6th
International Science, Social Sciences, Engineering and Energy Conference
17-19 December, 2014, Prajaktra Design Hotel, Udon Thani, Thailand
I-SEEC 2014
http//iseec2014.udru.ac.th
New types of Levitin-Polyak well-posedness for bilevel
vector equilibrium problem with equilibrium constraints
Kamonrat Sombuta,e1, Kulprapa Srimudb,e2
a,b
Department of Mathematics, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi,
Pathumthani, 12110, Thailand.
e1
jee_math46@hotmail.com,
e2
kulprapa_s@hotmail.com
Abstract
The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector
equilibrium problem with equilibrium constraints. Criteria and characterizations for these types of Levitin-Polyak
well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrcheckatescu’s measures of
noncompactness of approximate solution sets under suitable conditions, we prove that the Levitin-Polyak wellposedness for bilevel vector equilibrium problem with equilibrium constraints. We prove that the Levitin-Polyak
well-posedness for bilevel equilibrium problem with equilibrium constraints.
Keywords: Levitin-Polyak well-posedness, bilevel vector equilibrium problem.
1. Introduction
Well-posedness is one of most important topics for optimization theory and numerical methods
of optimization problems, which guarantees that, for approximating solution sequences, there is a
subsequence which converges to a solution. The will-posedness of unconstrained and constrained scalar
optimization problems was first introduced and studied by Tykhonov[1] and Levitin and Polyak[2],
respectively. The concept of Tykhonov well-posedness has also been generalized to several related
problems: variational inequalities, Nash equilibrium problems, optimization problems with variational
inequalities constrains, optimization problems with Nash equilibrium constrains, optimization problems
with Nash equilibrium constrains.
The study of Levitin-Polyak (LP for shot) well-posedness for scalar convex optimization
problems with functional constraints originates. Recently, this research was extended to nonconvex
optimization problems with abstract and functional constraints[3] and nonconvex vector optimization
2
problems with both abstract and functional constraints[4]. In 2009, Li and Li[5] introduced and
researched two types of LP well-posedness of vector equilibrium problems with variable domination
structures. Most recently, many papers appeared dealing with bilevel problems such as mathematical
programming with equilibrium constraints[6], optimization problems with Nash equilibrium
constraints[7], optimization problems with variational inequality constraints[8], optimization problems
with equilibrium constraints[9, 10], etc. In 2012, Anh et al.[11] considered the LP well-posedness of
bilevel equilibrium problems with equilibrium constraints and bilevel optimization problems with
equilibrium constraints. They introduced a relaxed level closedness and use it together with pseudo
monotonuity assumptions to establish sufficient conditions LP well-posedness.
Motivated and inspired by the above observations our consideration of LP well-posedness for
bilevel problems in this paper. We focus on vector equilibrium with equilibrium constraints. We propose
a generalized level closedness and use it together to study well-posedness in the LP sense. However, since
the existence topic has been intensively studied for vector equilibrium and bilevel problems, we focus on
LP well-posedness, assuming always that the mentioned solutions exist.
The layout of the paper is as follows. In Sect. 2, we state the bilevel problems under our
consideration and recall notions and preliminaries needed in the sequel. In Sect. 3, we study LP wellposedness of bilevel vector equilibrium problems with equilibrium constraints on diameters and measures
of noncompactness of approximate solution sets in the Kuratowski, Hausdorf, or Istratescu sense. The
results in this paper unify, generalize and extend some known results in[11, 12].
2. Preliminaries
Let ( X , d ) and Z be locally convex Hausdorff topological vector spaces, where d is a metric
Z
which is compatible with the topology of X . Throughout this paper, suppose C : X  Z  2 is a setvalued mapping such that for any x  X , C ( x ) is a pointed, closed and convex cone in Z with nonempty
interior int C ( x ), where C ( x ) depends on x . We also assume that e : X  Z is a continuous vectorvalued mapping and satisfies that for any x  X , e( x)  int C ( x). Let f : X  X  Z  Z be a vectorvalued mapping and
X
Ki : X  Z  2 , i  1, 2. The constraints appear in this paper are solution sets of
the following (parametric) vector quasi-equilibrium problem, for each z  Z ,
(VQEPZ ) find x  K1 ( x , z ) such that f ( x , y, z )   int C ( x), y  K 2 ( x , z )
(2.1)
Instead of writing { (VQEPZ )  Z } for the family of quasi-equilibrium problems, i.e., the parametric
problem, we will simply write (VQEP) in the sequel. Let S : Z  2
X
be the solution map of (VQEP ).
