Geoscience Reference
In-Depth Information
X
wPar
f
1
;:::;f
Q
, there exists
z
2
Proof
If x
62
R
2
such that f
q
.
z
/<f
q
.x/
for each q
2
Q
, that means,
Q
\
L
<
.f
q
;f
q
.x//:
z
2
qD1
Hence, we obtain that
Q
\
L
<
.f
q
;f
q
.x//
¤;
:
qD1
Since the implications above can be reversed the proof is concluded. The
remaining results can be proved analogously.
Remark 9.1
For the case Q
D
2 the previous result states that the set
X
wPar
.f
1
;f
2
/ coincides with tangential cusps between the level curves of
functions f
1
.
/ and f
2
.
/ union with
X
.f
2
/ (see Example
9.1
).
Corollary 9.1
If
f
1
;:::;f
Q
are strictly convex functions then
X
.f
1
/
[
X
Par
f
1
;:::;f
Q
D
X
sPar
f
1
;:::;f
Q
:
X
wPar
.f
1
;:::;f
Q
/
D
Example 9.1 (See Fig.
9.1
)
Let us consider the points a
1
D
.0;0/, a
2
D
.8;3/,
a
3
D
.
3;5/ and the functions f
1
.x/
Dk
x
a
1
k
1
, f
2
.x/
Dk
x
a
2
k
1
, f
3
.x/
D
k
x
a
3
k
1
. By Theorem
9.1
,
X
wPar
.f
1
;f
2
/ is the rectilinear thick path joining a
1
X
wPar
.f
1
;f
3
/ is the dark rectangle with a
1
and a
3
as opposite vertices.
In what follows, since we are dealing with general convex, inf-compact func-
tions, we will focus on providing information about the geometrical structure of
X
wPar
.f
1
;f
2
;f
3
/. This characterization will allow us to obtain a geometrical
description of
and a
2
and
X
sPar
f
1
;f
2
;f
3
in the next section for an
important family of functions. Actually, we will characterize
X
Par
f
1
;f
2
;f
3
and
X
wPar
.f
1
;f
2
;f
3
/ as
a kind of hull delimited by the chains of bicriteria solutions of any pair of functions
f
p
, f
q
p;q
D
1;2;3. This result enables us to obtain the set
X
wPar
f
1
;:::;f
Q
by union of three-criteria solution sets already characterized. In order to do that, let
2
/
WD
n
'
j
'
W
k
'.t/
k
2
D1
o
;
0
;
0
!
2
;'continuous; lim
t!1
C
1
.
R
R
R
R
0
;
2
/ is the set of
where
k
x
k
2
is the Euclidean norm of the point x. C
1
.
R
R
0
WD
Œ0;
1
/ into
continuous curves, which map the set of non-negative numbers
R
0
/ is unbounded in
2
.These
curves are introduced to characterize the geometrical locus of the points surrounded
by weak-Pareto and Pareto chains.
2
and whose image '.
the two-dimensional space
R
R
R
