- Research
- Open access
- Published:
Several types of well-posedness for generalized vector quasi-equilibrium problems with their relations
Fixed Point Theory and Applications volume 2014, Article number: 8 (2014)
Abstract
The conceptions of (generalized) Tykhonov well-posedness for generalized vector quasi-equilibrium problems, (generalized) Hadamard well-posedness for parametrically generalized vector quasi-equilibrium problems and (generalized) Tykhonov well-posedness for parametrical system of generalized vector quasi-equilibrium problems are introduced. The metric characterizations and/or sufficient criteria of the proposed well-posedness are presented, and the relations between (generalized) Tykhonov well-posedness for generalized vector quasi-equilibrium problems and that for constrained minimizing problems are discussed. Finally, the relations among several types of the well-posedness are exhibited in detail.
MSC:49K40, 90C31, 90C33.
1 Introduction and preliminaries
Well-posedness is very important for both theory and numerical method of many problems such as optimization problems, optimal control, variations, mathematical programming, fixed-point problems, variational inequality, variational inclusion problems and equilibrium problems (in short, EPs), since it guarantees that for any approximating solution sequence of one of mentioned problems, there must exist a subsequence converging to some correlative solution. The classical concept of well-posedness for unconstrained optimization problem was introduced by Tykhonov [1] in Banach space in 1966. In the same year, this notion was extended to the case of constrained optimization problems by Levitin and Polyak [2]. Ever since then, various types of well-posedness for scalar or vector optimization problems with unconstraint or constraints have been widely focused on. More details on well-posedness for optimization problems, optimal control, variations and mathematical programming and for vector optimization problems can be found in the monographs [3–5] and [6], respectively. In the other directions, some kinds of well-posedness were introduced for other problems, such as fixed-point problems [7–13], variational inequality problems [11–18], vector variational inequality problems [19], variational inclusion problems [10–12, 20–22], complementary problems [23, 24], Nash EPs in the game with two players [25–27] or n-players [13, 28–30] and Pareto-Nash EPs in the game with finite or infinite players [31], and many significant results related to them were obtained.
As understood by Blum and Oetti [32], EPs contain many problems as special cases, for example, optimization problems, fixed-point problems, variational inequality problems, complementary problems and Nash EPs. The discussion on various aspects, such as existence of solutions, iterative algorithms and stability of solutions, etc. for these problems can be classified to the corresponding discussion for general EPs. Some results on different types of well-posedness for EPs were obtained. For instance, Long et al. [33] and Zaslavski [34] introduced the notions of generalized Levitin-Polyak well-posedness for explicit constrained EPs and generic well-posedness for EPs, respectively. Bianchi et al. [35] defined Topt- and Tvi-well-posedness for EPs and proposed the conception of Hadamard well-posedness for parametrical EPs to unify two notions as above. Fang and Hu [36] and Wang and Cheng [37] defined well-posedness for parametrical systems of EPs which are the generalizations of Stampacchia/Minty type variational inequalities and quasi-variational-like inequalities. In addition, Fang et al. [38] introduced generalized well-posedness for a parametrical system of EPs. The sufficient and necessary conditions and metric characterizations of corresponding well-posedness were investigated in [33–38].
Recently, multifarious conceptions of well-posedness for vector equilibrium problems (in short, VEPs) and the related results have been recorded in many literature works. For example, the conceptions of (generalized) Levitin-Polyak well-posedness for VEPs [39, 40], convex symmetric vector quasi-equilibrium problems (VQEPs for brevity) [41], VQEPs without constraints [42] and VQEPs with functional constraints [43–45] were introduced respectively, and their criteria and/or metric characterizations were discussed. Besides, the notions of M- and B-well-posedness for VEPs were presented in [46] and their sufficient conditions were given. The generalized Tykhonov well-posedness for system of VEPs was studied by Peng and Wu [47]. Also, for the well-posedness of parametric strong VQEPs, refer to [48].
Up to the present, there are few literature works to record the well-posedness for EPs involving set-valued objective mappings. The aim of this article is to explore well-posed VEPs with set-valued objective mappings. This paper is organized as follows. A generalized nonlinear scalarization function, which will be used to construct gap functions of generalized vector quasi-equilibrium problems (in short, GVQEPs), is introduced in this section. The metric characterizations and sufficient criteria of (generalized) Tykhonov well-posedness, (G)TWPness for brevity, for GVQEPs are presented by applying Kuratowski noncompactness measure, and the relations between (G)TWPness for GVQEPs and that for constrained minimizing problems are exhibited in Section 2. The sufficient conditions of (generalized) Hadamard well-posedness ((G)HWPness, for brevity) for parametrically GVQEPs are proposed in Section 3. The metric characterizations and sufficient criteria of (G)TWPness for parametrical system of GVQEPs are presented in Section 4. Finally, the relations among the types of proposed well-posedness are illuminated in detail in Section 5.
We first recall some notions and concepts. ℝ, and ℕ denote the sets of real numbers, non-negative real numbers and positive integers, respectively, and denotes the collection of all open neighborhoods of ∗, where ∗ is a point or a set in a topological space.
Definition 1.1 Let X be a topological space and be a nonempty subset. A real-valued function is said to be upper semi-continuous on E if is open for each ; lower semi-continuous on E if is open for each .
