Levitin-Polyak Well-Posedness in Vector Quasivariational Inequality Problems with Functional Constraints
© J. Zhang et al. 2010
Received: 17 March 2010
Accepted: 6 July 2010
Published: 20 July 2010
We introduce several types of Levtin-Polyak well-posedness for a vector quasivariational inequality with functional constraints. Necessary and/or sufficient conditions are derived for them.
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 . The connection between generalized Nash games and quasivariational inequalities was first recognized by Bensoussan . 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 , 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  and are related to vector network equilibrium problems . 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.
Well-posedness for unconstrained and constrained optimization problems was first studied by Tikhonov  and Levitin and Polyak . 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 penalty-type methods and augmented Lagrangian methods, terminate when the constraint is approximately satisfied. These methods may generate sequences that may not be necessarily feasible .
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 Levitin-Polyak well posedness for scalar convex optimization problems with functional constraints originates from . Recently, this research was extended to nonconvex optimization problems with abstract and functional constraints  and nonconvex vector optimization problems with both abstract and functional constraints . Well-posedness of variational inequality problems, mixed variational inequality problems, and equilibrium problems without functional constraints was investigated in the literature (see, e.g., [27–30]). Well-posedness in variational inequality problems with both abstract and functional constraints was investigated in . Well-posedness of (generalized) quasivariational inequality and mixed quasivariational-like 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 Levitin-Polyak (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
(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 .
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 .
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.
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 .
First we have the following lemma.
The following lemma establishes some relationship between LP approximating solution sequence and LP minimizing sequence.
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).
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  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 Well-Posedness 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.
The next proposition can be straightforwardly proved.
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.
With the help of Lemma 2.7, analogously to [35, Theorems , and ], we can prove the following two theorems.
Next we give Furi-Vignoli type characterizations  for the (generalized) type I LP well posednesses of (VQVI).
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 .
The next lemma can be proved analogously to ([25, Theorem ]).
To continue our study, we make some assumptions below.
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.
The proof is similar to that of Theorem in  and thus omitted.
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.
The following proposition presents some sufficient conditions for type I LP well posedness of (VQVI)
Let Assumption 1 hold. Further assume that one of the following conditions holds.
which contradicts condition (3.27).
Similarly, we can prove the next proposition.
Let Assumption 1 hold. Further assume that one of the following conditions holds.
The next proposition follows immediately from Proposition 2.8(i), Lemma 3.6, and [12, Proposition (iv)].
Let be a normed space, let be a closed and convex cone with nonempty interior and . Let the set-valued 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.
This work is supported by the National Science Foundation of China.
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