DIEGO AVERNA. A point x 2 X is said to be weakly Pareto-optimal for the function f provided

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1 WEAKLY PARETO-OPTIMAL ALTERNATIVES FOR A VECTOR MAXIMIZATION PROBLEM: EXISTENCE AND CONNECTEDNESS DIEGO AVERNA Let X be a non-empty set and f =f 1 ::: f k ):X! R k a functon. A pont x 2 X s sad to be weakly Pareto-optmal for the functon f provded there s no x 2 X such that f x) >f x) for all =1 ::: k. Wth E w X f) we denote the set of all Pareto-optmal ponts for the functon f. Let us recall the followng Theorem, due to A.R.Warburton: Theorem A. [7], Theorem 4.1) Let X R n be a non-empty, compact and convex set and f =f 1 ::: f k ):X! R k such that f 1 ::: f k are contnuous and quasconcave on X. Then E w X f) s non-empty and connected. When the functon f depends also from a parameter t, whchvares n a non-empty set T,apont x 2 X s sad to be T -unform weakly Pareto-optmal for the functon f f x s weakly Pareto-optmal for all f: t), t 2 T. Wth E w X f T)= \ E w X f: t)) we denote the set of all T -unform weakly Pareto-optmal ponts for the functon f. The followng Theorem has been recently obtaned by B.Rccer, as an applcaton of hs very useful Alternatve Prncple for multfunctons [5]). Theorem B. [6], Theorem 7) Assume that X R n s non-empty, compact and convex and that f =f 1 ::: f k ):XR! R k s such that, for =1 ::: k, f s contnuous on X R and f : t) s quas-concave on X for all t 2 R. Moreover, suppose that, for each x 2 X, the set ft 2 R : x s not weakly Pareto-optmal for the functon f: t)g s connected. Then E w X f R) s non-empty. Ths note s dvded n two parts. In the rst part we gve a theorem on multfunctons and use t to deduce an extenson of Theorem A Theorem 1) n whch thelower semcontnuty assumpton s dropped. In the second part we study the problem of the exstence of unform weakly Pareto-optmal ponts from another pont of vew compared to Theorem B. 1

2 2 D.AVERNA 1. Non Parametrc case: Exstence and connectedness We begn by observng that when f =f 1 ::: f k ) s dened on a compact topologcal space X, then for the exstence of weakly Pareto-optmal ponts for f t suces to requre that only one of the f 's s upper semcontnuous on X. In fact, the pont x whch mzes f s weakly Pareto-optmal for f. In order to drop the lower semcontnuty restrcton of Theorem A and to extend t, we can proceed as follows. Lemma 1. Let X be a non-empty compact and convex subset of a real topologcal vector space, let Y be aconnected topologcal space andz d) ametrcspace. Let F : X Y! Z be a multfuncton wth non-empty values and x z 0 2 Z. Suppose that: a) dz 0 F: :)) s lower semcontnuous b) dz 0 Fx :)) s upper semcontnuous c) For each x 0 2 X and for each y 0 2 Y the set fx 2 X :dz 0 Fx y 0 )) dz 0 Fx 0 y 0 ))g s convex. Then the set fx 2 X : 9y 2 Y such that dz 0 Fx y)) dz 0 Fx 0 y)) 8x 0 2 Xg s non-empty and connected. Proof. Dene the multfuncton G : Y! X by puttng, for each y 2 Y, Gy) = \ x 0 2X fx 2 X :dz 0 Fx y)) dz 0 Fx 0 y))g : For each y 2 Y, Gy) s non-empty, because dz 0 F: y)) s lower semcontnuous and X compact, and Gy) s convex n vrtue of c). Moreover, GrG) s closed because dz 0 F: :)) ; dz 0 Fx 0 :)) s lower semcontnuous. Then, snce X s compact, G s upper semcontnuous [2], Theorem , pg.78). By usng Theorem 3.1 of [1], t follows that GY ), that s the set fx 2 X : 9y 2 Y such that dz 0 Fx y)) dz 0 Fx 0 y)) 8x 0 2 Xg s non-empty and connected. Remark. Under the hypotheses of Lemma 1, the connectedness of the set fx y) 2 X Y :suchthatdz 0 Fx y)) dz 0 Fx 0 y)) 8x 0 2 Xg can be also proved, by usng Theorem ) of [1] on the multfuncton G. Wth R k + we denote the strctly postve orthant ofr k,thatsf = 1 ::: k ) 2 R k : > 0 for all =1 ::: kg. Lemma 2. Let X be a non-empty set and Z d) a metrc space. Let F : X! Z, =1 ::: k,be k multfunctons wth non-empty values. Then, for each z 2 Z, the set