Definition 1.2 ([49])
Let X and Y be topological spaces and be a nonempty subset. A set-valued mapping is said to be upper semi-continuous at if for any , there exists such that for all ; lower semi-continuous at if for any and any , there exists such that for all ; upper semi-continuous (resp., lower semi-continuous) on E if G is upper semi-continuous (resp., lower semi-continuous) at each ; closed if its graph is closed in .
Definition 1.3
-
(i)
Let X be a topological space and be a nonempty subset. An extended real-valued function is said to be level-compact on E if is compact for any .
-
(ii)
Further suppose that is a finite-dimensional normed linear space. h is said to be level-bounded on E if E is bounded or
Lemma 1.1 Let X and Y be Hausdorff topological spaces and be a set-valued mapping.
-
(i)
([49]) If X is compact, and G is compact-valued and upper semi-continuous on X, then is compact.
-
(ii)
([50]) If G is upper semi-continuous with closed values, then G is closed.
Definition 1.4 Let be a metric space and be nonempty subsets. The excess of A to B and the Hausdorff distance of A and B are defined as
respectively, where is the distance from x to B.
Definition 1.5 Let be a complete metric space and be a nonempty bounded subset. The Kuratowski noncompactness measure [51] of A is defined as
where is the diameter of . It follows from [51] that
-
(i)
if A is compact;
-
(ii)
, where ;
-
(iii)
, where clA is the closure of A.
A subset D of a linear space Y is called a cone if for all and . Let D be a cone in Y and . D is called proper if . A is called D-closed [52] if is closed and D-bounded [52] if for each neighborhood U of zero in Y, there exists such that . Obviously, any compact subset in Y is both D-closed and D-bounded. Let X and Y be nonempty sets. A set-valued mapping is said to be strict if for any .
In order to construct gap functions of GVQEPs, a generalized nonlinear scalarization function of a set-valued mapping and its properties are listed.
From now on, let be a Hausdorff complete metric space, Y be a real Hausdorff topological vector space and Z be a Hausdorff topological space, let and be nonempty closed subsets, let be a set-valued mapping such that is a proper, closed and convex cone in Y with for each and let be a vector-valued mapping such that
In view of Lemma 3.1 in [53], we can define a general nonlinear scalarization function as follows.
Definition 1.6 Let be a strict compact-valued mapping. A generalized nonlinear scalarization function of G is defined by
It is easy to find differences between the generalized nonlinear scalarization function and the general nonlinear scalarization function given by Qu and Cheng [54]. But if and for all , then both and reduce simultaneously to the nonlinear scalarization function of a single-valued mapping introduced by Chen and Yang [55]. According to Proposition 3.1 in [53], we have the following.
Lemma 1.2 The following assertions are true for each , and :
-
(i)
.
-
(ii)
.
2 (G)TWPness for GVQEPs
In this section, the conceptions of (G)TWPness for GVQEPs are introduced, their metric characterizations are depicted by using Kuratowski noncompactness measure, and some sufficient criteria are presented. Besides, the relations between (G)TWPness for GVQEPs and that for constrained minimizing problems are exhibited. The GVQEP is defined as
where , and are strict set-valued mappings. Ω denotes the solution set of (GVQEP).
If , f is single-valued and for all , then (GVQEP) reduces to VEP described as:
For each , the following assumptions are introduced:
Definition 2.1 A type I ε-approximating solution set (resp., type II ε-approximating solution set) of (GVQEP) is defined by
(resp., ).
A sequence is called a type I approximating solution sequence, ASS1 for brevity (resp., type II approximating solution sequence, ASS2 for brevity) of (GVQEP) if there exists with such that (resp., ).
Definition 2.2 (GVQEP) is said to be generalized type I Tykhonov well-posed, GTWP1 for brevity (resp., generalized type II Tykhonov well-posed, GTWP2 for brevity) if and for any ASS1 (resp., ASS2) of (GVQEP), there exists a subsequence such that ; to be type I Tykhonov well-posed, TWP1 for brevity (resp., type II Tykhonov well-posed, TWP2 for brevity) if it is GTWP1 (resp., GTWP2) and Ω is a singleton.
When (GVQEP) reduces to (VEP), generalized type I Tykhonov well-posedness (GTWPness1 for brevity) and generalized type II Tykhonov well-posedness (GTWPness2 for brevity) for (GVQEP) become type I Levitin-Polyak well-posedness and type II Levitin-Polyak well-posedness for (VEP), respectively, which were discussed by Li and Li [39] in the case that X and Y are locally convex topological vector spaces, where X is equipped with a metric d compatible with its topology, F is a nonempty closed convex subset, and f is a continuous mapping.
Remark 2.1 (i) An ASS2 of (GVQEP) must be its ASS1. So GTWPness1 (resp., TWPness1) for (GVQEP) implies its GTWPness2 (resp., TWPness2), where TWPness1 and TWPness2 are the abbreviations of type I Tykhonov well-posedness and type II Tykhonov well-posedness, respectively.
(ii) Clearly, if P is closed-valued. In addition, for all . In fact, for any , we have and
Then for any . It follows from (2.4) and (1.1) that (2.2) holds.
