WEAKLY PARETO-OPTIMAL ALTERNATIVES FOR A VECTOR
MAXIMIZATION PROBLEM: EXISTENCE AND
CONNECTEDNESS
DIEGO AVERNA
Let X be a non-empty set and f = (f1; : : : ; f ) : X ! R a function.
k
k
A point x 2 X is said to be weakly Pareto-optimal for the function f provided
there is no x 2 X such that f (x) > f (x) for all i = 1; : : :; k.
With E (X; f ) we denote the set of all Pareto-optimal points for the function f .
Let us recall the following Theorem, due to A.R.Warburton:
Theorem A. ([7], Theorem 4.1) Let X R be a non-empty, compact and convex
set and f = (f1 ; : : :; f ) : X ! R such that f1; : : : ; f are continuous and quasiconcave on X .
Then E (X; f ) is non-empty and connected.
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When the function f depends also from a parameter t, which varies in a non-empty
set T , a point x 2 X is said to be T -uniform weakly Pareto-optimal for the function
f if x is weakly Pareto-optimal
for all f (:; t), t 2 T .
\
With E (X; f ; T ) = E (X; f (:; t)) we denote the set of all T -uniform weakly
2
Pareto-optimal points for the function f .
The following Theorem has been recently obtained by B.Ricceri, as an application
of his very useful Alternative Principle for multifunctions ([5]).
Theorem B. ([6], Theorem 7) Assume that X R is non-empty, compact and
convex and that f = (f1; : : : ; f ) : X R ! R is such that, for i = 1; : : : ; k , f
is continuous on X R and f (:; t) is quasi-concave on X for all t 2 R. Moreover,
suppose that, for each x 2 X , the set ft 2 R : x is not weakly Pareto-optimal for the
function f (:; t)g is connected.
Then E (X; f ; R) is non-empty.
This note is divided in two parts. In the rst part we give a theorem on multifunctions and use it to deduce an extension of Theorem A (Theorem 1) in which the lower
semicontinuity assumption is dropped. In the second part we study the problem of
the existence of uniform weakly Pareto-optimal points from another point of view
compared to Theorem B.
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D.AVERNA
1. Non Parametric case: Existence and connectedness
We begin by observing that when f = (f1; : : :; f ) is dened on a compact topological space X , then for the existence of weakly Pareto-optimal points for f it sucies
to require that only one of the f 's is upper semicontinuous on X . In fact, the point
x which maximizes f is weakly Pareto-optimal for f .
In order to drop the lower semicontinuity restriction of Theorem A and to extend
it, we can proceed as follows.
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Lemma 1. Let X be a non-empty compact and convex subset of a real topological
vector space, let Y be a connected topological space and (Z; d) a metric space. Let
F : X Y ! Z be a multifunction with non-empty values and x z0 2 Z .
Suppose that:
(a) d(z0; F (:; :)) is lower semicontinuous;
(b) d(z0; F (x; :)) is upper semicontinuous;
(c) For each x0 2 X and for each y0 2 Y the set fx 2 X : d(z0; F (x; y0)) d(z0; F (x0; y0))g is convex.
Then the set fx 2 X : 9y 2 Y such that d(z0 ; F (x; y )) d(z0; F (x0; y )) 8x0 2 X g
is non-empty and connected.
Proof. Dene the multifunction G : Y ! X by putting, for each y 2 Y ,
G(y) =
\
0 2X
fx 2 X : d(z0; F (x; y)) d(z0; F (x0; y))g :
x
For each y 2 Y , G(y) is non-empty, because d(z0; F (:; y)) is lower semicontinuous
and X compact, and G(y) is convex in virtue of (c). Moreover, Gr(G) is closed
because d(z0; F (:; :)) ; d(z0; F (x0; :)) is lower semicontinuous. Then, since X is compact, G is upper semicontinuous ([2], Theorem 7.1.16, pg.78). By using Theorem 3.1
of [1], it follows that G(Y ), that is the set fx 2 X : 9y 2 Y such that d(z0; F (x; y)) d(z0; F (x0; y)) 8x0 2 X g is non-empty and connected. Remark. Under the hypotheses of Lemma 1, the connectedness of the set f(x; y) 2
X Y : such that d(z0; F (x; y)) d(z0; F (x0; y)) 8x0 2 X g can be also proved, by
using Theorem 3.2 (3.2) of [1] on the multifunction G.