3 WEAKLY PARETO-OPTIMAL ALTERNATIVES... 3 \ k[ fx 2 X :dz F x)) dz F x 0 ))g x 0 2X =1 concdes wth the set [ \ dz F x)) + 1 x 2 X : 2R k =1 ::: k x 0 2X + =1 ::: k ) dz F x 0 )) + 1 : Proof. If x=2 \ X such that x 0 2X =1 then, for each = 1 ::: k ) 2 R k +, k[ fx 2 X :dz F x)) dz F x 0 ))g, that s there exsts x 0 2 dz F x 0 )) < dz F x)) for all =1 ::: k dz F x 0 ))+1 < dz F x)) + 1 Thus, for some j 2f1 ::: kg, dz F x 0 ))+1 = dz F jx 0 )) + 1 < dz F jx)) + 1 =1 ::: k j j hence x=2 [ 2R k + \ x 0 2X x 2 X : =1 ::: k dz F x)) + 1 To show the other ncluson, let x 2 \ x 0 2X =1 for all =1 ::: k: =1 ::: k =1 ::: k dz F x))+1 ) dz F x 0 )) + 1. k[ fx 2 X :dz F x)) dz F x 0 ))g and put = 1 ::: k ) = dz F x)) + 1 ::: dz F k x)) + 1). dz F x)) + 1 We clam that =1 ::: k =1 ::: k If not, there would exst x 0 2 X such that then dz F x 0 )) + 1, for each x 0 2 X. dz F x)) + 1 1= =1 ::: k dz F x)) + 1 > dz F x 0 )) + 1 =1 ::: k dz F x)) + 1 dz F x)) > dz F x 0 )) for all =1 ::: k thus x=2 \ k[ fx 2 X :dz F x)) df x 0 ))g, acontradcton. x 0 2X =1

4 4 D.AVERNA Theorem 1. Let X be a non-empty compact and convex subset of a real topologcal vector space and Z d) a metrc space. Let F : X! Z, =1 ::: k, be k multfunctons wth non-empty values such that: d) F s upper semcontnuous for each =1 ::: k e) for each =1 ::: k and each open ball B z r) of Z, the set F ; B z r)) s convex. \ k[ Then, for each z 2 Z, the set fx 2 X :dz F x)) dz F x 0 ))g s x 0 2X =1 non-empty and connected. Besdes, f Z s also connected, then the multfuncton G : Z! X, dened by Gz) = \ k[ fx 2 X :dz F x)) dz F x 0 ))g, for each z 2 Z, s such that x 0 2X =1 GZ) and GrG) are non-empty) and connected. Proof. Fx z 2 Z and denote wth the usual metrc on R. Denethemultfuncton F : X R k +! R by puttng, for each x 2 X and each = 1 ::: k ) 2 R k +, ) dz F x)) + 1 F x ) = : =1 ::: k dz F x))+1 For each x 2 X and each = 1 ::: k ), 0 Fx )) =. =1 ::: k For each x 2 X, dz F :)) s lower semcontnuous, n vrtue of Theorem 1.2 of [4] thus : :) slower semcontnuous. For each x 2 X, x :) scontnuous. Moreover, for each =1 ::: k and each 0 > 0, the functon dz F x))+1 s quas-convex, n vrtue of e). In fact, for each 2 R, x 2 X : dz F ) x))+1 = n x 2 X :dz F 0 x)) 0 ; 1o and ths last set s the empty set f 0 ; 1 < 0, whereas, f 0 ; 1 0, t concdes wth the set 1\ F ; B z 0 ; 1+1=n)): n=1 Therefore, for each = 1 ::: k ), 0 Fx :)), as mum of quas-convex functons, s quas-convex. In dentve, we can use Lemma 1 on F and then the set ) [ 2R k + \ x 0 2X x 2 X : =1 ::: k dz F x)) + 1 =1 ::: k 0 dz F x 0 ))+1