(iii) (GVQEP) is GTWP1 if and only if Ω is nonempty compact and for its any ASS1 . Assume that Ω is compact, (GVQEP) is GTWP2 if and only if and for its any ASS2 . In addition, (GVQEP) is TWP1 (resp., TWP2) if and only if and for any ASS1 (resp., ASS2) of (GVQEP).
The following example shows that neither the GTWPness1 for (GVQEP) nor the compactness of Ω can be deduced from the GTWPness2 for (GVQEP).
Example 2.1 Let , , , , for all and
(GVQEP) is to find such that
Obviously, is noncompact and so (GVQEP) is not GTWP1 by Remark 2.1(iii). However, (GVQEP) is GTWP2. In fact, for any ASS2 of (GVQEP), let with such that . For and , (2.1) and (2.2) hold trivially.
It is impossible that for sufficiently large . Otherwise, for some . By (2.3), there exists such that . We have
This is absurd for sufficiently large n. Therefore, without loss of generality, . It follows from (2.3) that there exists such that . Then
The fact proceeds from (2.5) and .
2.1 Metric characterization of (G)TWPness for (GVQEP)
The metric characterizations of (G)TWPness for GVQEPs are depicted by using Kuratowski noncompactness measure and the corresponding results are obtained as follows.
Lemma 2.1 Suppose that
-
(a1) f is lower semi-continuous on ;
-
(a2) P is compact-valued and upper semi-continuous on E;
-
(a3) Q is lower semi-continuous on E;
-
(a4) W is upper semi-continuous on E, where is defined as for all ;
-
(a5) e is continuous on E.
Then
-
(i)
is closed for each ;
-
(ii)
.
Proof (i) For each fixed , assume that with . Then
As a result of (a2) and Lemma 1.1(ii), P is closed. Letting in (2.6), we have
As a matter of fact, if we take , then A is compact and so is by Lemma 1.1(i). Since is compact, there exists such that and some subsequence of , still denoted by , converging to some point by , the compactness of and the closeness of P. Thus,
For any , there exists such that by virtue of (a3). Likewise, for any , there exists such that by (a1). This, together with (2.7), implies that , in other words,
It is easy to see that W is closed by Lemma 1.1. It follows that from (2.9), the continuity of e and the closeness of W, and so
Thus and is closed.
(ii) stems easily from Remark 2.1(ii). For any , (2.8) and (2.10) hold for any . Then by (a2), and for any and by (2.10). Letting , we have , and so . Consequently, and . □
Lemma 2.2 Suppose that (a1)-(a5) and
(a6) for all
hold. Then
-
(i)
is closed for each ;
-
(ii)
.
Proof (i) For each , let with . It is enough to testify that satisfies (2.3) by Lemma 2.1(i). As a matter of fact, for some according to (a6). As a result, owing to (1.1).
(ii) by Lemma 2.1(ii). We only need to show that satisfies (2.3) for any and , while this can be deduced easily from the proof of (i). □
Theorem 2.1 Suppose that E is bounded.
-
(i)
If (GVQEP) is GTWP 1, then
(2.11) -
(ii)
If (a1)-(a5) hold, then (2.11) implies that (GVQEP) is GTWP 1.
Proof (i) Since (GVQEP) is GTWP1, Ω is nonempty compact and so for all . Also by the compactness of Ω and by Remark 2.1(ii). This deduces that
It is enough to testify that as . Otherwise, there exist , and such that for all . Clearly, is an ASS1 of (GVQEP). Thus by Remark 2.1(iii), which contradicts for all .
(ii) For any ASS1 of (GVQEP), let with such that . In view of Lemma 2.1 and the boundedness of E, and is a nonempty bounded closed set. For any and ,
by (1.1). Therefore, , and so is increasing with . This, together with , implies that Ω is nonempty compact and
by Kuratowski theorem [51]. Resultingly, and (GVQEP) is GTWP1 by Remark 2.1(iii). □
Similarly, the following result can be proved by using Lemma 2.2.
Theorem 2.2 Assume that E is bounded and Ω is compact.
-
(i)
If (GVQEP) is GTWP 2, then
(2.12) -
(ii)
If (a1)-(a6) are satisfied, then (2.12) implies that (GVQEP) is GTWP 2.
When Ω is a singleton, the following corollary that shows the metric information of TWPness1 and TWPness2 for (GVQEP) follows from Theorems 2.1 and 2.2.
Corollary 2.1 Suppose that E is bounded and Ω is a singleton.
-
(i)
If (GVQEP) is TWP 1 (resp., TWP 2), then (2.11) (resp., (2.12)) holds.
-
(ii)
If (a1)-(a5) (resp., (a1)-(a6)) hold, then (2.11) (resp., (2.12)) implies that (GVQEP) is TWP 1 (resp., TWP 2).
2.2 Relations between (G)TWPness for (GVQEP) and that for constrained minimizing problem
First, we introduce the constrained minimizing problem described as follows:
where is a proper function and is a strict set-valued mapping. The optimal set and optimal value of (CMP) are denoted by argminϕ and , respectively. In this subsection, the equivalent relations between (G)TWPness for (GVQEP) and that for (CMP) are discussed, where a gap function of (GVQEP) is taken as the objective function ϕ of (CMP).
Definition 2.3 A sequence is called a type I minimizing sequence, MS1 for brevity (resp., type II minimizing sequence, MS2 for brevity) of (CMP) if the following (2.13) and (2.14) (resp., (2.13) and (2.15)) hold.
