Open Access

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

Fixed Point Theory and Applications20102010:984074

https://doi.org/10.1155/2010/984074

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 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 ,
(1.1)

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., ).

Let
(1.2)
The vector quasivariational inequality problem with functional and abstract set constraints considered in this paper is:
(VQVI)

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., [1318]). Studies on well posedness of optimization problems have been extended to vector optimization problems (see e.g., [1924]). 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., [2730]). 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 [3235]. The study of well posedness for (generalized) vector variational inequality, vector quasiequilibria and vector equilibrium problems can be found in [3639] 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 .

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
(2.1)
Let be fixed. Denote
(2.2)

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

Definition 2.1.
  1. (i)
    A sequence is called a type I Levtin-Polyak (LP in short) approximating solution sequence if there exists with such that
    (2.3)
     
(2.4)
(2.5)
 (ii) is called a type II LP approximating solution sequence if there exist with and with such that (2.3)–(2.5) hold and
(2.6)
 (iii) is called a generalized type I LP approximating solution sequence if there exists with such that
(2.7)
and (2.4), (2.5) hold.
  1. (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.
  1. (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].

     
  2. (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:
(P)
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
(2.8)

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.
  1. (i)
    A sequence is called a type I LP minimizing sequence for (P) if
    (2.9)
     
(2.10)
 (ii) is called a type II LP minimizing sequence for (P) if
(2.11)
and (2.10) hold.
  1. (iii)
    is called a generalized type I LP minimizing sequence for (P) if
    (2.12)
     
and (2.9) hold.
  1. (iv)

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

     
Definition 2.5.
  1. (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
(2.13)
From Lemma in [40], we know that is weak* compact. This fact combined with that when implies that
(2.14)
Recall the following nonlinear scalarization function (see, e.g., [9]):
(2.15)
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
(2.16)
Let be defined by
(2.17)

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.
  1. (i)

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

     
  2. (ii)
    Assume that . Suppose to the contrary that , then, there exists such that
    (2.18)
     
Thus,
(2.19)
It follows that
(2.20)
Hence, , contradicting the assumption, so . Conversely, assume that , then we have
(2.21)
As a result, for any , there exists such that
(2.22)

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 .

Proof.
  1. (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
(2.24)
such that
(2.25)
That is,
(2.26)
Now, we will show that (2.5) holds, otherwise there exists such that
(2.27)
As a result, for any , Since is a weak* compact set, we have
(2.28)
which contradicts (2.26).
  1. (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
(2.30)

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).
  1. (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).

     
  2. (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.

Consider the following statement:
(3.1)

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 real-valued function defined for sufficiently small such that
(3.2)

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 set-valued map is closed valued, then there exists a function c satisfying (3.2) such that
(3.3)

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 set-valued mapping is closed, then there exists a function satisfying (3.2) such that
(3.4)

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).

Let be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset of is defined as
(3.5)
where diam is the diameter of defined by
(3.6)
For any , define
(3.7)

Lemma 3.4.

Let be defined by (2.14) and . Let
(3.8)
(3.9)

then one has and .

Proof.

First, we prove the former result. For any satisfying
(3.10)

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
(3.11)

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
(3.12)

To continue our study, we make some assumptions below.

Assumption.
  1. (i)

    is a Banach space.

     
  2. (ii)

    The set-valued map is closed, and lower semicontinuous on .

     
  3. (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,
(3.13)
That is,
(3.14)
Namely,
(3.15)
which is impossible since is a finite function on . Second, we show that is l.s.c. on . Note that the function
(3.16)
is continuous on by the continuity of on and the continuity of . We also note that . Let . Suppose that the sequence satisfies
(3.17)
and . For any , by the lower semicontinuity of and continuity of , we have a sequence with converging to such that
(3.18)

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
(3.19)

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
(3.20)
hold and is l.s.c., for any , we can find that with such that
(3.21)

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
(3.22)

Proof.

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

Example 3.10.
  1. (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

(3.24)

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.
  1. (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.

     
  2. (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
(3.26)

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

(iii) is finite dimensional and
(3.27)

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.

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
(3.28)
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
(3.29)
Thus,
(3.30)

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
(3.31)
(3.32)
(3.33)
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
(3.34)
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,
(3.35)
Furthermore, by Lemmas 2.7 and 3.6, we have
(3.36)

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
(3.37)

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

(iii) is finite dimensional and
(3.38)

(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 .

Let and denote
(3.39)

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

(1)
School of Mathematics and Physics, Chongqing University of Posts and Telecommunications
(2)
Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong
(3)
School of Economics and Business Administration, Chongqing University

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