5 WEAKLY PARETO-OPTIMAL ALTERNATIVES... 5 s non-empty and connected. Thus the rst part follows by Lemma 2. For the second part, for each =1 ::: k, the functon d: F :)), s lower semcontnuous [4], Theorem 1.2), then, for each =1 ::: k and each x 0 2 X the functon d: F :)) ; d: F x 0 )) s lower semcontnuous then the set \ x 0 2X =1 k[ fz x) 2 Z X :dz F x)) dz F x 0 ))g that s GrG), s closed hence, snce X s compact, the multfuncton G dened on the connected Z and non-empty and connected valued) s upper semcontnuous [2], Theorem , pg.78) and compact-valued. Therefore GZ) and GrG) are non-empty) and connected, n vrtue of Theorem 3.1 and Theorem ) of [1]. Corollary 1. Let X be a non-empty compact and convex subset of a real topologcal vector space and f =f 1 ::: f k ):X! R k a functon such that, for all =1 ::: k, f s upper semcontnuous and quas-concave on X. Then E w X f) s non-empty and connected. Proof. For each =1 ::: k dene the multfuncton F : X! R by puttng F x) =];1 f x)], for each x 2 X. Wth denote, agan, the usual metrc on R. The multfunctons F, =1 ::: k, satsfy d) and e) of Theorem 1. Thus, n partcular, for z = 1=1 ::: k \ x2x x 0 2X =1 f x), the set k[ fx 2 X : z F x)) z F x 0 ))g that s E w X f), s non-empty and connected. 2. Parametrc case: Exstence Unlke the non parametrc case, the parametrc case s more dcult because, as very smple examples show, also a good regularty of f doesn't mply exstence. Besdes, the dculty seems to be prncpally n an adeguate resoluton of the exstence problem for the scalar case k = 1) ths s surely true n the case of one real functon f dened on X T. In ths case, n fact, E w X f T) s the set of all ponts whch mze f: t) T -unformly and ts convexty follows mmedately assumng the quas-concavty of the functon wth respect to x. We don't know, besdes Theorem B, other theorems whchgve sucent condtons for the exstence of T -unform weakly Pareto-optmal ponts, apart some trval ones. On the other hand, the connectedness of the set of all t such thatx s not weakly Pareto-optmal for f: t), requested n Theorem B, doesn't nclude some smple examples whch can be gven.

6 6 D.AVERNA So, n ths second part, we study the problem of the exstence of unform weakly Pareto-optmal ponts from another pont of vew respect to Theorem B. We begn wth the followng Lemma 3. Let X and T be two non-empty sets and f =f 1 ::: f k ):X T! R k. Suppose that x 2 X satses at least one of the followng condtons: 1) for some j 2f1 ::: kg and for each t 2 T, f j : t) attans ts supremum at x 2) for each t 2 T there exsts t) = 1 t) ::: k t)) 2 R k + such that the functon Lf:) t)) : X! R, dened bylfx) t)) = mn t)f x t), =1 ::: k attans ts supremum at x Then x s T -unform weakly Pareto-optmal for f. Proof. If x satses 1), then the concluson s obvous. For the case 2), suppose that, for some t 2 T, x s not weakly Pareto-optmal for f: t). Ths means that there exsts x 2 X such that: f x t) >f x t) for all =1 ::: k: Then, for each t) = 1 t) ::: k t)) 2 R k +,wehave hence t)f x t) > t)f x t) for all =1 ::: k Lfx) t)) = mn =1 ::: k t)f x t) > mn =1 ::: k t)f x t) =Lfx) t)): Thus, x can't verfy 2). It s clear how one can tackle n several ways the problem of the exstence of unform weakly Pareto-optmal ponts for the functon f, by usng the prevous Lemma 3 and an exstence theorem for the scalar case. Thus, n the followng Theorem 2, Lemma 4 and Corollary 2 we consder a functon f : X T! R. Theorem 2. Let X be a compact topologcal space and T be a non-empty set. Let f be areal functon on X T such that: ) f: t) s upper semcontnuous for all t 2 T ) for each x 0 x 1 2 X at least one of the followng condtons s satsed: 1 )fx 0 t) ; fx 1 t) 0 for all t 2 T 2 )fx 1 t) ; fx 0 t) 0 for all t 2 T 3 ) there exsts a contnuous functon ' :[0 1]! X such that '0) = x 0, '1) = x 1 and, for each s 2 [0 1], ether f's) t) ; fx 0 t) 0 for all t 2 T