Definition 2.4 (CMP) is said to be GTWP1 (resp., GTWP2) if and for any MS1 (resp., MS2) of (CMP), there exists a subsequence such that ; to be TWP1 (resp., TWP2) if it is GTWP1 (resp., GTWP2) and argminϕ is a singleton.
Definition 2.5 is called a gap function of (GVQEP) if
-
(i)
for all ;
-
(ii)
if and only if .
Further suppose that f is compact-valued in this subsection.
Lemma 2.3 If (a6) holds, then ϕ is a gap function of (GVQEP), where is defined by
and
Proof Clearly, for all . Otherwise, for some . Then for all , which contradicts the fact that is real-valued.
It follows from (a6) and Lemma 1.2(ii) that for each , for some . This deduces that
Finally, since
⟺ , for all (By (2.17));
⟺ , for all (By Lemma 1.2(i));
⟺ ,
ϕ is a gap function of (GVQEP). □
In general, ϕ is required to be lower semi-continuous. It is natural to expect the lower semi-continuity of the constructed gap function. Now assume that ϕ appearing in the rest of this section is defined as (2.16).
Proposition 2.1 If (a1)-(a5) hold, then ϕ is lower semi-continuous on E. If, further, (a6) holds and , then .
Proof In order to verify that ϕ is lower semi-continuous on E, it is enough to show that is closed for each . In fact, let with . Then
and (2.7) holds by Lemma 1.2(i). It is easy to see that (2.10) holds by (2.7) and a similar argument given in the proof of Lemma 2.1. Applying Lemma 1.2(i) again, we have
that is, . If, further, (a6) holds and , then by Lemma 2.3 and Definition 2.5. □
Theorem 2.3 Assume that (a6) holds. Then (GVQEP) is GTWP 1 (resp., GTWP 2) if and only if so is (CMP) with the objective function ϕ.
Proof ϕ is a gap function of (GVQEP) owing to Lemma 2.3. Thus if and only if argminϕ. Here, . Two equivalent relations are listed as follows:
-
1∘
is an ASS1 of (GVQEP).
⟺ There exists with such that ;
⟺ There exists with such that and for all . (By Lemma 1.2(i));
⟺ There exists with such that and
⟺ and ;
⟺ is an MS1 of (CMP).
-
2∘
is an ASS2 of (GVQEP).
⟺ There exists with such that ;
⟺ There exists with such that and for some ;
⟺ (A): There exists with such that and for some . (By Lemma 1.2(ii));
⟺ (B): There exists with such that and
⟺ and . (By 1∘);
⟺ is an MS2 of (CMP).
Now we shall prove the equivalence of (A) and (B). In fact, (A) implies (B) by taking . On the other hand, if (B) holds, then for fixed and for any , there exists such that
We can choose and is the corresponding point such that the above inequality holds. Therefore, (A) holds by taking .
It follows from 1∘ (resp., 2∘) that (GVQEP) is GTWP1 (resp., GTWP2) if and only if so is (CMP). □
Corollary 2.2 Assume that (a6) holds. Then (GVQEP) is TWP 1 (resp., TWP 2) if and only if so is (CMP) with the objective function ϕ.
When the assumptions in Theorem 2.3 are satisfied, we see that if (GVQEP) is GTWP1 (resp., GTWP2), then for any MS1 (resp., MS2) of (CMP) and for some , (resp., ) implies that , that is, . It is reasonable to try estimating a bound below of by using . For the sake of this intention, a forcing function with parameter is introduced.
A real-valued bifunction is called a forcing function with parameter (where S is a parameter set) if
Theorem 2.4 Suppose that (a6) holds and ϕ is the objective mapping of (CMP). Then the following assertions are equivalent:
-
(i)
(GVQEP) is GTWP 1;
-
(ii)
Ω is nonempty compact and there exists a forcing function with parameter (where S is the parameter set) such that
(2.20)
where
Proof By virtue of Lemma 2.3, ϕ is a gap function of (GVQEP).
Let (i) hold. Ω is nonempty compact by Remark 2.1(iii). Define as
where S and T are defined by (2.21). If , then by the compactness of Ω and so according to Definition 2.5(ii). So , that is, c satisfies (2.18). Let and with . Since
there exists such that , and by the definition of infimum. Since , is an MS1 of (CMP) and also an ASS1 of (GVQEP) in view of the proof of Theorem 2.3. Then (2.19) follows from the GTWPness1 for (GVQEP) and Remark 2.1(iii). Therefore, the assertion (ii) is true.
Suppose that (ii) holds. For any ASS1 of (GVQEP), (2.20) deduces
Setting and , we have . By the same argument given in the proof of Theorem 2.3, is an MS1 of (CMP). Therefore, . On the other hand, since for all . Thus and , and so by (2.19). This, together with the compactness of Ω and Remark 2.1(iii), implies that (GVQEP) is GTWP1. □
Similarly, we can prove the following result by using Remark 2.1(iii).
Theorem 2.5 If (a6) holds and Ω is compact, then the following assertions are equivalent:
-
(i)
(GVQEP) is GTWP 2;
-
(ii)
and there exists a forcing function with parameter (where S is the parameter set) such that (2.20) holds, where S and T are defined by (2.21).
