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NUMER. FUNCT. ANAL. AND OPTIMIZ., 23(3&4), 323–331 (2002)
ON ALMOST WELL-POSED MUTUALLY
NEAREST AND MUTUALLY FURTHEST
POINT PROBLEMS
Chong Li1 and Hong-Kun Xu2,*
1
Department of Applied Mathematics, Southeast University,
Nanjing, 210096, People’s Republic of China
E-mail: cli@seu.edu.cn
2
Department of Mathematics, University of Durban-Westville,
Private Bag X54001, Durban 4000, South Africa
E-mail: hkxu@pixie.udw.ac.za
ABSTRACT
Let G be a nonempty closed (resp. bounded closed) subset in a strongly
convex Banach space X. Let BðXÞ denote the space of all nonempty
bounded closed subsets of X endowed with the Hausdorff distance and
let BG ðXÞ denote the closure of the set fA 2 BðXÞ : A \ G ¼ ;g. We prove
that Eo(G) (resp. Eo(G)), the set of all A 2 BG ðXÞ (resp. A 2 BðXÞ) such
that the minimization (resp. maximization) problem min(A, G) (resp.
maxðA, GÞ) is well-posed, is residual in BG ðXÞ (resp. BðXÞ).
Key Words: Minimization problem; Maximization problem; Wellposed; Dense G-subset
AMS:
41A65; 46B20; 54E52
*Corresponding author.
323
Copyright
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LI AND XU
1. INTRODUCTION
Let X be a real Banach space and let BðXÞ denote the space of all
nonempty closed bounded subsets of X. For a closed subset G of X and
A 2 BðXÞ, we set
AG ¼ inffkx zk : x 2 A, z 2 Gg,
and if G is also bounded, we set
AG ¼ supfkx zk : x 2 A, z 2 Gg:
Let G be a nonempty closed (resp. closed bounded) subset of X.
Following Ref. [5] we say that a pair of (x0, z0) with x0 2 A, z0 2 G is a
solution of the minimization (resp. maximization) problem, denoted by
min(A, G) (resp. max(A, G)), if kx0 z0 k ¼ AG (resp. kx0 z0 k ¼ AG ).
Moreover, any sequence {(xn, zn)} with xn 2 A and zn 2 G for all n such
that limn!1 kxn zn k ¼ AG (resp. limn!1 kxn zn k ¼ AG ) is called a minimizing (resp. maximizing) sequence for min(A, G) (resp. max(A, G)). A minimization (resp. maximization) problem min(A, G) (resp. max(A, G)) is said to
be well-posed if it has a unique solution (x0, z0) and every minimizing (resp.
maximizing) sequence converges strongly to (x0, z0).
In the sequel, we suppose that the space BðXÞ is endowed with the
Hausdorff distance H(, ) which is defined by
HðA, BÞ ¼ max sup inf ka bk, sup inf ka bk , A, B 2 BðXÞ:
a2A b2B
b2B a2A
As is well known, ðBðXÞ, HÞ is a complete metric space.
In Ref. [5], the authors considered the well-posedness of the minimization and maximization problems for convex subsets. More precisely,
they proved that if X is a uniformly convex Banach space, then the set of
all convex subsets A 2 BG ðXÞ (resp. A 2 BðXÞ) such that the minimization
(resp. maximization) problem min(A, G) (resp. max(A, G)) is well-posed
forms a dense G-subset in the set of all convex subsets in BG ðXÞ (resp.
BðXÞ). (Here BG ðXÞ is the closure of the set fA 2 BðXÞ : AG > 0g in the
sense of Hausdorff distance H.) Recently, the first author of the
present article extended the above result to the framework of reflexive
locally uniformly convex Banach spaces for the class of compact convex
subsets.[12]
In the present paper, we continue to consider the same problem but for
the wider class of bounded subsets of X. Actually, if X is a strongly convex
Banach space and if G is a nonempty closed (resp. bounded closed) subset of
X, then we prove that Eo(G) (resp. Eo(G)) is a dense G-subset of BG ðXÞ (resp.
