1、-$(h,varphi)$不变广义凸函数的若干性质与$(h,varphi)$不变广义凸多目标规划的最优性及对偶性264Vol.26No.4Oct.,2003200310ACTAMATHEMATICAEAPPLICATAESINICA(h,?)-(h,?)-?(710071)(330047)(710071)Ben-Tal(h,?)-(h,?)-(h,?)-)(h,?)-Lagrange(h,?)-11?5.AvrielM1Ben-Tal(h,?)-f?(1?)?x1 ?x2?(1?)f(x1)+f(x2).6(h,?)Ben-Tal(h,?)-(h,?)?(h,?)?(h,?)?-(h,?)-(
2、h,?)-Lagrange?200110172003128(69972036),(2000SL03)(h,?)-)(4(h,?)-7272Ben-Tal(h,?)-H?RBen-Taln1,pp166,167.RnRhRn?1.m?=x1x2 xm,xiH,i=1,2,m;i,i=1,2,m;i=1m?i=1i=1+2+m,x?y=x(?1)?y; ? = +(?1).RRkk?+Rk=(x1,x2,xk)T:xi0,i=1,2,k;?+0Rk=(x1,x2,xk)T:xi0,i=1,2,kxi00;?+Rk=(x1,x2,xk)T:xi0,i=1,2,k;kk?+0+k=x Rk:xi=1;
3、k=xRk:xi=1.i=1i=1Ben-Tal2.1(a)xiH,i=1,2, ,m,yH,m?i=1?T?xiyh,?=m?i=1(xiTy)h,?.-(b)(i)(ii)kR,x,y H,2.2, R,(k?x)Ty)h,?=k(xTy)h,?.? = =();m?m?i=i;R,i,i=1,2,m,i=1i=1?(iii)(iv)(v)(vi); ;m?m?i i;i,i ,ii,i=1,2,m,0,0, , ,i=1i=1i, i,ii,i=1,2,m,m?m?ii.i=1i=1i0,i0i0,728262.1?(0)=0,h(0)=0,xyz=0,(xTw)h,?0,(yTw)h,
4、?0,(zTw)h,?0.x,y,z,wH,2.2m?i=1?T?i?xiyh,?m?=i(xiTy)h,?.i=1-2.1f(x)Rn?(t)t=h(x)f?fh?1(t),?(t)?.ffS?Rnt=h(x)?(t)=?f(h?1(t),f(h,?)-fx?(t)=f?f(x)=h?1?(h,?)-fS(h,?)-2.12.32.3fx(h,?)-?kf(x)=k?f(x).?T?kf(x)yh,?=k?2.1(ii)xf(x)?T?yh,?.2.32.1p?f(h,?)-Ben-Tal2.1fi(x),i=1,2,px(h,?)-Rp,g(x)=?p?i=1ifi(x),?g(x)=i=
5、1i?fi(x).2.2f(x)Rn?x,yRn,0,1 (x,y)RnRnRn?fy ?(x,y)f(x)+(1?)f(y),?f(x)(h,?)?2.2(i)(iii),(v)2.2?R(x,y)(h,?)?+Rm,fi(x),i=1,2,mm?ifi(x),g(x)=g(x)(h,?)?i=1-2.3f(x)Rnf(x)(h,?)?(h,?)-?(x,y)RnRnRnR?x,uRn,?f(x)?f(u) ?f(u)T (x,u)h,?.?R?fu?(x,u)f(x)+(1?)f(u),?R?fu?(x,u) ?f(x)+(1?)f(u)=?f(u)+?f(x)?f(u),?fh?1h(u
6、)+h(x,u)?fh?1h(u)+?f(x)?f(u),4(h,?)-729?=?fh?1,f?h(u)+h(x,u)f?h(u)+?f(x)?f(u),f(0,1)?h(u)+h(x,u)?f?h(u)/?f(x)?f(u),f?T?(t)?f0h(x,u) ?f(x)?f(u),?f(u)=t=h(u)?T?(t)?h?1?fh?f(u)h(x,u)?f(x)?f(u),?t=h(u)?T?f(u)T(x,u)h,?1h(?f(u)h(x,u)?1?f(x)?f(u).f(x)?f(u).2.1-2.4?2.2R2.3(h,?)?+Rm,fi(x),i=1,2, ,mm?i fi(x),
7、g(x)=(h,?)-i=1?x,yRn,g(x)?g(y)m?i=1i?fi(y)?T(x,y)?h,?.2.3f(x)Rn?x,yRn,0,1 (x,y)RnRnRn?fy?(x,y)maxf(x),f(y)f(x)(h,?)?(?1)f(x)(h,?)?f(x)(h,?)?2.4f(x)Rn(h,?)-f(x)(h,?)?R?f(u)T(x,u)h,?1(0).f(x)f(u)(x,y)RnRnRn?x,u Rn,2.32.2(i)(iii),2.32.5f(x),(?1)f(x)2.3(h,?)?f(x)?f(y)=?f(y)T(x,y)h,?.2.4f(x)f(x)Rn(h,?)?(
8、h,?)-(x,y)?x,u Rn,?RnRnRn?f(y)T(x,y)h,? ?1(0),-f(x)f(y).(1?)(h,?)?(2?) (x,u)=x?uu ?(x,u)=u?(x?u)=?x(1?)?u,(h,?)-2.22.52.4(x,u)=x?u,(h,?)-3(h,?)-()73026(VP)?Tminf(x)=f1(x),f2(x),fp(x)s.t.gi(x)0,i=1,2,hi(x)=0,i=1,2,l.(VP)?X=xRn:gi(x)0,i=1,2, ,hi(x)=0,i=1,2,l;X,I()=i:gi(=0,i=1,2,;2.2(vi)?Rp,(SP)=minxXp