Corollary 2.3 If (a6) holds, then the following assertions are equivalent:
-
(i)
(GVQEP) is TWP 1 (resp., TWP 2);
-
(ii)
Ω is a singleton and there exists a forcing function with parameter (where S is the parameter set) such that (2.20) holds, where S and T are defined by (2.21).
2.3 Sufficient criteria of (G)TWP for (GVQEP)
In this subsection, we shall list some sufficient criteria of (G)TWPness for (GVQEP).
Theorem 2.6 Let (a1)-(a5) hold and . If
(b1) is compact for some
holds, then (GVQEP) is GTWP 1 and also GTWP 2.
Proof For any ASS1 of (GVQEP), let with such that .
Since , for sufficiently large . So has a subsequence, still denoted by , such that . It follows from (a2) that . For any , there exists such that by (a3). For each , we have . In view of the lower semi-continuity of f, for any , can be chosen to satisfy and . By letting , by the closeness of W, and so . Therefore, . (GVQEP) is GTWP1 and also GTWP2. □
It is easy to see that the conclusion of Theorem 2.6 still holds if (b1) is replaced by ‘E is compact’. In addition, if f is compact-valued, then (b1) can also be substituted by:
(b2) ϕ is compact-level on for some , or
(b3) X is a finite-dimensional normed linear space and ϕ is level-bounded on E.
Indeed is compact for each by (b2) and so (b1) holds. Define for each . Then is bounded by (b3), otherwise, there exists such that and . This is absurd according to (b3). is closed since ϕ is lower semi-continuous by Proposition 2.1 and so it is compact. Clearly, . Thus, (b1) is satisfied by Lemma 2.1(i).
In fact, GTWPness1 or GTWPness2 for (GVQEP) can fail without the lower semi-continuity of f. The following example only states the fact under the assumption that ‘E is compact’.
Example 2.2 Let , , , , , , for all , and
(GVQEP) is to find such that
It is easy to know . (GVQEP) is neither GTWP2 nor GTWP1. As a matter of fact, if and . Again, taking , we have
Thus , in other words, is an ASS2 of (GVQEP), but . It is worth noting that (a2)-(a5) are satisfied, but f is not lower semi-continuous at . Indeed, let . For and for any , does not converge to .
By the argument given in the proof of Theorem 3.6, it is easy to yield the following.
Theorem 2.7 If (a2) and (a4) are substituted by ‘E is closed and for all ’ and ‘W is closed’ in Theorem 2.6, respectively, then the conclusion still holds.
Corollary 2.4 If, further, assume that Ω is a singleton in Theorem 2.6 (resp., 2.7), then (GVQEP) is TWP 1 and also TWP 2.
Furthermore, if is a locally convex topological space and E is a closed convex subset and if f is single-valued and for all in Theorem 2.7, then Theorem 2.7 reduces to Corollary 3.1 in [39].
3 (G)HWPness for parametrically GVQEPs
In this section, the conceptions of HWPness and GHWPness for parametrically GVQEPs are introduced and their sufficient criteria are proposed. Consider the following parametrically GVQEP: For any given ,
where , and are strict set-valued mappings, and is a Hausdorff metric space (parametric space). denotes the solution set of (GVQEP) p for each .
If , , , for all and h is single-valued, then (GVQEP) p reduces to the following parametrical EP:
Definition 3.1 (GVQEP) p is said to be generalized Hadamard well-posed (in short, GHWP) at if and for any with and , there exists a subsequence of such that ; to be Hadamard well-posed (in short, HWP) at if it is GHWP at and is a singleton.
Remark 3.1 (i) Obviously, (GVQEP) p is GHWP at if and only if is nonempty compact and for any with and any , . (GVQEP) p is HWP at if and only if and for any with and any , .
(ii) If , , , for all and h is single-valued, then the HWPness at for (GVQEP) p reduces to the HWPness at for (EP) p , which was investigated by Bianchi et al. [35].
In general, HWPness for (GVQEP) p at implies that as , since
where . However the converse fails to be true. See the following example.
Example 3.1 Let , , , for all and
Then
Set . for any with , but (GVQEP) p is not HWP at . In fact, by taking , .
Theorem 3.1 Let E be compact, and . If (a2)-(a4) and
(a7) h is lower semi-continuous at for each
hold, then (GVQEP) p is GHWP at . If, further, is a singleton, then (GVQEP) p is HWP at .
Proof For any and , we have and
that is,
for all . Without loss of generality, assume that . For any , there exists such that by (a3) and for any , there exists such that by (a7). Since both P and W are closed, and by (3.1), and so
This deduces that and (GVQEP) p is GHWP at . The second conclusion follows directly from Definition 3.1. □
The conclusion in Theorem 3.1 is still true if (a2) is replaced by the assumption that P is closed-valued and upper semi-continuous on E. Theorem 3.1 can be false without (a7). See the instance as follows.
Example 3.2 Let , , , , for all , and h is defined by
(resp.,
(GVQEP) p is not HWP (resp., GHWP) at . It is worth noting that h is not lower semi-continuous at . Indeed, take . For and for any , it is impossible that the case happens.
