VECTORIAL OPTIMIZATION IN LOCALLY CONVEX SPACES ORDERED BY SUPERNORMAL CONES AND EXTENSIONS

' This research work was conceived as a compret ion of [27]with existence :iäå:# :ffi:Lfiiå':lJ;;"ftîii"ïi]i ;;"""x spaces rdered bv supernormar cones, Key word's: supernormal (nuctear) cone, vectorial optimization, efficient (pareto) point.

the origi,n of thc spar:e is conta'i,ned i,n the tra'nslation of the prilar cone to K bg sorne l,it'¿ear and cont,inuous functional (.,yr)p, Þ e L The first existence result fol the efficient point is based on supernormal- ity of 11, the boundedness and completeness of conical (extension) sections induced by non-empty sets and the following important ilrcorcm of [B].Tupon¡ir¿

[B]
If K is a supernor-m,al cone in a rlausd,orff locaLly con'uer space E and S 'is a non-emytty subset of E haui,ng th,e property tho,t tl¿ere erists a boun,rled and complete set,90 C,S with S n (Á-+ r) ç Ss:for euery íx Ç Ss, tlte,n th,ere erists rs € S such that Sñ(K *16) : {re}.
Tn¡;on.ur¡ 2.2.126l Let A c B c A r K.If K i,s sr,,pernormal anrl B a (A0 -K) is bounded, and, complete for sotn,e non-ent,tty set A6 C A ür,en MIN7 (A) I ø.
In [27]wefindmanyexamplesofSupernormalconestogetherwithproper ,"-urk. (Section 2) ancl we gave an extension of supernormalitv to sets ac- companieà by its immediate connections rvith a generalization of approximate .,rbáiff"r"rrtialused by J.B. Hiriart-Uruty [6] (Section 3).Here we present some of our existence results for the efficient (Pareto minimum) points which generated their corresponding implications for Pareto maximum points and recent and best extensions' 2. EXIS'fENCE RESULTS FOR, EFFICIENT POINTS IN HAUSDORFF ORDER,ED LOCALLY CONVEX SPACE Itisknownthat,ingeneral,tosolveavectoroptimizationproblemin an ordered vector upu."-ãons to find the efficient (Pareto optimums) points of an adequate non-empty set.For this reason' we selected in this section some illustrative e*istencå results of the efficient points in separated locally convexSpàcesorcÌeredtlysupernormalconesanclbytherecentconesdefined and studìed in [4], f b].il fact, these cones and a great part of the o'iginal and beautif'l existence råsults on the efÊcient points given in the above mentioned ,-esearch works wet'e suggestecl by the existence results of the efficient points obtained through the agency of supernormal cones (see [B], [12]' [i3], [22]' 1241,126) ar.cl otlter .orr"ãt"dpapers) and by the largest class of convex cones ensuring the existence of the elficient points in compact sets defined in [30]' Let X be a real Flausdorff iocally convex space with the topology induced byafarnilyP:{po:ar-I}ofseminormsorderedbyaconvexconeK'its tåpotogicai dual spaãe X*, A a uon-empty subset of X and a € A' DpnrNrflorq 2.1.we say that a is an efficient po'int (Pareto min'imurn) for A with respect to K, in notat'ior¿, a e MINç(Ð ''f a sati's'fies one of the f ollowi,ng equiu alent condit'ions (i) AÀ(a-N) Ca* K; (i'i') (A+ K) n (o.-K) Çd+ K; '(m¡ rà1a -'t)C-x, (*) KÀ(d-A-K)c-K' WerecallthatKispoi,nted,ifKÀ(-K):{0}andacute\fitsclosure K is pointed.clearlv, vtiNr< (A) :MIN¡< (K + A) and if r{ is pointed' then a €MIN/{ (,4) if uttá ä,.tv tf in @-K): {a)' or equivalently' Kr\(d-A): {0}.In a sìmila, manner one defines the Pareto rnaximum points' Dpr,n¡rrroN 2.2.17) we søy that K 'is suTternormal or nuclear i'f for euery pe P there erists I e X* suchthatp(") ( f (r) for aIIrÇK' Proof.Since -4 is bounded and K is supernormal (Proposition 5 of [B], by Theorem 2.2),it is sufficient to prove that every section of ,4 with respect to K is complete.Let a € ,4 be an arbitrary element and let (ø¡)r., be a Cauchy net in AÀ(a-I{).Because K is wellbased by a complete set, there exists a non-empty, convex, bounded and complete set B such that 0 Ç B and K: U À8. Hence, for each a¡ (j e ..I), there exist À¡ ) 0 and b¡ € B with À>0 aj: a-\¡b¡.Therefore (\¡b¡)¡q is a bounded Cauchy net' Since the set B is closed, bounded and 0 ( B,there exists a convex and closed neighbourhood I/ of the zero element in X and a ) 0 such that V o B : Ø and B C aV.If p, is the Minkowski functional of V, then 1 < p"(b) ( a for every b € B and there exists M > 0 with À¡ { p,(\¡b¡) ( M for all Ày, that is, (\¡)¡e¡ is bounded.When (\¡)¡r, contains at least a subnet convergent to zero, then it is clear that a¡ tends to a; otherwise, because it is bounded, we can find a subnet (Àr)"es convergent to À6 ) 0. Since (or)res is a Cauchy net, (br)"., is a Cauchy net in B. Therefore (b")"."converges to ó6 € B and (ør)r." is convergent to a -Àobo which implies that (À¡)r.J convelges to a -Àsó6.So, we have proved the corollary.!
CoRolr-aRv 2.2.2 126l If A is a non-empty, ltounded and closed subset of X and K is pointed closed and locally compact, then MIN¡ç (A) + Ø and A ÇMrNx Ø) + K.
Proof.This follows from the above corollary because in a Hausdorff locally convex space a pointed cone is locally compact if and only if it has a compact generating base.¡ Dp¡'INtrIoN 2.3.A non-empty set B C X is K-bounded [20] i,f there erists a bounded set Bs C X su,ch that B C Bo f K and B is sai,d to be K-closed [16] ,Í the con'ical ertension B + K is closed.'We recall that X is called quasi-complete if every non-empty, bounded and closed subset in X is complete.
1'upoRptr¿ 2.3.126l Let X be quas'i-complete.If K is a closed and supernormal cone in X, then (i) for euery non-empty K-bounded and K-closed, subset A in X we haue MINK Ø) I Ø and A Ç MINrc (A) + K ; (i,i,) ,Í the set B À(A0-K) is K-bounded, and K-closed for sorne nonempty subsets B and As wi,th Aç B ç A+K and AsC A, then MINTa(A) *ø; (i,ä,) for euery K-bounded and K-closed set A ç X, MIN6(A) + X : A+ K and, MIN¡ç (A) i,s K-bounded and, K-closed.