With R+ we denote the strictly positive orthant of R , that is f = (1 ; : : :; ) 2
R : > 0 for all i = 1; : : : ; kg.
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Lemma 2. Let X be a non-empty set and (Z; d) a metric space. Let F : X ! Z ,
i = 1; : : :; k, be k multifunctions with non-empty values.
Then, for each z 2 Z , the set
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WEAKLY PARETO-OPTIMAL ALTERNATIVES ...
\ [
k
x
coincides with the set
[
(
\
fx 2 X : d(z; F (x)) d(z; F (x0))g
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)
max d(z; F (x0)) + 1 :
x 2 X : =1
max d(z; F (x)) + 1 =1
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2Rk+ x0 2X
Proof. If x 2=
X such that
0 2X i=1
3
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\ [
k
0 2X i=1
;:::;k
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;:::;k
i
fx 2 X : d(z; F (x)) d(z; F (x0))g, that is there exists x0 2
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x
d(z; F (x0)) < d(z; F (x)) for all i = 1; : : : ; k;
then, for each = (1; : : : ; ) 2 R+ ,
d(z; F (x0)) + 1 < d(z; F (x)) + 1 for all i = 1; : : : ; k:
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Thus, for some j 2 f1; : : : ; kg,
d(z; F (x0)) + 1 = d(z; F (x0)) + 1 < d(z; F (x)) + 1 max d(z; F (x)) + 1 ;
max
=1
=1
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;:::;k
j
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hence x 2=
[
\
(
j
j
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j
;:::;k
)
d(z; F (x0)) + 1 .
x 2 X : =1
max d(z; F (x)) + 1 =1
max
i
2Rk+ x0 2X
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;:::;k
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;:::;k
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\ [
To show the other inclusion, let x 2
fx 2 X : d(z; F (x)) d(z; F (x0))g
0 2 =1
and put = (1; : : :; ) = (d(z; F (x)) + 1; : : :; d(z; F (x)) + 1).
We claim that =1
max d(z; F (x)) + 1 =1
max d(z; F (x0)) + 1 , for each x0 2 X .
If not, there would exist x0 2 X such that
z; F (x)) + 1 > max d(z; F (x0)) + 1 ;
1 = =1
max d(
d(z; F (x)) + 1 =1 d(z; F (x)) + 1
then
d(z; F (x)) > d(z; F (x0)); for all i = 1; : : : ; k;
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X i
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;:::;k
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;:::;k
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;:::;k
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\ [
thus x 2=
k
0 2X i=1
x
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;:::;k
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fx 2 X : d(z; F (x)) d(F (x0))g, a contradiction. i
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D.AVERNA
Theorem 1. Let X be a non-empty compact and convex subset of a real topological vector space and (Z; d) a metric space. Let F : X ! Z , i = 1; : : : ; k , be k
i
multifunctions with non-empty values such that:
(d) F is upper semicontinuous for each i = 1; : : : ; k;
(e) for each i = 1; : : : ; k and each open ball B (z; r) of Z , the set F ;(B (z; r)) is
convex.
\ [
Then, for each z 2 Z , the set
fx 2 X : d(z; F (x)) d(z; F (x0))g is
0 2 =1
non-empty and connected.
Besides, if Z is also connected, then the multifunction G : Z ! X , dened by
\ [
G(z) =
fx 2 X : d(z; F (x)) d(z; F (x0))g, for each z 2 Z , is such that
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X i
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0 2X i=1
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G(Z ) and Gr(G) are (non-empty) and connected.
Proof. Fix z 2 Z and denote with the usual metric on R. Dene the multifunction F : X R+ ! R by putting, for each x 2 X and each = (1; : : : ; ) 2 R+ ,
(
)
d(
z;
F
(
x
))
+
1
:
F (x; ) = =1
max
For each x 2 X and each = (1; : : : ; ), (0; F (x; )) = =1
max d(z; F (x)) + 1 .
