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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
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