Vectorial in Locallv Convex Soaces
Proof.(i), (ii).In the conditions of the theorem every conical extension section of A is bounded and closed and the results follows by Theorem 2.2 and Remark 2.2.(äi) is based on the inclusion A c MIN¡a Øi +1l for every If-bounded and K-closed subset .4. n Before we give an extension of this theorem to orderecl Hausdorff topo- logical vector spaces, we recall two basic ciefinitions.Dnrrruruolr 2.4.[4] We say that K has property (*) iÍ the set (M + K)ì (N -K) is bounded wheneuer M and, N are boundid,'subiets in x.
DpprNrrrolr 2.5.[4] we say that K has the property (*x) i/ one of the following equiualent, cond,itions holds: (i') ang bounded'increas'i,ng net whi,ch is contained, in K and, ,in a complete subset of X has a li,mit; (ä') any bounded monotone net which is conta,ined, i,n a complete subset ofXhasalimi,t.
Remarlç 2.3.Euery supernorTnal cone has p,operti,es (*) and, (**) but there erist conue:x cones hauing the properti,es (*), (*+) which are not supernormal.Thus, in the class'ical Banach spaces L, ([o,t]) (p > r) the usual pisitiue cone is closed, connet, it has the properti,es (*) and, (**) but it i,s not supernormal The same conclus'ion is ualid, for the cone of nonnegatdue functions in an orlicz space (see Erample B giuen in Section p of tZTl ) Remarlc 2.4.under appropriate conrlitions, in [5] it is shown that in euery separated topological aector space the largest class of conuer cones ensuri,ng the eristence of the fficient po'ints in any bound,ed and complete subset co,inc,ides wi,th the class of cones hau,ing the property (x*).
The announced generalization of Theorem 2.3 is THpoRprvr 2.4.[4] .Let x be a Hausd,orff topolog'ical uector space, K c x a conuer cone and, A c x a non-empty set.suppose that thi foliowing conditi,ons hold: (i,) X is quas'i-complete; (ä) the cone K has the properties (x) and (**); (äi) the set A is K-bounded and K-closed,.
Following the final remark of Ha T.X.D. in [5] we must mention here that each of the conditions (i)-(iii) in the above theorem cannot be weakened.
Through the agency of the natural extensions with respect to convex cones of upper semicontinuit¡ boundedness and completeness for multitunc- tions, the above theorem leads in [4] to obtain a criterion for the existence of