For each x 2 X , d(z; F (:)) is lower semicontinuous, in virtue of Theorem 1.2 of
[4]; thus (:; :) is lower semicontinuous.
For each x 2 X , (x; :) is continuous.
Moreover, for each i = 1; : : :; k and each 0 > 0, the function d(z; F (x0 )) + 1 is
quasi-convex, in virtue of (e). In fact, for each 2 R,
) n
(
o
d(
z;
F
(
x
))
+
1
0 ; 1
=
x
2
X
:
d(
z;
F
(
x
))
x2X :
0
and this last set is the empty set if 0 ; 1 < 0, whereas, if 0 ; 1 0, it concides
x
k
k
k
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;:::;k
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k
i
;:::;k
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with the set
i
1
\
=1
i
F ;(B (z; 0 ; 1 + 1=n)):
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Therefore, for each = (1; : : : ; ), (0; F (x; :)), as maximum of quasi-convex
functions, is quasi-convex.
In denitive, we can use Lemma 1 on F and then the set
(
)
[ \
d(
z;
F
(
x
))
+
1
d(
z;
F
(
x
0)) + 1
x 2 X : max
max
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2Rk+ x0 2X
=1
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;:::;k
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=1
;:::;k
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WEAKLY PARETO-OPTIMAL ALTERNATIVES ...
5
is non-empty and connected. Thus the rst part follows by Lemma 2.
For the second part, for each i = 1; : : : ; k, the function d(:; F (:)), is lower semicontinuous ([4], Theorem 1.2), then, for each i = 1; : : : ; k and each x0 2 X the function
d(:; F (:)) ; d(:; F (x0)) is lower semicontinuous; then the set
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\ [
k
0 2X i=1
f(z; x) 2 Z X : d(z; F (x)) d(z; F (x0))g;
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x
that is Gr(G), is closed; hence, since X is compact, the multifunction G (dened
on the connected Z and non-empty and connected valued) is upper semicontinuous
([2], Theorem 7.1.16, pg.78) and compact-valued. Therefore G(Z ) and Gr(G) are
(non-empty) and connected, in virtue of Theorem 3.1 and Theorem 3.2 (3.2) of [1].
Corollary 1. Let X be a non-empty compact and convex subset of a real topological
vector space and f = (f1; : : :; f ) : X ! R a function such that, for all i = 1; : : : ; k ,
k
f is upper semicontinuous and quasi-concave on X .
Then E (X; f ) is non-empty and connected.
Proof. For each i = 1; : : : ; k dene the multifunction F : X ! R by putting
F (x) =] ; 1; f (x)], for each x 2 X . With denote, again, the usual metric on R.
The multifunctions F , i = 1; : : : ; k, satisfy (d) and (e) of Theorem 1. Thus, in
particular, for z = 1=1
max max
f (x), the set
2
k
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;:::;k x
\ [
k
0 2X i=1
X
i
fx 2 X : (z; F (x)) (z; F (x0))g;
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x
that is E (X; f ), is non-empty and connected. w
2. Parametric case: Existence
Unlike the non parametric case, the parametric case is more dicult because, as
very simple examples show, also a good regularity of f doesn't imply existence. Besides, the diculty seems to be principally in an adeguate resolution of the existence
problem for the scalar case (k = 1); this is surely true in the case of one real function f dened on X T . In this case, in fact, E (X; f; T ) is the set of all points
which maximize f (:; t) T -uniformly and its convexity follows immediately assuming
the quasi-concavity of the function with respect to x.
We don't know, besides Theorem B, other theorems which give sucient conditions
for the existence of T -uniform weakly Pareto-optimal points, apart some trivial ones.
On the other hand, the connectedness of the set of all t such that x is not weakly
Pareto-optimal for f (:; t), requested in Theorem B, doesn't include some simple examples which can be given.
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D.AVERNA
So, in this second part, we study the problem of the existence of uniform weakly
Pareto-optimal points from another point of view respect to Theorem B.
We begin with the following
Lemma 3. Let X and T be two non-empty sets and f = (f1; : : :; f ) : X T ! R .