7 or WEAKLY PARETO-OPTIMAL ALTERNATIVES... 7 f's) t) ; fx 1 t) 0 for all t 2 T: Then, there exsts some x 2 X such that, for all t 2 T, the functons f: t) attan ther suprema at x. Proof. Dene G : X! X by puttng, for each x 2 X, Gx) = y 2 X :nffy t) ; fx t)) 0 : For each x 2 X we have Gx) = \ fy 2 X : fy t) fx t)g : thus, by ), Gx) s a closed subset of X. We clam that Gz 1 ) \\Gz n ) 6= for each nte subset fz 1 ::: z n g of X. In fact, for n = 1 t s true because, for each x 2 X, x 2 Gx). Suppose the property true for any subset of X consstng of n elements and let fz 1 ::: z n z n+1 gx. Then, by the nductve hypothess, there exsts x 0 2 Gz 1 ) \ \Gz n ). Let us put x 1 = z n+1 and use ). If 1 ) holds, then x 0 2 Gz 1 ) \\Gz n ) \ Gz n+1 ). If 2 ) holds, then z n+1 2 Gx 0 ), thus z n+1 2 Gz 1 )\\Gz n )\Gz n+1 ), because Gx 0 ) Gz 0 ) \\Gz n )andz n+1 2 Gz n+1 ). If 3 ) holds, then '0) = x 0, '1) = z n+1 and 's) 2 Gx 0 ) [ Gz n+1 ) for each s 2 [0 1]. Now '[0 1]) s connected, so t follows that Gx 0 ) \ Gz n+1 ) 6=. A fortor, Gz 1 ) \\Gz n ) \ Gz n+1 ) 6=, because Gx 0 ) Gz 1 ) \\Gz n ). Hence the clam s proved. Therefore, by the compactness of X, there exsts x 2 X such thatx 2 \ In x, obvously, all the f: t)'s, t 2 T, attan ther suprema. x2x Gx). Lemma 4. Let X be a non-empty convex subset of a real topologcal vector space and T be a non-empty set. Let f be areal functon on X T. Then the followng two statements are true: I) If, for each x 0 x 1 2 X and each x 2 cofx 0 x 1 g, one has ether or then the set X n G y = s convex for all y 2 X. nf fx t) ; fx 0 t)) 0 nf fx t) ; fx 1 t)) 0 z 2 X : nf fy t) ; fz t)) < 0

8 8 D.AVERNA II) Let x 0 x 1 2 X and x 2 cofx 0 x 1 g. If the set X n G x = z 2 X : nf fx t) ; fz t)) < 0 s convex, then ether or nf fx t) ; fx 0 t)) 0 nf fx t) ; fx 1 t)) 0: Proof. I). Suppose that, for some y 2 X, the set X n G y s not convex. Then there exst x x 0 x 1 2 X, wth x 2 cofx 0 x 1 g,andt 0 t 1 2 T such that and whereas and and Thus n contrast wth the hypothess. II). If and fy t 0 ) ; fx 0 t 0 ) < 0 fy t 1 ) ; fx 1 t 1 ) < 0 fy t 0 ) ; fx t 0 ) 0 fy t 1 ) ; fx t 1 ) 0: fx t 0 ) ; fx 0 t 0 ) < 0 fx t 1 ) ; fx 1 t 1 ) < 0 nf fx t) ; fx 0 t)) < 0 nf fx t) ; fx 1 t)) < 0 then x 0 x 1 2 X n G x, whereas, obvously, x 62 X n G x.thus, X n G x s not convex. Takng nto account II) of Lemma 4, we can formulate the followng partcular case of Theorem 2.

9 WEAKLY PARETO-OPTIMAL ALTERNATIVES... 9 Corollary 2. Let X a non-empty compact and convex subset of a real topologcal vector space andt be a non-empty set. Let f be areal functon on X T such that: ) f: t) s upper semcontnuous for all t 2 T ) for each x 0 x 1 2 X at least one of the followng condtons 1 ), 2 ), 0) or 3 ) s satsed: 1 ) nf fx 0 t) ; fx 1 t)) 0 2 ) nf fx 1 t) ; fx 0 t)) 0 0 ) for each x 2 cofx 3 0 x 1 g one has ether or nf fx t) ; fx 0 t)) 0 00 nf fx t) ; fx 1 t)) 0: ) for each x 2 cofx 3 0 x 1 g the set X n G x = z 2 X : nf fx t) ; fz t)) < 0 s convex. Then, there exsts some x 2 X such that, for all t 2 T, the functons f: t) attan ther suprema at x. Proof. For x 0 x 1 2 X, 00 3) ) 0 3) ) 3 )oftheorem2. The exstence of T -unform weakly Pareto-optmal ponts for a functon f = f 1 ::: f k ):X T! R k can be easly obtaned also wthout any preventve "scalarzaton", but, n ths case, not all the condtons are n terms of smple propertes gven explctly on the f 's. When X s a non-empty compact and convex subset of a real Hausdor topologcal vector space, the followng Theorem can be easly deduced by a very famous Lemma of Ky Fan [3], Lemma 4). Theorem 3. Let X a non-empty compact and convex subset of a real Hausdor topologcal vector space and T be a non-empty set. Let f =f 1 ::: f k ):X T! R k be a functon such that: ) f : t) s upper semcontnuous for all t 2 T and for all =1 ::: k ) for each y 2 X, the set X n A y = fz 2 X : there exsts t 2 T such that f z t) >f y t) for all 2f1 ::: kgg s convex. Then, there exsts some y 0 2 X whch s T -unform weakly Pareto-optmal for the functon f.

