- Research Article
- Open Access
- Published:
Saddle Point Problems, Bilevel Problems, and Mathematical Program with Equilibrium Constraint on Complete Metric Spaces
Journal of Inequalities and Applications volume 2010, Article number: 306403 (2010)
Abstract
We apply an existence theorem of variational inclusion problem on metric spaces to study optimization problems, set-valued vector saddle point problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces. We study these problems without any convexity and compactness assumptions. Our results are different from any existence results of these types of problems in topological vector spaces.
1. Introduction
Let 
and 
be two metric spaces, let 
be a real Hausdorff topological vector space ordered by a nonempty pointed closed convex 
in 
with nonempty interior, and let 
be a real Banach space ordered by a nonempty pointed closed convex 
with nonempty interior. Let 
, 
, and 
be multivalued maps. Throughout this paper, we use these notations unless specified otherwise. In this paper, the following vector mathematical programs with equilibrium constraint on metric spaces are considered.
(MPEC-1)
is subject to 
, and 
for all 
.
(MPEC-2)
is subject to 
, and 
for all 
.
If 
, 
, and 
is a real function, then problems (MPEC-1) and (MPEC-2) are reduced to the following problem:
(MPEC-3)
is subject to 
, and 
for all 
.
We also study the following bilevel problems.
(BL-1)
is subject to 
, for 
, 
is a solution of problem 
: 
(i.e., for 
, 
), for 
, and 
is a solution of problem 
: 
. (i.e., for 
, 
).
(BL-2)
is subject to 
, 
for all 
, and 
for all 
(BL-3)
is subject to 
, for 
, 
is a solution of problem 
: 
and (i.e., for 
, 
), for 
, 
is a solution of problem 
: 
. (i.e., for 
, 
).
If 
is a single valued function, then (BL-2) is reduced to the following problem.
(BL-4)
is subject to 
, for 
, and 
is a solution of problem and 
: 
; and for 
, 
is a solution of problem 
: 
.
Problem (BL-1) has applications in real world. Let 
be the set of government's agricultural policies and let 
be the set of government's industrial policies. For each 
and 
, let 
be the amount of money that the government uses to promote agriculture and industrial developments, and let 
be the degree of industrial development. We suppose that as the industry becomes more and more developed, the losses from the agriculture sector rise accordingly. Therefore, the solution of problem (BL-1) represents the government's best policy to promote the development of the industry so that the losses from agriculture sector will be as minimal as possible, while the amount of money that the government uses to promote the policies can be the lowest possible.
Mathematical program with equilibrium constraint and bilevel problem represent two important classes of optimization problems which have been investigated in a large number of articles and books. We find in the literatures that Luo et al. [1], Stein and Still [2], Stein [3], Birbil et al. [4], Liou et al. [5], Lin and Still [6],Lin [7], as well asLin and Hsu [8] have studied mathematical program with equilibrium constraint and bilevel problem on topological vector spaces. As usual in linear and nonlinear optimization, these studies mainly deal with optimality conditions and numerical methods to solve these problems and typically the existence of feasible points is tacitly assumed. Besides, the domains of the functions they consider are subsets of topological vector spaces and certain convexity assumptions on the functions are needed. In this paper, we study these problems with functions defined on metric spaces, so we do not need any convexity assumptions on the functions we consider. We study the existence theorems of solutions for (MPEC-1), (MPEC-2), (BL-1), (BL-2), and (BL-3). To the best of our knowledge, there is no result of these types of problems on metric space.
In this paper, we also study the following loose set-valued vector saddle point problems.
(LSP-1)For each 
, find 
such that 
, 
, and 
.
(LSP-2)For each 
, find 
such that 
, 
and 
for all 
and all 
.
If 
is a map, then problems (LSP-1) and (LSP-2) are reduced to the following vector saddle point problems (VSP).
(i)Find 
such that 
.
