Existence and porosity for a class of perturbed optimization problems in Banach spaces
Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup z ∈ Z { J ( z ) + ‖ x − z ‖ } , which is denoted by ( x , J ) -sup. We shall prove in the present pap...
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| Vydáno v: | Journal of mathematical analysis and applications Ročník 325; číslo 2; s. 987 - 1002 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
San Diego, CA
Elsevier Inc
15.01.2007
Elsevier |
| Témata: | |
| ISSN: | 0022-247X, 1096-0813 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Let
X be a Banach space and
Z a nonempty closed subset of
X. Let
J
:
Z
→
R
be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem
sup
z
∈
Z
{
J
(
z
)
+
‖
x
−
z
‖
}
, which is denoted by
(
x
,
J
)
-sup. We shall prove in the present paper that if
Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all
x
∈
X
for which the problem
(
x
,
J
)
-sup has a solution is a dense
G
δ
-subset of
X. In the case when
X is uniformly convex and
J is bounded, we will show that the set of all points
x in
X for which there does not exist
z
0
∈
Z
such that
J
(
z
0
)
+
‖
x
−
z
0
‖
=
sup
z
∈
Z
{
J
(
z
)
+
‖
x
−
z
‖
}
is a
σ-porous subset of
X and the set of all points
x
∈
X
∖
Z
0
such that there exists a maximizing sequence of the problem
(
x
,
J
)
-sup which has no convergent subsequence is a
σ-porous subset of
X
∖
Z
0
, where
Z
0
denotes the set of all
z
∈
Z
such that
z is in the solution set of
(
z
,
J
)
-sup. |
|---|---|
| ISSN: | 0022-247X 1096-0813 |
| DOI: | 10.1016/j.jmaa.2006.02.055 |