10 10 D.AVERNA Proof. The set A = fx y) 2 X X : for each t 2 T there exsts j 2f1 ::: kg such thatf j x t) f j y t)g satses Lemma 4 of [3] whch holds also when only the x-sectons of A, but not necessarly A, are closed). Thus, there exsts y 0 2 X such thatxfy 0 ga, that s, for each x 2 X and for each t 2 T there exsts j 2f1 ::: kg such that f j y 0 t) f j x t). In consequence of the followng Proposton, we have that, n the partcular case when X =[a b], Theorem B s contaned n Theorem 3. Proposton 1. Let X and T be two topologcal spaces. Let f =f 1 ::: f k ):X X! R k be a functon such that: j) f : t) s quas-concave for all t 2 T and for all =1 ::: k jj) f x :) s contnuous for all x 2 X and for all =1 ::: k Then, for each y 2 X such that the set T y) =ft 2 T : y s not weakly Paretooptmal for the functon f: t)g s connected, the set X n A y = fz 2 X : there exsts t 2 T such that f z t) >f y t) for all 2f1 ::: kgg s connected. Proof. Let y 2 X such that T y) s connected. Dene H : T! X by puttng, for each t 2 T, Ht) =fz 2 X : f z t) >f y t) for all 2f1 ::: kgg : For each t 2 T y), the set Ht) s non-empty and,by j), connected, because Ht) = 1[ Moreover, for each X, H ; ) = [ z2 k\ n=1 =1 z 2 X : f z t) f y t)+ 1 : n H ; fzg) = [ z2 =1 k\ ft 2 T : f z t) >f y t)g hence, by jj), H s lower semcontnuous. Thus, by usng Theorem 3.1 of [1], we obtan that the set X n A y = HT y)) s connected. More n partcular, when X =[a b] andk = 1, then also Corollary 2 contans Theorem B, because, n ths context, the set X n G y dened n Lemma 4) concdes wth X n A y for all y 2 X. The nclusons before stated are strct n fact: Example. Let f :[;1 1] R! R dened by

11 WEAKLY PARETO-OPTIMAL ALTERNATIVES fx t) = mn n ; t jtj x xt o sn 2 t,ft 6= 0 0,ft =0: f s contnuous and f: t) sconcave for all t 2 R). Moreover, for each x 2 [;1 1] nf0g, T x) =R nf2h : h 2 Zg. Hence f does not satsfy Theorem B. f satses Corollary 2 and Theorem 3. In fact, for each x y 2 [;1 1]: 1) f 0 x y, then y ; x) t fx t) ; fy t) = jtj sn2 t,ft>0 y ; x);t)sn 2 t,ft0 thus fx t) ; fy t) s non-negatve for all t 2 R 2) If y x 0, then x ; y)t sn 2 t,ft0 fx t) ; fy t) = x ; y) ; jtj t sn 2 t,ft<0 so, agan, fx t) ; fy t) s non-negatve for all t 2 R. Hence, takng nto account the prevous two steps, for each x 0 x 1 2 [;1 1] wehave that: 3) f x 0 x 1 0, then 1 )or 2 )and,nany case, 0 3) of Corollary 2 hold 4) f x 0 x 1 < 0, then 0 3) of Corollary 2 holds. Furthermore, n vrtue of I) of Lemma 4, ) of Theorem 3 holds. References 1. Hrart-Urruty J.B., Images of connected sets by semcontnuous multfunctons, J. Math. Anal. Appl., ), pp Klen E. and Thompson A.C., Theory of correspondences, John Wley & Sons, Fan K., A generalzaton of Tychono's xed pont theorem, Math. Ann., ), pp Rccer B., On multfunctons wth convex graph, Att Accad. Naz. Lnce Rend. Cl. Sc. Fs. Mat. Natur., ), pp Rccer B., Some topologcal mn- theorems va an alternatve prncple for multfunctons, Arch. Math., ), pp Rccer B., A theorem on sets wth connected sectons and some of ts applcatons. 7. Warburton A. R., Quasconcave vector mzaton: Connectedness of the sets of Paretooptmal and weak Pareto-optmal alternatves, J. Opt. Th. Appl., ), pp