There are many results on loose saddle point problems, vector saddle point, and saddle point problems (see, e.g., [9–17]). But to the best of our knowledge, there are no existence theorems in the literatures for loose saddle point and vector saddle point problems for functions defined on the product of metric spaces. We study the loose saddle point problems and vector saddle point problems for functions defined on the product of metric spaces. We do not assume any compact assumptions on the spaces and convexity assumptions on the maps we consider. As for applications of our existence theorems on saddle point problems, we study bilevel problems on metric spaces. We also study mathematical programs with equilibrium constraint. Our results on mathematical programs with equilibrium constraint, bilevel problems, loose saddle point problems, and vector saddle point problems are different from any existence results of these types of problems in the literatures.
2. Preliminaries
Let 
and 
be topological spaces (in short t.s.), 
be a multivalued map. 
is said to be u.s.c. (resp., l.s.c.) at 
if for every open set 
in 
with 
(resp., 
), there exists an open neighborhood 
of 
such that 
(resp., 
) for all 
; 
is said to be u.s.c. (resp., l.s.c.) on 
if 
is u.s.c. (resp., l.s.c.) at every point of 
; 
is continuous at 
if 
is both u.s.c. and l.s.c. at 
; 
is said to be closed if 
=
is a closed set in 
; 
is said to be open if 
is an open set in 
. For a subset 
of topological space 
, let 
denote the closure of 
.
Lemma (see [18]).
Let 
and 
be topological spaces, and let 
be a multivalued map. Then 
is l.s.c. at 
if and only if for any 
and any net 
in 
converges to 
, there exists a net 
such that 
for all 
and 
.
Lemma (see [19]).
Let 
and 
be Hausdorff topological spaces, and let 
be a multivalued map. (i) If 
is an u.s.c. multivalued map with nonempty closed values, then 
is closed; (ii) if 
is a compact set and 
is an u.s.c. multivalued map with nonempty compact values, then 
is compact.
Definition.
Let 
be a nonempty subset of a t.v.s. 
ordered by a nonempty pointed closed convex 
. An element 
is said to be a minimal (resp., maximal) point of 
if 
(resp., 
). Here, 
and 
denote the sets of minimal point of 
and maximal point of 
, respectively.
Definition.
Let 
be a nonempty subset of a t.v.s. 
ordered by a nonempty pointed closed convex 
with nonempty interior. An element 
is said to be a weakly minimal (resp., weakly maximal) point of 
if 
(resp., 
). Here, 
and 
denote the sets of weakly minimal point of 
and weakly maximal point of 
, respectively.
Theorem (see [20]).
Let 
be a Hausdorff t.v.s. ordered by a nonempty pointed closed convex 
with nonempty interior. If 
is a nonempty compact subset of 
, then 
and 
.
Definition (see [21]).
Let 
be pointed closed convex 
with nonempty interior in a Banach space 
. Then 
is called normal if there exists 
such that if 
and 
, then 
. Here, it is called 
-normal pointed closed convex 
.
Theorem (see [22]).
Let 
be a complete metric space, and let 
be a Hausdorff t.v.s.. Let 
be a multivalued map. Assume that
(i)
for each 
(ii)for each 
, 
is a closed subset of 
(iii)for each 
, if 
and 
, then 
,
(iv)for each sequence 
in 
, if 
, then 
as 
.
Then there exists 
such that 
for all 
.
Theorem.
Let 
be a complete metric space. Let 
be a multivalued map with nonempty values. Assume that
(i)
is closed,
(ii)for each sequence 
in 
, if 
and 
for all 
, then 
and 
as 
.
Then there exists 
such that 
.
Proof.
Let 
be fixed. Clearly, 
is a complete metric space. Let 
be defined by 
for each 
. Clearly, 
for each 
. For each 
, we know that
If 
is a sequence in 
and 
as 
, then 
and 
for all 
. Clearly, 
and 
. By (ii), condition (iv) of Theorem 2.7 is satisfied. By Theorem 2.7, there exists 
such that 
for all 
. Clearly, 
. If 
and 
, then there exists 
such that 
and 
. Then 
and this implies that 
. This leads to a contradiction. Hence, 
. This implies that 
.
