functional Analysis and Optimization 1994,15(7–8):889–907. 10.1080/01630569408816598
Huang XX, Yang XQ: LevitinPolyak wellposedness of constrained vector optimization problems. Journal of Global Optimization 2007,37(2):287–304. 10.1007/s108980069050z
Fang YP, Huang NJ, Yao JC: Wellposedness of mixed variational inequalities, inclusion problems and fixed point problems. Journal of Global Optimization 2008,41(1):117–133. 10.1007/s1089800791696
Fang YP, Huang NJ, Yao JC: Wellposedness by perturbations of mixed variational inequalities in Banach spaces. European Journal of Operational Research 2010,201(3):682–692. 10.1016/j.ejor.2009.04.001
Lignola MB, Morgan J: wellposedness for Nash equilibria and for optimization problems with Nash equilibrium constraints. Journal of Global Optimization 2006,36(3):439–459. 10.1007/s1089800690205
Lignola MB, Morgan J: Generalized variational inequalities with pseudomonotone operators under perturbations. Journal of Optimization Theory and Applications 1999,101(1):213–220. 10.1023/A:1021783313936
Huang XX, Yang XQ, Zhu DL: LevitinPolyak wellposedness of variational inequality problems with functional constraints. Journal of Global Optimization 2009,44(2):159–174. 10.1007/s1089800893101
Ceng LC, Hadjisavvas N, Schaible S, Yao JC: Wellposedness for mixed quasivariationallike inequalities. Journal of Optimization Theory and Applications 2008,139(1):109–125. 10.1007/s1095700894289
Lignola MB: Wellposedness and wellposedness for quasivariational inequalities. Journal of Optimization Theory and Applications 2006,128(1):119–138. 10.1007/s1095700575612
Huang XX, Jiang B, Zhang J: LevitinPolyak wellposedness of quasivariational inequality problems with functional constraints. Natural Science Jourmal of Xiangtan University 2008, 30: 1–11.
Jiang B, Zhang J, Huang XX: LevitinPolyak wellposedness of generalized quasivariational inequalities with functional constraints. Nonlinear Analysis: Theory, Methods & Applications 2009,70(4):1492–1503. 10.1016/j.na.2008.02.029
Anh LQ, Khanh PQ, Van DTM, Yao JC: Wellposedness for vector quasiequilibria. Taiwanese Journal of Mathematics 2009,13(2):713–737.
Huang XX, Yang XQ: LevitinPolyak wellposedness of vector variational inequality problems with functional constraints. Numerical Functional Analysis and Optimization 2010,31(4):440–459. 10.1080/01630563.2010.485296
Li SJ, Li MH: LevitinPolyak wellposedness of vector equilibrium problems. Mathematical Methods of Operations Research 2009,69(1):125–140. 10.1007/s0018600802140
Wang G, Huang XX, Zhang J, Chen GY: LevitinPolyak wellposedness for vector equilibrium problems with functional constraints. Acta Mathematica Scientia 2010,30(5):1400–1412. 10.1016/S02529602(10)601324
Huang XX, Teo KL, Yang XQ: Calmness and exact penalization in vector optimization with cone constraints. Computational Optimization and Applications 2006,35(1):47–67. 10.1007/s1058900664415
Furi M, Vignoli A: About wellposed minimization problems for functionals in metric spaces. Journal of Optimization Theory and Applications 1970, 5: 225–229. 10.1007/BF00927717
Acknowledgment
This work is supported by the National Science Foundation of China.
Author information
Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhang, J., Jiang, B. & Huang, X. LevitinPolyak WellPosedness in Vector Quasivariational Inequality Problems with Functional Constraints. Fixed Point Theory Appl 2010, 984074 (2010). https://doi.org/10.1155/2010/984074
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/984074
Keywords
 Variational Inequality
 Equilibrium Problem
 Inequality Problem
 Variational Inequality Problem
 Functional Constraint