Suppose that x 2 X satises at least one of the following conditions:
(1) for some j 2 f1; : : :; kg and for each t 2 T , f (:; t) attains its supremum at x;
(2) for each t 2 T there exists (t) = (1(t); : : :; (t)) 2 R+ such that the
function L(f (:); (t)) : X ! R, dened by L(f (x); (t)) = =1
min (t)f (x; t),
attains its supremum at x;
Then x is T -uniform weakly Pareto-optimal for f .
Proof. If x satises (1), then the conclusion is obvious.
For the case (2), suppose that, for some t 2 T , x is not weakly Pareto-optimal for
f (:; t). This means that there exists x 2 X such that:
f (x; t) > f (x; t) for all i = 1; : : : ; k:
Then, for each (t) = (1(t); : : :; (t)) 2 R+, we have
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k
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k
k
(t)f (x; t) > (t)f (x; t) for all i = 1; : : : ; k;
i
hence
;:::;k
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L(f (x); (t)) = =1min (t)f (x; t) > =1min (t)f (x; t) = L(f (x); (t)):
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;:::;k
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;:::;k
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Thus, x can't verify (2). It is clear how one can tackle in several ways the problem of the existence of uniform
weakly Pareto-optimal points for the function f , by using the previous Lemma 3 and
an existence theorem for the scalar case. Thus, in the following Theorem 2, Lemma 4
and Corollary 2 we consider a function f : X T ! R.
Theorem 2. Let X be a compact topological space and T be a non-empty set. Let f
be a real function on X T such that:
() f (:; t) is upper semicontinuous for all t 2 T ;
( ) for each x0; x1 2 X at least one of the following conditions is satised:
(1)f (x0; t) ; f (x1; t) 0 for all t 2 T ;
(2)f (x1; t) ; f (x0; t) 0 for all t 2 T ;
(3) there exists a continuous function ' : [0; 1] ! X such that '(0) = x0,
'(1) = x1 and, for each s 2 [0; 1], either
f ('(s); t) ; f (x0; t) 0 for all t 2 T
WEAKLY PARETO-OPTIMAL ALTERNATIVES ...
7
or
f ('(s); t) ; f (x1; t) 0 for all t 2 T:
Then, there exists some x 2 X such that, for all t 2 T , the functions f (:; t) attain
their suprema at x.
Proof. Dene G : X ! X by putting, for each x 2 X ,
G(x) = y 2 X : inf
(
f
(
y;
t
)
;
f
(
x;
t
))
0
:
2
For each x 2 X we have
\
G(x) = fy 2 X : f (y; t) f (x; t)g :
t
T
2
t
T
thus, by (), G(x) is a closed subset of X .
We claim that G(z1) \ \ G(z ) 6= ; for each nite subset fz1; : : :; z g of X .
In fact, for n = 1 it is true because, for each x 2 X , x 2 G(x).
Suppose the property true for any subset of X consisting of n elements and let
fz1; : : :; z ; z +1g X . Then, by the inductive hypothesis, there exists x0 2 G(z1) \
\ G(z ). Let us put x1 = z +1 and use ( ).
If (1) holds, then x0 2 G(z1) \ \ G(z ) \ G(z +1 ).
If (2) holds, then z +1 2 G(x0), thus z +1 2 G(z1) \ \ G(z ) \ G(z +1 ), because
G(x0) G(z0) \ \ G(z ) and z +1 2 G(z +1 ).
If (3) holds, then '(0) = x0, '(1) = z +1 and '(s) 2 G(x0) [ G(z +1 ) for each
s 2 [0; 1]. Now '([0; 1]) is connected, so it follows that G(x0) \ G(z +1 ) 6= ;. A
fortiori, G(z1) \ \ G(z ) \ G(z +1 ) 6= ;, because G(x0) G(z1) \ \ G(z ).
Hence the claim is proved.
\
Therefore, by the compactness of X , there exists x 2 X such that x 2 G(x).
2
In x, obviously, all the f (:; t)'s, t 2 T , attain their suprema. Lemma 4. Let X be a non-empty convex subset of a real topological vector space and
T be a non-empty set. Let f be a real function on X T .
Then the following two statements are true:
(I) If, for each x0; x1 2 X and each x 2 cofx0; x1g, one has either
inf
(f (x; t) ; f (x0; t)) 0
2
or
inf
(f (x; t) ; f (x1; t)) 0;
2
then the set
X n G = z 2 X : inf
(f (y; t) ; f (z; t)) < 0
2
is convex for all y 2 X .