Example.
Let 
, 
, 
, and 
. Let 
be defined by 
for each 
. Then by Theorem 2.8, there exists 
such that 
.
Example.
Let 
, 
, and 
. Let 
be defined by 
for each 
. Clearly, 
is closed. But condition (ii) of Theorem 2.8 does not hold. Indeed, let 
. Then 
for all 
, and 
and 
for all 
. Furthermore, there is no 
such that 
. Hence, condition (ii) of Theorem 2.8 is essential in Theorem 2.8.
Theorem.
Let 
be a metric space. Let 
be a multivalued map with nonempty values. Assume that
(i)
is a nonempty closed subset of 
(ii)for each sequence 
in 
, if 
, then 
as 
.
Then there exists 
such that 
.
Proof.
Let 
be fixed. Clearly, 
is a complete metric space. Let 
be defined by 
for each 
. Clearly, 
for each 
. Besides, for each 
, let 
. Then 
is a closed set. Next, for each 
, if 
and 
, then 
. By (ii), for each sequence 
in 
with 
, we have 
as 
. By Theorem 2.7, there exists 
such that 
.
Example.
Let 
, 
, 
, and 
. Let 
be defined by
Thus by Theorem 2.11, there exists 
such that 
. Note that 
is not closed, not open, and not convex.
Example.
Let 
, 
, and 
. Let 
be defined by
By Theorem 2.11, there exists 
such that 
. Indeed, 
.
Remark.
Example 2.10 also shows that condition (ii) of Theorem 2.11 is essential in Theorem 2.11. Next, the following result is a special case of Theorem 2.11. Note that it is different from Theorem 2.5 since we do not assume that 
is a compact set, but we assume that 
is a Banach space.
Corollary.
Let 
be a nonempty closed subset of 
. Suppose that for each sequence 
in 
if 
, then 
as 
. Then 
.
Proof.
Let 
be a singleton subset of 
. Then 
is a complete metric space. Let 
be defined by 
for each 
. By Theorem 2.11, there exists 
such that 
and this implies that 
.
Example.
Let 
and 
. It is easy to see that all conditions of Corollary 2.15 are satisfied. Hence, 
. Indeed, 
. Note that 
is not a compact subset of 
.
Corollary.
Let 
be a Banach space, and let 
be a 
-normal pointed closed convex 
with nonempty interior 
. If 
is a nonempty bounded closed subset of 
, then 
.
Proof.
Take any sequence 
in 
with 
. This implies that 
By assumption, we get
Since 
is bounded, there exists 
such that 
This implies that 
as 
and Corollary 2.17 follows from Corollary 2.15.
Remark.
Theorem 2.11 and Corollary 2.15 are equivalent.
Remark.
From the above results and examples, we observe that if 
is a closed set and there exists 
such that 
, then 
. Indeed, for each sequence 
in 
with 
, it is easy to see that, for each 
, 
is a decreasing sequence in 
and bounded from below. Then 
as 
and 
.
3. Saddle Point Problems
Theorem.
Let 
and 
be two complete metric spaces. Assume that
(i)for each 
, 
is a closed set,
(ii)for each 
, if 
and 
, then 
,
(iii)for each sequence 
in 
, if 
, then 
and 
as 
.
Then, for each 
, there exists 
such that 
, 
, and 
.
Proof.
Let 
. Then 
is a complete metric space. Take any 
. Define the set 
Clearly, 
is a complete metric space. Let 
be defined by 
. By Theorem 2.7, there exists 
such that 
for all 
. By (ii) and the definition of 
, it is easy to see that 
for all 
. Hence,
Since 
is a nonempty compact set, by Theorem 2.5, there exist 
such that 
and 
. Next, it is easy to see that 
and 
. Therefore, 
and 
.
Remark.
If 
is an u.s.c. multivalued map with nonempty compact values, then condition (i) of Theorem 3.1 holds.
Example.