BðXÞ), where Eo(G) (resp. Eo(G)) denotes the set of all subsets A 2 BG ðXÞ
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MUTUALLY NEAREST AND FURTHEST POINT PROBLEMS
325
(resp. BðXÞ) such that the minimization (resp. maximization) problem
min(A, G) (resp. max(A, G)) is well-posed.
It should be noted that the problems considered here are in the spirit of
Steckin.[16] Some other research in this direction can be found in Refs. [2, 3, 4,
8, 10, 11, 13] and in the monograph.[7] The generic results in spaces of convex
sets can be found in Refs. [5, 6, 12]. The results of the present paper generalize some results from Refs. [1, 5, 10, 12, 14, 15, 16].
2. PRELIMINARIES
Let A be a subset of Banach space X. As usual, we denote the closure
and diameter of A by A and diam A, respectively.
Let >0 and let G X, F 2 BðXÞ and x 2 X be arbitrary. We set
dðx, GÞ ¼ inffkx gk : g 2 Gg,
LG ðF, Þ ¼ fg 2 G : dðg, FÞ FG þ g,
and, if G is bounded,
eðx, GÞ ¼ supfkx gk : g 2 Gg,
MG ðF, Þ ¼ fg 2 G : eðg, FÞ FG g:
Then the sets LG(F, ), MG(F, ) are nonempty, closed and satisfy LG ðF, Þ LG ðF, 0 Þ, MG ðF, Þ MG ðF, 0 Þ for < 0 .
Proposition 2.1. Let G X be closed and F 2 BðXÞ arbitrary.
(i) The problem min(F, G) is well-posed if and only if
inf diam LG ðF, Þ ¼ 0 and
>0
inf diam LF ðG, Þ ¼ 0:
>0
(ii) If G is bounded, the problem max(F, G) is well-posed if and only if
inf diam MG ðF, Þ ¼ 0 and
>0
inf diam MF ðG, Þ ¼ 0:
>0
Proof. This is similar to that of Proposition 2.2. of Ref. [5].
g
Definition 2.1. A Banach space X is said to be (sequentially) Kadec provided
for each sequence fxn g X which converges weakly to x with limn!1 kxn k ¼
kxk we have limn!1 kxn xk ¼ 0.
Definition 2.2. A. Banach space X is said to be strongly convex if it is reflexive, Kadec and strictly convex.
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LI AND XU
The next result concerning the characterization of strongly convex
spaces is due to Konjagin.[9] (See also Borwein and Fitzpatrick[3]).
Proposition 2.2. A Banach space X is strongly convex if and only if for every
closed nonempty subset G of X there is a dense set of points in X \G possessing
unique nearest points.
We also need a result due to Lau.[10]
Proposition 2.3. Suppose that X is a strongly convex Banach space and G a
nonempty closed (resp. closed bounded ) subset of X. Then the set of all points
x 2 X such that the minimization (resp. maximization) problem min(x, D)
(resp. max(x, G)) is well-posed is a dense G-subset of X \G (resp. X).
3. MINIMIZATION PROBLEM
In this section, let G be a fixed nonempty closed subset of a Banach
space X and we put, for convenience, F ¼ FG. Define
BG ðXÞ ¼ fA 2 BðXÞ : A > 0g,
where the closure is taken in the sense of the Hausdorff metric H.
For k 2 N, set k ¼ 1=k and define
Lk ¼ F 2 BG ðXÞ : inf diam LG ðF, Þ < k and inf diam LF ðG, Þ < k :
>0
>0
We need two lemmas which ensure that Lk is open dense in BG ðXÞ
provided X is a strongly convex Banach space. The proof of the first
lemma can be found in Ref. [5] and is thus omitted.
Lemma 3.1. For any Banach space X, Lk is open in BG ðXÞ.