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D.AVERNA
(II) Let x0; x1 2 X and x 2 cofx0; x1g. If the set
X n G = z 2 X : inf
(f (x; t) ; f (z; t)) < 0
2
x
t
is convex, then either
T
inf
(f (x; t) ; f (x0; t)) 0
2
t
or
T
inf
(f (x; t) ; f (x1; t)) 0:
2
t
T
Proof. (I). Suppose that, for some y 2 X , the set X n G is not convex. Then there
exist x; x0; x1 2 X , with x 2 cofx0; x1g, and t0; t1 2 T such that
f (y; t0) ; f (x0; t0) < 0
and
f (y; t1) ; f (x1; t1) < 0;
whereas
f (y; t0) ; f (x; t0) 0
and
f (y; t1) ; f (x; t1) 0:
Thus
f (x; t0) ; f (x0; t0) < 0
and
f (x; t1) ; f (x1; t1) < 0;
in contrast with the hypothesis.
(II). If
inf
(f (x; t) ; f (x0; t)) < 0
2
and
inf
(f (x; t) ; f (x1; t)) < 0;
2
y
t
t
T
T
then x0; x1 2 X n G , whereas, obviously, x 62 X n G . Thus, X n G is not convex. x
x
x
Taking into account (II) of Lemma 4, we can formulate the following particular
case of Theorem 2.
WEAKLY PARETO-OPTIMAL ALTERNATIVES ...
9
Corollary 2. Let X a non-empty compact and convex subset of a real topological
vector space and T be a non-empty set. Let f be a real function on X T such that:
() f (:; t) is upper semicontinuous for all t 2 T ;
( 0) for each x0; x1 2 X at least one of the following conditions (1), (2), (30 ) or
(300) is satised:
(1) inf
(f (x0; t) ; f (x1; t)) 0;
2
(2) inf
(f (x1; t) ; f (x0; t)) 0;
2
(30 ) for each x 2 cofx0; x1g one has either
inf
(f (x; t) ; f (x0; t)) 0
2
or
inf
(f (x; t) ; f (x1; t)) 0:
2
t
T
t
T
t
t
T
T
(300) for each x 2 cofx0; x1g the set
X n G = z 2 X : inf
(f (x; t) ; f (z; t)) < 0
2
x
t
T
is convex.
Then, there exists some x 2 X such that, for all t 2 T , the functions f (:; t) attain
their suprema at x.
Proof. For x0; x1 2 X , (300) ) (30 ) ) (3) of Theorem 2. The existence of T -uniform weakly Pareto-optimal points for a function f =
(f1; : : :; f ) : X T ! R can be easily obtained also without any preventive "scalarization", but, in this case, not all the conditions are in terms of simple properties
given explicitily on the f 's.
When X is a non-empty compact and convex subset of a real Hausdor topological
vector space, the following Theorem can be easily deduced by a very famous Lemma
of Ky Fan ([3], Lemma 4).
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Theorem 3. Let X a non-empty compact and convex subset of a real Hausdor
topological vector space and T be a non-empty set. Let f = (f1 ; : : : ; f ) : X T ! R
be a function such that:
(i) f (:; t) is upper semicontinuous for all t 2 T and for all i = 1; : : :; k;
(ii) for each y 2 X , the set X n A = fz 2 X : there exists t 2 T such that
f (z; t) > f (y; t) for all i 2 f1; : : : kgg is convex.
Then, there exists some y0 2 X which is T -uniform weakly Pareto-optimal for the
k
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function f .
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D.AVERNA
Proof. The set A = f(x; y) 2 X X : for each t 2 T there exists j 2 f1; : : : ; kg
such that f (x; t) f (y; t)g satises Lemma 4 of [3] (which holds also when only
the x-sections of A, but not necessarily A, are closed). Thus, there exists y0 2 X
such that X fy0g A, that is, for each x 2 X and for each t 2 T there exists
j 2 f1; : : : ; kg such that f (y0; t) f (x; t). j
j
j
j
In consequence of the following Proposition, we have that, in the particular case
when X = [a; b], Theorem B is contained in Theorem 3.
