# Levitin-Polyak Well-Posedness in Vector Quasivariational Inequality Problems with Functional Constraints

- J Zhang
^{1}, - B Jiang
^{2}and - XX Huang
^{3}Email author

**2010**:984074

**DOI: **10.1155/2010/984074

© J. Zhang et al. 2010

**Received: **17 March 2010

**Accepted: **6 July 2010

**Published: **20 July 2010

## Abstract

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.

## 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 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 set-valued map (i.e., ).

Denote by the solution set of (VQVI).

Well-posedness 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 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 [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 Levitin-Polyak 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]. 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 [31]. 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

Let , be two normed spaces. A set-valued 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
.

Throughout this paper, we always assume that the feasible set is nonempty and the function is continuous on .

- (i)A sequence is called a type I Levtin-Polyak (LP in short) approximating solution sequence if there exists with such that(2.3)

- (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 .

- (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.

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].

- (i)A sequence is called a type I LP minimizing sequence for (P) if(2.9)

- (iii)is called a generalized type I LP minimizing sequence for (P) if(2.12)

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

- (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 .

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 .

- (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)

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 set-valued 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 .

- (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 .

- (ii)Let be any sequence, we can check that(2.29)

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.

- (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 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.

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.

With the help of Lemma 2.7, analogously to [35, Theorems , and ], we can prove the following two theorems.

Theorem 3.2.

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.

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 Furi-Vignoli type characterizations [41] for the (generalized) type I LP well posednesses of (VQVI).

Lemma 3.4.

then one has and .

Proof.

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 .

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.

To continue our study, we make some assumptions below.

- (i)
is a Banach space.

- (ii)
The set-valued 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.

It follows that . Hence, is l.s.c. on . Furthermore, if , by Lemma 2.6, we see that .

Theorem 3.7.

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.

Hence, is closed.

Theorem 3.9.

Proof.

The proof is similar to that of Theorem in [35] and thus omitted.

- (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 set-valued 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.

- (i)
Let be a topological space, and let be nonempty. Suppose that is an extended real-valued 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.

(ii)the function defined by (2.14) is level compact on ,

where is defined by (2.14).

(iv)There exists such that is level-compact on defined by (3.26). Then, (VQVI) is type I LP well posed.

Proof.

which contradicts condition (3.27).

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.

(ii)the function defined by (2.14) is level compact on ,

(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 "level-compactness" condition in Propositions 3.12 and 3.13 can be replaced by the "level-boundedness" condition.

Now, we consider the case when is a normed space, is a closed and convex cone with nonempty interior and let .

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 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.

Remark 3.16.

If is finite dimensional, then the level-compactness condition of can be replaced by the level boundedness of .

## Declarations

### Acknowledgment

This work is supported by the National Science Foundation of China.

## Authors’ Affiliations

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