Let 
, 
, 
, and 
. Let 
be defined by 
for each 
. By Theorem 3.1, for each 
, there exists 
such that 
, 
, and 
. Indeed, for each 
, 
.
In Theorem 3.1, if 
is singleton, then we have the following result.
Corollary.
Let 
be a multivalued map with nonempty compact values. Assume that
(i)for each 
, 
is a closed set,
(ii)for each 
, if 
and 
, then 
,
(iii)for each sequence 
in 
, if 
, then 
as 
.
Then, for each 
, there exists 
such that 
and 
.
Theorem.
Assume that
(i)for each 
, 
is closed,
(ii)for each 
, if 
and 
, then 
,
(iii)for each sequence 
in 
, if 
, then 
and 
as 
.
Then, for each 
, there exists 
such that 
, and 
for all 
.
Proof.
Let 
. Take any 
. Define the set 
. Clearly, 
is a complete metric space. Let 
be defined by 
. For each 
, by (i), 
is a closed set. Then by Theorem 2.7, there exists 
such that 
for all 
.
Now, we want to show that 
for all 
. In fact, we only need to consider that 
. Suppose that 
. By (ii) and 
, 
and this is a contradiction. Therefore, 
for all 
.
Remark.
Condition (i) of Theorem 3.5 can be replaced by 
being l.s.c. and 
being an u.s.c. multivalued map with nonempty compact values.
Example.
Let 
, 
, 
, and 
, and let 
be defined by
This is an example for Theorem 3.5.
Example.
Let 
, 
, 
, and 
. Let 
be defined by 
for each 
. By Theorem 3.5, for each 
, there exists 
such that 
, and 
for all 
. Indeed, 
for each 
.
4. Bilevel Problems
Proposition.
Let 
be defined by
If 
is a continuous multivalued map with nonempty compact values, then 
is closed, and 
is a closed set.
Proof.
Here, we only need to show that 
is closed. If 
and 
, for each 
, we have then
(1)
,
(2)
,
(3)
.
By (1), for each 
, there exists 
and 
such that 
. Let 
, 
, 
, and 
. Then 
, 
, 
, and 
are compact sets. By Lemma 2.2, 
and 
are compact sets. Hence, we may assume that 
and 
. By Lemma 2.2, 
and 
. Clearly, 
and 
. By (2), for each 
, there exists 
. By Lemma 2.2, 
is a compact set. Hence, we may assume that 
. By Lemma 2.2, 
.
For each 
, 
Take any 
. Then there exists 
such that 
. There exists a net 
in 
such that 
, 
Therefore, 
for all 
. That is, 
. Hence, 
. Similarly, we can prove that 
. So, 
and 
is closed.
Lemma.
Let 
and 
be topological spaces, and let 
be a multivalued map. Let 
be defined by 
for each 
. If 
is an u.s.c. multivalued map with nonempty compact values and 
is a nonempty closed subset of 
, then 
is a closed subset of 
.
Proof.
If 
, then there exist a net 
such that 
and a net 
in 
such that 
for all 
. Hence, 
for all 
. Let 
. Then 
is a compact set and 
is a compact set. We may assume that 
. Since 
is a closed set, 
. By Lemma 2.2, 
and 
. Therefore, 
is a closed subset of 
.
Theorem.
In Theorem 3.1, let 
be a multivalued map with nonempty values, and further assume that
(a)
is an u.s.c. multivalued map with nonempty compact values,
(b)
is a continuous multivalued map with nonempty compact values,
(c)for each sequence 
in 
, if 
, then 
as 
.
Then there is a solution of problem (BL-1).
Proof.
Let 
and 
be defined as in Proposition 4.1. By Theorem 3.1 and Proposition 4.1, 
is a nonempty closed subset of 
. By Lemma 4.2, 
is a nonempty closed subset of 
. By Theorem 2.11, there exists 
such that 
.
Example.