Let Y  X  Z , F : Y  Y  Z and g : X  Z  Z be two functions with g  x, z   D. We consider the
following bilevel vector equilibrium problem with equilibrium constraints:
(2.2)
( BVEPEC ) finding y  grS such that f ( y , y )   int C ( y ), y  grS
where grS denotes the graph of S, i.e grS : {( x, z ) | x S ( z )}.We denote by  the set of solutions of
( BVEPEC ) . We first defined LP well-posedness notions.
*
Definition 2.1. A sequence { xn }  {( xn , z n )}  X  Z is called type I LP approximating sequence for
( BVEPEC ) if and only if there exists a sequence of nonnegative real number { n } with  n  0 such that
d ( xn , grS )   n ;
(2.3)
3
F ( xn , y )   n e( xn )   int C ( xn ), y  grS ( z ) and z  Z , where y  ( y , z );
*
*
*
*
(2.4)
and
{ xn } is an approximating sequence for the parametric problem (VQEP) corresponding to { zn }. (2.5)
*
Definition 2.2. A sequence { xn }  {( xn , z n )}  X  Z is called generalized type I LP approximating
sequence for ( BVEPEC ) if and only if there exists a sequence of nonnegative real number { n } with
 n  0 such that (2.4) – (2.5) hold and
d ( g ( xn , D )   n ;
*
(2.6)
Definition 2.3. Problems ( BVEPEC ) is called type I (resp., generalized type I) LP well-posed if and
only if
(i) the solution set of ( BVEPEC ) is nonempty;
(ii) for any type I (resp., generalized type I) LP approximating sequence of ( BVEPEC ) has a
subsequence converging to a solution.
Z
Recall now some notions. Let X and Z be as above and T : X  2 be a multifunction. T is
called lower semicontinuous (lsc) at x0 iff T ( x0 )  U   for some open subset U  Z implies the
existence of a neighborhood N of
x0 such that T ( x)  U
  for all x  N . T is upper semicontinuous
(usc) at x0 iff, for each open subset U  T ( x0 ), there is a neighborhood N of
x0 such that U  T ( N ).
T is called closed at x0 iff, for each net ( x , y )  grT with ( x , y )  ( x0 , y0 ), one has y0  T ( x0 ). We
say that T satisfies a certain property in a subset A  X iff T satisfies it at every point of A. If
A  domT : {x | T ( x)  } we omit "in domT " in the saying. The following assertions are known.
(i) T is lsc at
x0 if and only if, x
 x , y  T ( x ), y  T ( x ), y  y.
0
0



(ii) T is closed if and only if grT is closed.
(iii) T is usc at x0 if T ( A) is compact for any compact subset A of domT and T is closed at
x0 .
(iv) T is usc at x0 if Z is compact and T is closed at x0 .
Definition 2.4. Let X be Hausdorff topological spaces and Z be a topological vector space, f : X  Z .
The function f is called upper 0  level closed at x0  X  Y iff, for any
xn converging to x0 ,
[ f ( xn )   int C , n]  [ f ( x0 )   int C ].
Definition 2.5.[13] Let A, B be nonempty subsets of metric space X . The Hausdorff distance  (, )
between A and B is defined by
( A, B)  max{e( A, B), e( B, A)},
where e( A, B)  sup aA d ( a, B ) with
d (a, B)  infbB d ( a, b). Let { An } be a sequence of nonempty
subsets of X . We say that An converges to A in the sense of Hausdorff metric if  ( An , A)  0. It is easy
to see that e( An , A)  0 if and only if d ( an , A)  0 for all selection an  An . For more details on this
topic, we[13].
4
3. Bilevel Vector Equilibrium Problems with Equilibrium Constraints (BVEPEC)
In this section, we give some criteria and characterizations for LP well-posedness of ( BVEPEC )
with using noncompactness. Now we need the following notions of measures of noncompactness.
Definition 3.1. Let M be a nonempty subset of a metric space X .