4 P(G)TWPness for parametrical system of GVQEPs
The main topic of this section is (G)TWPness for parametrical system of GVQEPs, which is a common extension of both (G)TWPness and (G)HWPness, and its the metric characterizations and sufficient conditions. The parametrical system of GVQEPs is
where (GVQEP) p for each is described at the beginning of Section 4 and is a Hausdorff metric space (parametric space).
For any and , we list some conditions as follows:
and denote
Incidentally, for any given , define as
Then the GVQEP related to is
Clearly, , and are just the solution set, the type I ε-approximating solution set and the type II ε-approximating solution set of p-(GVQEP), respectively, for each and .
Definition 4.1 For each and , the type I -approximating solution set related to (resp., type II -approximating solution set related to ) of is defined as
A sequence is called a type I approximating solution sequence related to , for brevity (resp., type II approximating solution sequence related to , for brevity) of if there exist with such that (resp., ).
Remark 4.1 (i) An of is its for each .
(ii) Apparently,
for each and . Also, for each and ,
on the assumption of
(a8) for all and .
Definition 4.2 is said to be GTWP1 (resp., GTWP2) if for each , and for any (resp., ) of , there exists a subsequence such that ; to be TWP1 (resp., TWP2) if it is GTWP1 (resp., GTWP2) and is a singleton for each .
Remark 4.2 (i) The GTWPness1 for implies its GTWPness2 according to Remark 4.1(i).
(ii) is GTWP1 if and only if for each , is nonempty compact and for any for , . When is compact for each , is GTWP2 if and only if for any , and for any for , . In addition, is TWP1 (resp., TWP2) if and only if for each , and for any (resp., ) for , .
Lemma 4.1 Assume that (a2)-(a5) hold and
(a9) h is lower semi-continuous on ;
(a10) Φ is compact.
Then the following facts are true:
-
(i)
is closed for each and .
-
(ii)
for each .
Proof (i) Let with for each . Then there exists with such that . Without loss of generality, . Thus, and
By (a10), without loss of generality, , which implies that . Applying the analogous argument given in the proof of Lemma 2.1(i), we have and
Accordingly, and so by .
(ii) Obviously, on account of (4.4). Let for all . Then there exist and such that
Therefore, there exists such that and . As a result, , and for all . Then for all proceeds from (a2), (a3) and (a10) and so . □
By a resemblant argument given in the proof of Lemma 2.2, we have the following.
Lemma 4.2 Let (a2)-(a5) and (a9)-(a10) hold. Then the following assertions hold:
-
(i)
is closed for each and .
-
(ii)
for each .
Theorem 4.1 Let E be bounded.
-
(i)
If is GTWP 1, then for each ,
(4.6) -
(ii)
Suppose that (a2)-(a5) and (a9)-(a10) hold. If (4.6) holds for each , then is GTWP 1.
Proof (i) Since is GTWP1, is nonempty compact for each . Then and
for any and . It is enough to testify as for each . Otherwise, there exist , and with and such that . This says that is an of . By Remark 4.2(ii), , which contradicts .
(ii) For any of , there exist with such that by Remark 4.2(ii). In view of Lemma 4.1 and Kuratowski theorem [51], is nonempty compact and . Thus . It follows that is GTWP1. □
Theorem 4.2 Let E be bounded.
-
(i)
Suppose that for each , is compact. If is GTWP 2, then for ,
(4.7) -
(ii)
Assume that (a2)-(a5) and (a8)-(a10) hold. If (4.7) holds for each , then is GTWP 2.
Proof This proof is completed by using Lemma 4.2 and a similar argument proposed in the proof of Theorem 2.2 and is omitted. □
Corollary 4.1 Assume that E is bounded and is a singleton for each .
-
(i)
If is TWP 1 (resp., TWP 2), then (4.6) (resp., (4.7)) holds for each .
-
(ii)
Assume that (a2)-(a5) and (a9)-(a10) (resp., (a2)-(a5) and (a8)-(a10)) hold. If (4.6) (resp., (4.7)) holds for each , then is TWP 1 (resp., TWP 2).
By a similar method of the proof in Theorem 2.6, we have the following.
Theorem 4.3 Let (a2)-(a5) and (a9)-(a10) hold and for each . If
(b4) For each , is compact for some ; or
(b5) X is a finite-dimensional normed linear space and for each , is bounded for some
holds, then is GTWP 1 and also GTWP 2.
Corollary 4.2 Further suppose that is a singleton for each in Theorem 4.3. Then is TWP 1 and also TWP 2.
5 Relations among the types of proposed well-posedness
In this section we are interested in the comparison among the types of proposed well-posedness defined in previous sections.
It seems on the surface to have no relations between the (G)HWPness for (GVQEP) p and (G)TWPness for (GVQEP). However, if there are some connections between their objective mappings, we may discuss the relations.
Example 5.1 Let , and let be the objective mapping of (GVQEP) p and be the objective mapping of (GVQEP), where
And let Ω be the solution set of (GVQEP) and (resp., ) be the type I p-approximating solution set (resp., type II p-approximating solution set) of (GVQEP). It follows that
and
if (a6) holds. If, further, for all , then both (5.2) and (5.3) are indeed equalities.
Figure 1 illuminates the relations among GHWPness for (GVQEP) p , GTWPness1 and GTWPness2 for (GVQEP) when the relation of their objective mapping is defined by (5.1). If, further, is a singleton, then Figure 1 illuminates the relations among HWPness for (GVQEP) p , TWPness1 and TWPness2 for (GVQEP) when the relation of their objective mapping is defined by (5.1).