In Example 3.3, let 
, 
, and 
for each 
. Clearly, 
is a closed subset of 
. Besides, we have
and 
is an u.s.c. multivalued map with nonempty compact values. By Theorem 4.3, there is a solution of problem (BL-1). Indeed, the solution set is 
.
The following theorem is similar to Theorem 4.3. Note that the conditions of Theorems 4.3 and 4.5 are different.
Theorem.
In Theorem 3.1, let 
be a multivalued map, and further assume that
(a)
is an u.s.c. multivalued map with nonempty compact values,
(b)for each 
, if 
and 
, then 
,
(c)
is a continuous multivalued map with nonempty compact values,
(d)for each 
in 
, if 
, then 
as 
.
Then there is a solution of problem (BL-1).
Proof.
Let 
and 
be defined as in Proposition 4.1. By Theorem 3.1 and Proposition 4.1, 
is a nonempty closed subset of 
. Hence, 
is a complete metric space. Now, for each 
, let 
. If 
, then there exists a net 
in 
such that 
. Then for each 
, there exists 
. Since 
is a compact set, we may assume that 
. There exists 
such that 
. Let 
. Clearly, 
and 
are compact sets. Hence, we may assume that 
. By Lemma 2.2, 
. Clearly, 
and this implies that 
. Therefore, 
and 
is a closed set. By Corollary 3.4, there exists 
such that 
.
Furthermore, we have the following result which is different from Theorems 4.3 and 4.5. In Theorem 4.6, 
is a Hausdorff t.v.s., and 
and 
are compact metric spaces. In Theorems 4.3 and 4.5, 
is a Banach space, and 
and 
are complete metric spaces.
Theorem.
In Theorem 3.1, let 
be a multivalued map, and further assume that
(a)
is an u.s.c. multivalued map with nonempty compact values,
(b)
is a continuous multivalued map with nonempty compact values,
(c)
and 
are compact.
Then there is a solution of problem (BL-1).
Proposition.
Let 
be defined by
If 
is a continuous multivalued map with nonempty compact values, then 
is closed, and 
is a closed set.
Proof.
Here, we only need to show that 
is closed. If 
and 
, then we have
(1)
,
(2)
for all 
,
(3)
for all 
.
Take any 
; there exists a net 
such that 
for all 
and 
. There exists 
such that 
. Let 
, 
, 
, and 
. Then 
, 
, 
, and 
are compact sets and 
is a compact set. Hence, we may assume that 
. By Lemma 2.2, 
. Clearly, 
. Hence, 
, and 
.
Take any 
there exists 
such that 
. 
is a compact set, and we may assume that 
; By Lemma 2.2, 
. Take any 
, there exists a net 
such that 
for all 
and 
, 
for all 
. Clearly, 
for all 
. Hence, 
for all 
. Similarly, 
for all 
. Therefore, 
is closed.
Applying Proposition 4.7 and following the similar argument as in the proof of Theorems 4.3–4.6, we can get the following similar results.
Theorem.
In Theorem 3.5, let 
be a multivalued map with nonempty values. Further assume that conditions (a)–(c) of Theorem 4.3 (resp., conditions (a)–(d) of Theorem 4.5) are satisfied. Then there is a solution of problem (BL-2).
Proposition.
Let 
be a map with 
. Let 
be defined by
Suppose that, for each 
, 
is continuous, 
is one to one, and 
is one to one. Then 
is closed, and 
is a closed set.
Proof.
Let 
and 
as 
. Then 
, 
for all 
, and 
for all 
. Since 
is continuous, 
. Furthermore, for each 
, we have
Take any 
. There exists 
such that 
for all 
. Furthermore, for each 
with 
, there exists 
such that 
Indeed, if not, there exists 
with 
such that 
for all 
. Since 
and 
is one to one, 
for all 
. Hence, 
. This leads to a contradiction.
Therefore, there exist 
and 
, 
of 
and 
, respectively, such that 
Suppose that 
Then 
, 
Hence, 
. This leads to a contradiction since 
is one to one. Therefore, 
Similarly, we have 
Therefore, 
is closed.