Lemma 3.2. Suppose that X is a strongly convex Banach space. Then Lk is
dense in BG ðXÞ.
Proof. Let F 2 BG ðXÞ be arbitrary. Without loss of generality, we may
assume that F >0. For any 0<r<F, take x 2 F such that dðx , GÞ <
F þ r=4. It follows from Proposition 2.3 that there exists x~ 2 X such that
kx x~ k < r=4 and the minimization problem minðx~ , GÞ is well-posed. Take
g~ 2 G such that kx~ g~k ¼ dðx~ , GÞ and set
r
r
x~ þ
g~:
u¼ 1
kx~ g~k
kx~ g~k
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MUTUALLY NEAREST AND FURTHEST POINT PROBLEMS
327
Then ku x~ k ¼ r and the minimization problem min(u, G) is also wellposed. Define Y ¼ F [ fug. Then
HðF, YÞ ku x k ku x~ k þ kx~ x k 5r
:
4
So to complete the proof it suffices to show that Y 2 Lk . Since
ku g~k ¼ kx~ g~k kx~ uk
kx~ x k þ dðx , GÞ r
< r=4 þ F þ r=4 r
¼ F r=2,
we obtain
Y ku g~k < F r=2:
Next, if < r/2, then for y 2 LY ðG, Þ, we derive that
dð y, GÞ Y þ < F r=2 þ < F :
It follows that y 2
= F. This means that we have proved that
LY ðG, Þ ¼ fug,
for all < r=2:
ð3:1Þ
We now show that
LG ðY, Þ LG ðu, 2Þ,
for all < r=4:
ð3:2Þ
Indeed, for any g 2 LG ðY, Þ, we have g 2 G and dðg, YÞ Y þ . Let
y 2 Y satisfy kg yk Y þ 2. It follows that
kg yk Y þ 2 < F r=2 þ 2 < F ,
which implies that y 2
= F and we must have y ¼ u. Hence,
kg uk Y þ 2 dðu, GÞ þ 2:
That is, g 2 LG ðu, 2Þ and Eq. (3.2) is proved.
Now since the minimization problem min(u, G) is well-posed, it follows
from Proposition 2.1 that diam LG ðu, 0 Þ < k for some 0 < 0 < r=4. Thus
we obtain by Eq. (3.2) that
inf diam LG ðY, Þ diam LG ðY, 0 =2Þ diam LG ðu, 0 Þ < k :
>0
This together with Eq. (3.1) implies that Y 2 Lk .
g
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Theorem 3.1. Suppose that X is strongly convex and G a nonempty closed
subset of X. Then Eo(G), the set of all subsets F 2 BG ðXÞ such that the minimization problem min(F, G) is well-posed, is a dense G-subset of BG ðXÞ.
Proof. By Proposition 2.1, we have
\
E o ðGÞ ¼
Lk :
k2N
Thus, Lemmas 3.1 and 3.2 imply that Eo(G) is a dense G-subset of BG ðXÞ and
the proof is complete.
g
Theorem 3.2. Let X be a Banach space. Then the following statements are
equivalent:
(i) X is strongly convex;
(ii) for every closed nonempty subset G of X, the set of all A 2 BG ðXÞ
such that the minimization problem min(A, G) is well posed contains
a dense G-subset of BG ðXÞ;
(iii) for every closed nonempty subset G of X, the set of all A 2 BG ðXÞ
such that the minimization problem min(A, G) is well-posed contains
a dense subset of BG ðXÞ.
Proof. By Theorem 3.1, it suffices to prove that (iii) implies (i). For any fixed
x 2 XnG and 0 < < dG ðxÞ, it follows from (iii) that there exists B 2 BG ðXÞ
such that Hðx, B Þ < =2 and minðB , GÞ is well-posed so that minðB , GÞ has
a unique solution (x0 , z0 ). Thus,
kx x0 k Hðx, B Þ þ =2 < and x0 has a unique nearest point z0 in G. Thus from Proposition 2.2 follows
the implication ðiiiÞ ) ðiÞ.
g
4. MAXIMIZATION PROBLEM
In this section, we let G be a fixed nonempty closed bounded subset of a
Banach space X and put, for convenience, F ¼ FG.