(i) The Kuratowski measure of M is
 ( M )  inf{  0 | M 
n
k 1
M k and diamM k   , k  1,
, n, for some n  }.
(ii) The Hausdorff measure of M is
 ( M )  inf{  0 | M 
n
k 1
B ( xk ,  ), x  X , for some n  }.
k
(iii)The Istratescu measure of M is
 ( M )  inf{  0 | M have no infinite   discrete subset}.
The following inequalities are obtained in[15]
(3.1)
 ( M )   ( M )   ( M )  2 ( M ).
The measures  ,  and  share many common properties and we will use  in sequel to denote either one
of them  is a regular measure[16,17], i.e., it enjoys the following properties :
(i)  ( M )   if and only if the set M is unbounded;
(ii)  ( M )   (clM );
(iii) if  ( M )  0 then M is a totally bounded set;
(iv) if X is a complete space and if { An } is a sequence of closed subset of X such that
An 1  An for each n 
and lim  ( An , K )  0 where  is the Hausdorff metric;
n 
(v) if M  N , then  ( M )   ( N ).
As above, S ( z ) denotes the solution set of (VQEPz ). For positive  the  -solution set of
(VQEPz ) is defined by
S ( z ,  )  { x  K1 ( x, z ) | f ( x, y , z )   e   int C ( x ), y  K 2 ( x, z )}.
For positive  and  the corresponding approximate solution sets for ( BVEPEC ) are defined,
respectively, by
1 ( ,  ) : {x*
*
*
*
*
 ( x , z )  K ( x , z )  Z | d ( x , grS )   , F ( x , y )   e   int C , y  geS and
1
f ( x, y , z )   e   int C , y  K 2 ( x, z )}.
 2 ( ,  ) : { x*
*
*
*
*
 ( x , z )  K ( x , z )  Z | d ( g ( x ), D )   , F ( x , y )   e   int C , y  geS and
1
f ( x, y , z )   e   int C , y  K 2 ( x, z )}.
In terms of a measure   { , , } of noncompactness we have the following result.
Theorem 3.2. Let X and Z be complete and   { ,  , }. Assume that
(i) in X  Z , K1 is closed and K 2 is lsc;
(ii) f is upper 0  level closed in K1 ( X , Z )  K 2 ( X , Z )  Z ;
5
*
*
(iii) F (, y ) is upper 0  level closed in X  Z , for all y  grS ;
Z
(iv) the mapping W : X  Z  2 defined by W ( x)  Z \  int C ( x) is closed.
Then, ( BVEPEC ) is type I (resp., generalized type I) LP well posed if and only if
 (1 ( ,  ))  0
(3.2)
Proof. By the relationship (3.1), the proof is similar for the three mentioned measures of noncompactness.
We discuss only the case    , the Kuratowski measure. Assume that ( BVEPEC ) is type I LP wellposed. The solution set  of ( BVEPEC ) clearly the relation   1 ( ,  ). Hence,
 (1 ( ,  ), )  max{e(1 ( ,  ), ), e(, 1 ( ,  ))}  e(1 ( ,  ), ).
*
*
Let { xn }  {( xn , z n )} be in . Since { xn } is an approximating sequence, it has a subsequence convergent
to some point of . Therefore,  is compact. Assume that  
n
M with diamM k   , k  1,
k
k 1
Setting N k  { z  X | d ( z , M k )   (1 ( ,  ),  )}, it obvious that 1 ( ,  ) 
, n.
n
N and
k 1
k
diamN k    2 (1 ( ,  ),  ). Since  is compact and  ()  0, then we get
 (1 ( ,  ))  2 (1 ( ,  ), )   ()  2e(1 ( ,  ), ).
Next, we show that  (1 ( ,  ), )  0 as ( ,  )  (0, 0). By contradiction, suppose the exist
  0, ( ,  )  (0, 0) and xn  1 ( n ,  n ) such that d ( xn ,  )   , n  . This contradicts the type I LP
well-posed. So,  (1 ( ,  ), )  0 as ( ,  )  (0, 0). It follows that (3.2) holds. Conversely, First, we
show that for all  ,   0, 1 ( ,  ) is closed. Assume that  ( 1 ( ,  ))  0 as ( ,  )  (0, 0). Let
*
*
*
{ xn }  {( xn , z n )}  1 ( ,  ) with { xn }  x : ( x , z ). Then for all y  grS and yn  K 2 ( xn , z n ),
*
d ( xn , grS )   ;
*
(3.3)
F ( xn , y )   e( xn )   int C ( xn ),
(3.4)
f ( xn , yn , z n )   e( xn )   int C ( xn ).