It follows from (4.4) and (4.5) that GTWPness1 and GTWPness2 for imply that GTWPness1 and GTWPness2 for p-(GVQEP) for each , respectively, and GTWPness1 for also implies GHWPness for (GVQEP) p at each , while GTWPness2 implies GHWPness for (GVQEP) p at each if (a10) holds. But these converses fail to hold. See the following example.
Example 5.2 Let , , , , for all and h defined by
Then ,
and
for each and . It is clear that (GVQEP) p is GHWP at each , and p-(GVQEP) is GTWP1 and GTWP2 for each , while is neither GTWP1 nor GTWP2. In fact, take , , and . It is easy to see that is an ASS1(0) (resp., ASS2(0)) of , but it has no convergent subsequence.
Figure 2 illuminates the relation between GTWPness for and GTWPness for p-(GVQEP) for each , and that between GTWPness for and GHWPness for (GVQEP) p at each .
References
Tykhonov AN: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 1966, 6: 28–33. 10.1016/0041-5553(66)90003-6
Levitin ES, Polyak BT: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 1966, 7: 764–767.
Dontchev AL, Zolezzi T Lecture Notes in Mathematics 1543. In Well-Posed Optimization Problems. Springer, Berlin; 1993.
Zaslavski AJ Springer Optimization and Its Applications. In Optimization on Metric and Normed Spaces. Springer, New York; 2010.
Zaslavski AJ Springer Optimization and Its Applications. In Nonconvex Optimal Control and Variational Problems. Springer, New York; 2013.
Papalia, M: On well-posedness in vector optimization. Università degli studi di Bergamo, Bergamo, Italy (2010)
Reich S, Zaslavski AJ: A note on well-posed null and fixed point problems. Fixed Point Theory Appl. 2005, 2005: 207–211.
Chifu C, Petruşl G: Well-posedness and fractals via fixed point theory. Fixed Point Theory Appl. 2008., 2008: Article ID 645419
Petruşel A, Rus IA, Yao JC: Well-posedness in the generalized sense of the fixed-point problems for multivalued operators. Taiwan. J. Math. 2007, 3(11):903–914.
Lemaire B, Ould Ahmed Salem C, Revalski JP: Well-posedness by perturbations of variational problems. J. Optim. Theory Appl. 2002, 2(115):345–368.
Fang YP, Huang NJ, Yao JY: Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. J. Glob. Optim. 2008, 41: 117–133. 10.1007/s10898-007-9169-6
Ceng LC, Yao JC: Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed point problems. Nonlinear Anal., Theory Methods Appl. 2008, 69: 4585–4603. 10.1016/j.na.2007.11.015
Yu J, Yang H, Yu C: Well-posed Ky Fan’s point, quasi-variational inequality and Nash equilibrium problems. Nonlinear Anal., Theory Methods Appl. 2007, 66(4):777–790. 10.1016/j.na.2005.10.018
Lignola MB: Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl. 2006, 1(128):119–138.
Ceng LC, Hadjisavvas N, Schaible S, Yao JC: Well-posedness of mixed quasivariational-like inequalities. J. Optim. Theory Appl. 2008, 139: 109–125. 10.1007/s10957-008-9428-9
Huang XX, Yang YQ, Zhu DC: Levitin-Polyak well-posedness of variational inequalities problems with functional constraints. J. Glob. Optim. 2009, 44: 159–174. 10.1007/s10898-008-9310-1
Jiang B, Zhang J, Huang XX: Levitin-Polyak well-posedness of generalized quasivariational inequalities with functional constraints. Nonlinear Anal., Theory Methods Appl. 2009, 70: 1492–1530. 10.1016/j.na.2008.02.029
Zhang J, Jiang B, Huang XX: Levitin-Polyak well-posedness in vector quasivariational inequality problems with functional constraints. Fixed Point Theory Appl. 2010., 2010: Article ID 984074
Xu Z, Zhu DC, Huang XX: Levitin-Polyak well-posedness in generalized vector quasivariational inequality problems with functional constraints. Math. Methods Oper. Res. 2008, 3(67):505–524.
Wang SH, Huang NJ: Levitin-Polyak well-posedness for generalized quasi-variational inclusion and disclusion problems and optimization problems with constraints. Taiwan. J. Math. 2012, 1(16):237–257.
Wang SH, Huang NJ, Wong MM: Strong Levitin-Polyak well-posedness for generalized quasi-variational inclusion problems with applications. Taiwan. J. Math. 2012, 2(16):665–690.
Lin LJ, Chuang CS: Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint. Nonlinear Anal., Theory Methods Appl. 2009, 70: 3609–3617. 10.1016/j.na.2008.07.018
Heemels WPMH, Schumacher JM, Weiland S: Well-posedness of linear complementarity systems. 3. Decision and Control 1999, 3037–3042.
Heemels PMH, Çamlibel MKC, Van der Schaft AJ, Schumacher JM: Well-posedness of the complementarity class of hybrid systems. Proc. IFAC 15th Triennial World Congress 2002.
Margiocco M, Patrone F, Chicco LP: A new approach to Tikhonov well-posedness for Nash equilibria. Optimization 1997, 4(40):385–400.