For k 2 N, set k ¼ 1=k and define
Mk ¼ F 2 BðXÞ : inf diam MG ðF, Þ < k and inf diam MF ðG, Þ < k :
>0
>0
As counterparts of Lemmas 3.1 and 3.2, we have
Lemma 4.1. For any Banach space X, Mk is open in BðXÞ.
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Lemma 4.2. Suppose that X is a strongly convex Banach space. Then Mk is
dense in BðXÞ.
Proof. Let F 2 BðXÞ be arbitrary. Since the case F ¼ 0 is trivial, we may
assume that F>0. For any 0 < r< F, take x 2 F such that eðx , GÞ >
F r=4. From Proposition 2.3 it follows that there exists x~ 2 X such that
kx x~ k < r=4 and the maximization problem maxðx~ , GÞ is well-posed.
Let g~ 2 G with kx~ g~k ¼ eðx~ , GÞ and set
r
r
x~ g~:
u¼ 1þ
kx~ g~k
kx~ g~k
Then ku x~ k ¼ r and the maximization problem max(u, G) is also wellposed. Define Y ¼ F [ fug. Then
HðF, YÞ ku x k ku x~ k þ kx~ x k 5r
:
4
Hence to complete the proof it suffices to show that Y 2 Mk . Observing
ku g~k ¼ ku x~ k þ kx~ g~k
r kx~ x k þ eðx , GÞ
> r r=4 þ F r=4
¼ F þ r=2:
we deduce that
Y ku g~k > F þ r=2:
Since for <1/2 and y 2 MY ðG, Þ,
eð y, GÞ Y > F þ r=2 > F ,
we must have y 62 F and thus y ¼ u. This proves that inf >0 diam MY ðG, Þ ¼
0 < k . It remains to show that
inf diam MG ðY, Þ < k :
ð4:1Þ
>0
To prove Eq. (4.1) we need to verify the inclusion:
MG ðY, Þ MG ðu, 2Þ,
for all < r=4:
ð4:2Þ
Take g 2 MG ðY, Þ; i.e., g 2 G and eðg, YÞ Y . Let y 2 Y be such that
kg yk Y 2. It follows that
kg yk Y 2 > F þ r=2 2 > F :
This implies that y ¼ u; hence
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LI AND XU
kg uk Y 2 eðu, GÞ 2:
That is, g 2 MG ðu, 2Þ and Eq. (4.2) is proven.
Now since the maximization problem max(u, G) is well-posed, it follows
from Proposition 2.1 that diam MG ðu, 0 Þ < k for some 0< 0<r/4. Thus
by Eq. (4.2) we get
inf diam MG ðY, Þ diam MG ðY, 0 =2Þ diam MG ðu, 0 Þ < k :
>0
This is Eq. (4.1) and the proof is complete.
g
Theorem 4.1. Suppose that X is a strongly convex Banach space and G is a
nonempty bounded closed subset of X. Then Eo(G), the set of all subsets
F 2 BðXÞ such that the maximization problem max(F, G) is well-posed, is a
dense G-subset of BðXÞ.
Proof. By Proposition 2.1, we have
\
Eo ðGÞ ¼
Mk :
k2N
Thus, Lemmas 4.1 and 4.2 show that Eo(G) is a dense G-subset of BðXÞ,
completing the proof.
g
Remark. The example given in the end of Ref. [12] shows that if X is not
strongly convex then Theorem 4.1 is no longer true.
ACKNOWLEDGMENTS
Supported in part by the National and Jiangsu Provincial Natural
Science Foundations of China (Grant No. 19971013 and BK99001). This
work was completed when the first author was visiting the University of
Durban-Westville as an NRF Research Fellow.
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