Theorem.
Let 
be a map with 
, 
. Let 
be a multivalued map with nonempty values. Further assume that
(i)for each 
, 
is continuous, 
is one to one, and 
is one to one,
(ii)for each 
, if 
and 
, then 
,
(iii)for each sequence 
in 
with 
, 
and 
as 
,
(iv)
is an u.s.c. multivalued map with nonempty compact values,
(v)for each sequence 
in 
, if 
, then 
as 
.
Then there is a solution of problem (BL-3).
Proof.
Applying Proposition 4.9 and following the similar argument as in the proof of Theorem 4.3, we can get the proof of Theorem 4.10.
Remark.
The conditions of Theorem 4.10 and Theorem 3.5 in Liou et al. [5] are different. Note that Liou et al. [5] assumed that the feasible set 
is nonempty, and let the considered multivalued map 
be proper, lower semicontinuous, and weakly coercive on 
.
5. Equilibrium Problems and Mathematical Program with Equilibrium Constraint on Complete Metric Spaces
Theorem.
Let 
and 
be a complete metric spaces, and let 
be a multivalued map. Assume that
(i)for each 
, 
,
(ii)for each 
, 
is a closed subset of 
,
(iii)for each 
and 
, if 
and 
, then 
,
(iv)for each sequence 
in 
with 
, 
and 
as 
.
Then there exists 
such that 
for all 
.
Proof.
Let 
be defined by 
for each 
. Then Theorem 5.1 follows from Theorem 2.7.
Theorem.
Let 
and 
be a complete metric spaces, and let 
be a multivalued map. Assume that
(i)
is a closed set; and for each 
, 
,
(ii)for each 
, 
is a closed subset of 
,
(iii)for each 
and 
, if 
and 
, then 
,
(iv)for each sequence 
in 
with 
, 
and 
as 
,
(v)
is closed; and for each 
, 
is l.s.c.,
(vi)for each sequence 
in 
with 
, 
and 
as 
.
Then there is a solution of problem (MPEC-1).
Proof.
By Theorem 5.1, there exists 
such that 
for all 
. Since 
, 
for all 
.
Let 
for all 
. Clearly, 
. If 
, then there exists a net 
in 
such that 
. Then, for each 
, 
for all 
. That is, 
Take any 
and any 
; there exists a net 
such that 
for all 
and 
. Clearly, 
. Hence, 
for all 
. Then 
is a closed subset of a complete metric space 
. Furthermore, 
is a complete metric space. Let 
. Then 
is a closed subset of 
. By Theorem 2.8, there is a solution of problem (MPEC-1).
Theorem.
Let 
and 
be a complete metric spaces, and let 
be a multivalued map. Assume that
(i)for each 
, 
,
(ii)for each 
, 
is a closed subset of 
,
(iii)for each 
and 
, if 
and 
, then 
,
(iv)for each sequence 
in 
with 
, 
and 
as 
.,
(v)
is a closed set,
(vi)for each 
, 
is an u.s.c. multivalued map with nonempty compact values,
(vii)
is closed,
(viii)for each sequence 
in 
with 
, 
and 
as 
.
Then there is a solution of problem (MPEC-2).
Proof.
Let 
be defined by 
for each 
. By Theorem 5.1 and condition (i), there exists 
such that 
for all 
.
Let 
for all 
. If 
, then there exists a sequence 
in 
such that 
. Let 
. Then 
is a compact set. For each 
, since 
, 
For each 
and 
, there exists 
such that 
and 
. Then 
. By (vi), we may assume that 
. Since 
is a closed set, 
. Hence, 
and 
is closed. Following the similar argument as in the last part of the proof of Theorem 5.2, we get the proof of Theorem 5.3.
References
Luo Z-Q, Pang J-S, Ralph D: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, UK; 1996:xxiv+401.