9、?i=1ifi(x).-3.1?R+Rp(+0Rp),(SP)(VP)().3.1?R?(0)=0fi(x),i=1,2,p,gi(x),i=1,2,hi(x),i=1,2,l(h,?)-(VP)jI(?+0+?I()?=k(), Rp(Rp),Rk,Rlp?i=1i?fi()?k?i=1i?gi()(h,?)?l?i=1-i?hi(=0,?(1)?i:i0,(x,y),gi(x)?J=i:i0,M=i Jhi(x)(h,?)?ifi(x)(h,?)?p?i M(VP)hi(x)(h,?)?().i=1xXiI(),?gi(x)g i(),-?h,?2.4?gi(T(x,0,(2)Ben-Ta
10、l?k?i=1i?gi(?T(x,)?h,?=?1k?i=1i?gi(x,)?T?h,?,(3)-i0,?1?1(0)=0?(2),(3)?Tk?i=1i?gi()(x,?h,?0,hi(x)(h,?)?(4)hi(x)=h()=0hi(x)h(?i Ji J?Ti 0,?h,?i?hi()?(x,)0.(5)-4(h,?)-731hi(x)=hi(=0(?1)hi(x)(h,?)-(?1)hi(x)(?1)hi(iMi0?i M(?)?(?1)hi(?T(x,)?h,?0,?(?i)?(?1)hi()=i?hi(?i Mi?hi(?T(x,)?h,?-0,(6)(1),(4)(6)2.1p?
11、i=1p?i?fi()?T(x,?h,?0.i=1ifi(x)(h,?)?2.1p?i=1ifi(x)p?i=1ifi(,3.14(h,?)-Lagrange(VP)-?Tminf(x)=f1(x),f2(x),fp(x)s.t.gi(x)0,i=1,2,m,hi(x)=0,i=1,2,l.(VP)Lagrange?(VD)?f1(y)+i=1uigi(y)+i=1vihi(y)?m?l?f(y)+uigi(y)+vihi(y)?2maxG(y,u,v)=?i=1i=1?.?m?l?fp(y)+uigi(y)+vihi(y)s.t.p?i=1+,uRm?m?l?i?fi(y)?i=1m?ui?
12、gi(y)-?l?i=1i=1vi?hi(y)=0,?+p?i=1+.p4.1()(?1)hk(x),k=1,2,lfi(x),i=1,2,p,gj(x),j=1,2,m,hk(x)(h,?)-(h,?)?73226x?w=(y,u,v),?(0)=0,h(0)=0,(VP)(VD)p?i=1p?i fi(x)iGi(y,u,v).i=12.2?+p(?+p)+uRmp?i fi(x)i=1m?uigi(x)i=1(h,?)?2.4-p?p?p?T?i?fi(y)(x,y)ifi(x)?ifi(y)i=1m?i=1i=1m?i=1i=1m?i=1h,?,(7)(8)uigi(x)?uigi(y
13、)ui?gi(y)?T(x,y)?h,?,2.5?hi(x)?hi(y)=?hi(y)T(x,y)h,?.l?l?l?vihi(x)?vihi(y)=vi?hi(y)T(x,y)h,?,i=1i=1i=12.2l?i=1l?l?T?vi?hi(y)(x,y)vihi(x)?vihi(y)=i=1i=1h,?,(9)(7)(9)2.2(v)p?i=1p?m?ifi(x)?ifi(y)+uigi(x)?i=1i=1m?l?l?uigi(y)+vihi(x)?vihi(y)i=1p?i=1i?fi(y)?Ti=1(x,y)?h,?+i=1m?i=1ui?gi(y)?T(x,y)?h,?l?T?+vi
14、?hi(y)(x,y)-i=1h,?pml?T?=i?fi(y)uj?gj(y)vk?hk(y)(x,y)?=0T(x,y)h,?=?1(0)=0.i=1j=1k=1h,?4(h,?)-733p?m?l?ifi(x)+uigi(x)+vihi(x)i=1p?i=1i=1i=1m?l?ifi(y)+uigi(y)+vihi(y),i=1i=1(10)ui0,gi(x)0?(0)=0m?m?1ui?(gi(x)0,uigi(x)=?i=1i=1(11)hi(x)=0,?(0)=0l?l?1vi?hi(x)=0,vihi(x)=?i=1i=1(12)(10)(12)p?p?m?l?ifi(x)ifi
15、(y)+uigi(y)+vihi(y).i=1p?(13)i=1i=1i=1i=1i=1-p?m?l?ifi(y)+uigi(y)+vihi(y)i=1p?i=1i=1i=1m?l?ifi(y)+uigi(y)+vihi(y).i=1i=1=(13),(14)(14)p?i=1p? ifi(x)iGi(y,u,v).i=1(VP)(VD)?+ p,=()(VD)()(SD)()p?maxGi(y,u,v)i=1p?i=1m?i=1l?i=1s.t.i?fi(y)?ui?gi(y)?vi?hi(y)=0,?u+Rm,Gi(y,u,v) Gi(,i=1,2,p.73426m?l?uigi(y)+v
16、ihi(y).Gi(y,u,v)=fi(y)+i=1i=1(SD)(?+p(-W(?+(VP)p),Lagrange(VD)()?m?l?f1(y)+i=1uigi(y)+i=1vihi(y)?m?l?f(y)+uigi(y)+vihi(y)?2maxG(y,u,v)=?i=1i=1?.?m?l?fp(y)+uigi(y)+vihi(y)?s.t.p?i=1+uRm.i?fi(y)?i=1m?i=1ui?gi(y)?l?-i=1i=1vi?hi(y)=0,?2.2(vi)4.1?+p,W(VD)(VP)(SD)(4.2()?R?+p,Rm,Rl,h(0)=0,?(0)=0,p?i?fi(?m?