Proposition 1. Let X and T be two topological spaces. Let f = (f1; : : :; f ) : X X ! R be a function such that:
(j) f (:; t) is quasi-concave for all t 2 T and for all i = 1; : : : ; k;
(jj) f (x; :) is continuous for all x 2 X and for all i = 1; : : : ; k;
Then, for each y 2 X such that the set T (y) = ft 2 T : y is not weakly Paretooptimal for the function f (:; t)g is connected, the set X n A = fz 2 X : there exists
t 2 T such that f (z; t) > f (y; t) for all i 2 f1; : : : kgg is connected.
Proof. Let y 2 X such that T (y) is connected. Dene H : T ! X by putting, for
each t 2 T ,
H (t) = fz 2 X : f (z; t) > f (y; t) for all i 2 f1; : : : ; kgg :
For each t 2 T (y), the set H (t) is non-empty and, by (j), connected, because
1 \
[
1
H (t) =
z 2 X : f (z; t) f (y; t) + :
k
k
i
i
y
i
i
i
i
k
i
=1 =1
n
Moreover, for each X ,
H ; (
) =
[
z
2
n
i
i
H ; (fzg) =
[ \
k
z
2
=1
ft 2 T : f (z; t) > f (y; t)g;
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hence, by (jj), H is lower semicontinuous.
Thus, by using Theorem 3.1 of [1], we obtain that the set X n A = H (T (y)) is
connected. y
More in particular, when X = [a; b] and k = 1, then also Corollary 2 contains
Theorem B, because, in this context, the set X n G (dened in Lemma 4) coincides
with X n A for all y 2 X .
y
y
The inclusions before stated are strict; in fact:
Example. Let f : [;1; 1] R ! R dened by
WEAKLY PARETO-OPTIMAL ALTERNATIVES ...
(
n
11
o
2
0
f (x; t) = 0min ; j j x; xt sin t ,, ifif tt 6=
= 0:
f is continuous (and f (:; t) is concave for all t 2 R). Moreover, for each x 2
[;1; 1] n f0g, T (x) = R n f2h : h 2 Zg. Hence f does not satisfy Theorem B.
f satises Corollary 2 and Theorem 3. In fact, for each x; y 2 [;1; 1]:
(1) if 0 x y, then
(
2t
(
y
;
x
)
sin
0
j
j
f (x; t) ; f (y; t) = (y ; x)(;t) sin2 t ,, ifif tt >
0;
thus f (x; t) ; f (y; t) is non-negative for all t 2 R;
(2) If y x 0, then
(
(x ; y)t sin2 t
, if t 0
f (x; t) ; f (y; t) = (x ; y) ; sin2 t , if t < 0;
jj
so, again, f (x; t) ; f (y; t) is non-negative for all t 2 R.
Hence, taking into account the previous two steps, for each x0; x1 2 [;1; 1] we have
that:
(3) if x0 x1 0, then (1) or (2) and, in any case, (30 ) of Corollary 2 hold;
(4) if x0 x1 < 0, then (30 ) of Corollary 2 holds.
Furthermore, in virtue of (I) of Lemma 4, (ii) of Theorem 3 holds.
t
t
t
t
t
t
References
1. Hiriart-Urruty J.B., Images of connected sets by semicontinuous multifunctions, J. Math. Anal.
Appl., 111 (1985), pp.407-422.
2. Klein E. and Thompson A.C., Theory of correspondences, John Wiley & Sons, 1984.
3. Fan K., A generalization of Tychono's xed point theorem, Math. Ann., 142 (1961), pp.305310.
4. Ricceri B., On multifunctions with convex graph, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis.
Mat. Natur., 77 (1984), pp.64-70.
5. Ricceri B., Some topological mini-max theorems via an alternative principle for multifunctions,
Arch. Math., 60 (1993), pp.367-377.
6. Ricceri B., A theorem on sets with connected sections and some of its applications.
7. Warburton A. R., Quasiconcave vector maximization: Connectedness of the sets of Paretooptimal and weak Pareto-optimal alternatives, J. Opt. Th. Appl., 40 (1983), pp.537-557.