Stein O, Still G: On generalized semi-infinite optimization and bilevel optimization. European Journal of Operational Research 2002, 142(3):444–462. 10.1016/S0377-2217(01)00307-1
Stein O: Bilevel Strategies in Semi-Infinite Programming, Nonconvex Optimization and Its Applications. Volume 71. Kluwer Academic Publishers, Boston, Mass, USA; 2003:xxviii+202.
Birbil Şİ, Bouza G, Frenk JBG, Still G: Equilibrium constrained optimization problems. European Journal of Operational Research 2006, 169(3):1108–1127. 10.1016/j.ejor.2004.07.075
Liou YC, Yang XQ, Yao JC: Mathematical programs with vector optimization constraints. Journal of Optimization Theory and Applications 2005, 126(2):345–355. 10.1007/s10957-005-4720-4
Lin L-J, Still G: Mathematical programs with equilibrium constraints: the existence of feasible point. Optimization 2006, 55(3):205–219. 10.1080/02331930600703635
Lin L-J: Mathematical programming with system of equilibrium constraints. Journal of Global Optimization 2007, 37(2):275–286. 10.1007/s10898-006-9049-5
Lin L-J, Hsu H-W: Existences theorems of systems of vector quasi-equilibrium problems and mathematical programs with equilibrium constraint. Journal of Global Optimization 2007, 37(2):195–213. 10.1007/s10898-006-9044-x
Luc DT, Vargas C: A saddlepoint theorem for set-valued maps. Nonlinear Analysis: Theory, Methods & Applications 1992, 18(1):1–7. 10.1016/0362-546X(92)90044-F
Tanaka T: Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. Journal of Optimization Theory and Applications 1994, 81(2):355–377. 10.1007/BF02191669
Kim I-S, Kim Y-T: Loose saddle points of set-valued maps in topological vector spaces. Applied Mathematics Letters 1999, 12(8):21–26. 10.1016/S0893-9659(99)00116-0
Lin L-J: System of coincidence theorems with applications. Journal of Mathematical Analysis and Applications 2003, 285(2):408–418. 10.1016/S0022-247X(03)00406-2
Ansari QH, Lin LJ, Su LB: Systems of simultaneous generalized vector quasiequilibrium problems and their applications. Journal of Optimization Theory and Applications 2005, 127(1):27–44. 10.1007/s10957-005-6391-6
Lin L-J: Existence theorems of simultaneous equilibrium problems and generalized vector quasi-saddle points. Journal of Global Optimization 2005, 32(4):613–632. 10.1007/s10898-004-2697-4
Fu JY: Stampacchia generalized vector quasiequilibrium problems and vector saddle points. Journal of Optimization Theory and Applications 2006, 128(3):605–619. 10.1007/s10957-006-9034-7
Kimura K, Tanaka T: Existence theorem of cone saddle-points applying a nonlinear scalarization. Taiwanese Journal of Mathematics 2006, 10(2):563–571.
Gong X-H: The strong minimax theorem and strong saddle points of vector-valued functions. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(8):2228–2241. 10.1016/j.na.2007.01.056
Tan NX: Quasivariational inequalities in topological linear locally convex Hausdorff spaces. Mathematische Nachrichten 1985, 122: 231–245. 10.1002/mana.19851220123
Aubin JP, Cellina A: Differential Inclusion. Springer, Berlin, Germany; 1994.
Luc DT: An existence theorem in vector optimization. Mathematics of Operations Research 1989, 14(4):693–699. 10.1287/moor.14.4.693
Huang L-G, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007, 332(2):1468–1476. 10.1016/j.jmaa.2005.03.087
Lin L-J, Chuang C-S: Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(7):2665–2672. 10.1016/j.na.2008.03.053
Author information
Authors and 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
Lin, LJ., Chuang, CS. Saddle Point Problems, Bilevel Problems, and Mathematical Program with Equilibrium Constraint on Complete Metric Spaces. J Inequal Appl 2010, 306403 (2010). https://doi.org/10.1155/2010/306403
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/306403
Keywords
- Saddle Point
- Mathematical Program
- Equilibrium Problem
- Existence Result
- Existence Theorem