17、i=1i?gi(?-l?i=1i?hi(=0,?(15)(16)(17)i=1m?i=1igi(=0,+Rm.=(=(VD)(f(=G(15),(17)(SD)(=()()(SD)(SD)(=(w?=(y?,u?,v?)p?i=1p?i=1?)+fi(ym?i=1l?)+?)u?igi(yv?ihi(yi=1-m?l?igi(ihi(),fi(i=1i=1(18)4(h,?)-735(y?,u?,v?)(SD)()fi(y?)+m?i=1l?)+?)u?igi(yv?ihi(yi=1m?l?fi()+igi()+hi(),i=1i=1(19)(18),(19)i0m?l?fi0(y?)+?)+
18、?)u?igi(yv?ihi(yi=1i=1m?i=1l?i=1fi0()+i0,i=1,2,pigi()+ihi().2.2(iii)(vi)p?m?l?ifi(y?)+?)+?)u?igi(yv?ihi(yi=1p?i=1p?i=1i=1m?l?ifi(igi()+hi(.i=1i=1i=1i=1,(14)p?m?l?ifi(y?)+?)+?)u?igi(yv?ihi(y-i=1p?i=1i=1i=1m?l?ifi(igi()+ihi(.i=1i=1hi(=0(20)?(0)=0l?l?1ihi()=?i?hi(=0,i=1i=1(21)(16),(20),(21)p?p?m?l?ifi(
19、y?)+?)+?)ifi().u?igi(yv?ihi(yi=1i=1i=1i=13.1(VD)()=(SD)()4.1=(?T(16)(21)G(=f1(,f2(,fp()=f().736261AvrielM.NonlinearProgramming:AnalysisandMethod.Cli?s,19762NewJersey:Printice-Hall,Englewood1992Changchun:Jilin(LinCuoyun,DongJiali.EducationPress,1992)259604MethodsandTheoryinMultiobjectiveOptimization.
20、3HansonMA.InvexityandtheKuhn-TuckerTheorem.J.Math.AnalysisAppl.,1999,256:J.Math.Analysis4WeirT,MondB.Pre-invexFunctionsinMultipleObjectiveOptimization.Appl.,1988,136:2938-J.Optim.TheoryandAppl.,1998,98(3):651661(h,?)-65Osuna-GomezR,etal.InvexFunctionsandGeneralizedConvexityinMultiobjectiveProgrammin
21、g.,2001,24(1):129138(ZhangQingxiang.OnSu?ciencyandDualityofSolutionsforNonsmooth(h,?)-Semi-in?nitePro-gramming.ActaMath.Appl.Sinica,2001,24(1):129138)SOMEPROPERTIESFOR(h,?)-GENERALIZEDINVEXFUNCTIONSANDOPTIMALITYANDDUALITYOF(h,?)-GENERALIZEDINVEXMULTIOBJECTIVEPROGRAMMINGXUYihong(DepartmentofAppliedMa
22、thmatics,XidianUniversity,Xian710071)(DepartmentofMathematics,NanchangUniversity,Nanchang330047)LIUSanyang(DepartmentofAppliedMathmatics,XidianUniversity,Xian710071)AbstractSomepropertiesforBen-Talsgeneralizedalgebraicoperationsarestudied,(h,?)-generalizedinvexfunctionsareintroduced,somepropertiesof
23、whicharediscussed,thesu?-cientconditionisgivenfor(h,?)-generalizedinvexsemi-in?nitemultiobjectiveprogrammingtoattainitse?cientsolution(orweaklye?cientsolution)andtheLagrangiandualitytheoryof(h,?)-generalizedinvexmultiobjectiveprogrammingareestablished.Keywords(h,?)-generalizedinvexfunction,multiobjectiveprogramming,e?cientsolution,optimalitycondition,duality