Margiocco M, Patrone F, Chicco LP: Metric characterizations of Tikhonov well-posedness in value. J. Optim. Theory Appl. 1999, 2(100):377–387.
Lignola MB, Morgan J: α -Well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints. J. Glob. Optim. 2006, 3(36):439–459.
Margiocco M, Patrone F, Chicco LP: On the Tikhonov well-posedness of concave games and Cournot oligopoly games. J. Optim. Theory Appl. 2002, 2(112):361–379.
Morgan J: Approximations and well-posedness in multicriteria games. Ann. Oper. Res. 2005, 137: 257–268. 10.1007/s10479-005-2260-9
Scalzo W: Hadamard well-posedness in discontinuous non-cooperative games. J. Math. Anal. Appl. 2009, 360: 697–703. 10.1016/j.jmaa.2009.07.007
Peng JW, Wu SY: The well-posedness for multiobjective generalized games. J. Optim. Theory Appl. 2011, 150: 416–423. 10.1007/s10957-011-9839-x
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.
Long XJ, Huang NJ, Teo KL: Levitin-Polyak well-posedness for equilibrium problems with functional constraints. J. Inequal. Appl. 2008., 2008: Article ID 657329
Zaslavski AJ: Generic well-posedness for a class of equilibrium problems. J. Inequal. Appl. 2008., 2008: Article ID 581917
Bianchi M, Kassay G, Pini R: Well-posed equilibrium problems. Nonlinear Anal., Theory Methods Appl. 2010, 72: 460–468. 10.1016/j.na.2009.06.081
Fang YP, Hu R: Parametric well-posedness for variational inequalities defined by bifunctions. Comput. Math. Appl. 2007, 53: 1306–1316. 10.1016/j.camwa.2006.09.009
Wang HJ, Cheng CZ: Parametic well-posedness for quasivariational-like inequalities. Far East J. Math. Sci. 2011, 55: 31–47.
Fang YP, Hu R, Huang NJ: Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput. Math. Appl. 2008, 1(55):89–100.
Li SJ, Li MH: Levitin-Polyak well-posedness of vector equilibrium problems. Math. Methods Oper. Res. 2009, 69: 125–140. 10.1007/s00186-008-0214-0
Peng JW, Wang Y, Zhao LJ: Generalized Levitin-Polyak well-posedness of vector equilibrium problems. Fixed Point Theory Appl. 2009., 2009: Article ID 684304
Zhang WY: Well-posedness for convex symmetric vector quasi-equilibrium problems. J. Math. Anal. Appl. 2012, 387: 909–915. 10.1016/j.jmaa.2011.09.052
Wang G, Huang XX, Zhang J, Chen GY: Levitin-Polyak well-posedness of generalized vector equilibrium problems with functional constraints. Acta Math. Sci. 2010, 30(5):1400–1412. 10.1016/S0252-9602(10)60132-4
Peng JW, Wang Y, Wu SY: Levitin-Polyak well-posedness for vector quasi-equilibrium problems with functional constraints. Taiwan. J. Math. 2012, 2(16):635–649.
Peng JW, Wang Y, Wu SY: Levitin-Polyak well-posedness of generalized vector equilibrium problems. Taiwan. J. Math. 2011, 5(15):2311–2330.
Peng JW, Wang Y, Wu SY: Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints. J. Glob. Optim. 2012, 52: 779–795. 10.1007/s10898-011-9711-4
Bianchi M, Kassay G, Pini P: Well-posedness for vector equilibrium problems. Math. Methods Oper. Res. 2009, 70: 171–182. 10.1007/s00186-008-0239-4
Peng JW, Wu SY: The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems. Optim. Lett. 2010, 4: 501–512. 10.1007/s11590-010-0179-9
Li QY, Wang SH: Well-posedness for parametric strong vector quasi-equilibrium problems with applications. Fixed Point Theory Appl. 2011., 2011: Article ID 62
Aubin JP, Ekeland I: Applied Nonlinear Analysis. Wiley, New York; 1984.
Aubin JP, Cellina A: Differential Inclusion. Springer, Berlin; 1994.
Kuratowski C 1. In Topologie. Panstwowe Wydawnictwo Naukove, Warsaw; 1952.
Luc DT Lecture Notes in Economics and Mathematical Systems 319. In Theory of Vector Optimization. Springer, Berlin; 1989.
Sach PH, Tuan LA: New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems. J. Optim. Theory Appl. 2013, 157: 347–364. 10.1007/s10957-012-0105-7
Qu DN, Cheng CZ: Existence of solutions for generalized vector quasi-equilibrium problems by scalarization method with applications. Abstr. Appl. Anal. 2013., 2013: Article ID 916089
Chen GY, Yang XQ, Yu H: A nonlinear scalarization function and generalized vector quasi-equilibrium problems. J. Glob. Optim. 2005, 32: 451–466. 10.1007/s10898-003-2683-2
Acknowledgements
This research is supported by the Doctoral Fund of innovation of Beijing University of Technology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
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
Qu, Dn., Cheng, Cz. Several types of well-posedness for generalized vector quasi-equilibrium problems with their relations. Fixed Point Theory Appl 2014, 8 (2014). https://doi.org/10.1186/1687-1812-2014-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-8