(3.5)
*
*
*
*
and
As K 1 is closed at ( x , z ), we have x  K1 ( x, z ). From (3.3), (3.4) and (3.5), we obtain that d ( x , grS )   .
*
By the upper 0  level closedness of F in first argument and assumption (iv), one obtain
F ( x , y )   e( x )  W ( x ) that is, F ( x , y )   e( x )   int C ( x ), y  grS .
*
*
*
*
*
*
*
*
*
Next, we show that by contrapositive that f ( x, y , z )   e( x )   int C ( x ), y  K 2 ( x, z ). Suppose
that there exist y  K 2 ( x , z ) such that f ( x, y, z )   e( x)   int C ( x). Since K 2 is lsc at ( x , z ), there exist
yn  K 2 ( x , z ) such that yn  y. By upper 0  level closedness of f at ( x , y , z ), there is n0 
f ( xn , yn , z n )   e( xn )   int C ( xn ), n  n0 . That is a contradiction.
such that
6
Thus, we have f ( x, y , z )   e( x )   int C ( x ), y  K 2 ( x, z ). Therefore, x  1 ( ,  ) and so 1 ( ,  ) is
*
closed.
Secondly, we show that  
*
( ,  )  (0, 0) and x 
n n
 ( ,  ). It is obvious that  
1
 , 0
 ( ,  ). Now, suppose that
 , 0
1
 ( ,  ). Thus, we have
 , 0
1
n
n
d ( x , grS )   n ;
*
(3.6)
F ( x , y )   n e( x )   int C ( x ), y  grS ;
(3.7)
f ( x, y , z )   n e( x )   int C ( x ), y  K 2 ( x, z ).
(3.8)
*
*
*
*
*
and
*
*
*
*
*
By (3.7) and (3.8) and closedness of W ( x ), we obtain that F ( x , y )  W ( x ), y  grS and
*
f ( x, y , z )  W ( x ), y  K 2 ( x, z ). That is, x  1 ( ,  ). Hence,  
 ( ,  ) holds.
 , 0
1
We know that  ( 1 ( ,  ))  0 as ( ,  )  (0, 0). Then, by properties of  , we see that . is
*
compact and  (1 ( ,  ), )  0 as ( ,  )  (0, 0). Let { xn }  {( xn , z n )} be a type I LP approximating
solution sequence for ( BVEPEC ). There is {( n ,  n )}  (0, 0) such that for all y  grS and y  K 2 ( x, z ),
*
d ( xn , grS )   n ;
*
F ( xn , y )   n e( xn )   int C ( xn ),
*
*
*
*
and
f ( xn , yn , z n )   n e( xn )   int C ( xn ).
*
Therefore, xn  1 ( n ,  n ). It follows that from  ( 1 ( ,  ))  0 as ( ,  )  (0, 0) that
*
d ( xn , )  (1 ( ,  ), )  0.
By the compactness of  , there is a subsequence of { xn } convergent to some point of . Hence,
( BVEPEC ) is type I LP well-posed. This completes the proof.
Similar to Theorem 3.2, we can prove that the following results.
Theorem 3.3. Let X and Z be complete and   { ,  , }. Assume that
(i) in X  Z , K1 is closed and K 2 is lsc;
(ii) f is upper 0  level closed in K1 ( X , Z )  K 2 ( X , Z )  Z ;
*
*
(iii) F (, y ) is upper 0  level closed in X  Z , for all y  grS ;
Z
(iv) the mapping W : X  Z  2 defined by W ( x)  Z \  int C ( x) is closed.
Then, ( BVEPEC ) is generalized type I LP well posed if and only if  ( 2 ( ,  ))  0 as ( ,  )  (0, 0).
7
Acknowledgements
The authors would like to thanks Faculty of Science and Technology, Rajamangala University of
Technology Thanyaburi for financial support.
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