中职 优化计算方法及其MATLAB程序实现第11章电子课件 高教版 .pdf

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1、1/73JJIIJIBackClosezO9MATLABSy1?z?12/73JJIIJIBackClose110?g5yK?).l?m,?5zK?).k0?)?zK?;1.1a?n?zK?,?g:zS“?)?|=8I?e,?51,=S“:ovk?.?1?O3u?1dk?,p0?Zoutendijk1!KF?Fn1.11.1 Zoutendijk1Zoutendijk155y5(|e1?,dZoutendijk31960cJ5?.?!5?5?/5?n.3/73JJIIJIBackClose11.1.15?e?11.?ne?5zKmin f(x),x Rn,s.t.Ax b,Ex=e,(11.1

2、):f(x)Y;Am n?;El n?;b Rm;e Rl.=K(11.1)km5?l5?.e?n?K(11.1)?e1dAv?.n11.1?xK(11.1)?1:,3 x?k4/73JJIIJIBackCloseA1 x=b1,A2 x b2,A=A1A2,b=b1b2.Kd Rn:x?e1?A1d 0,Ed=0,f(x)Td 0.yJuy,df(x)3 x?e?f(x)Td b1:x?k?,?A2 x b2L?A2x b2:x?k?.d,31:x?A?A?A1A2.5/73JJIIJIBackClose5.?A1d 0,Ed=0.x1:A1 x=b1,E x=e,?0kA1(x+d)=A1

3、 x+(A1d)A1 x=b1,E(x+d)=E x+(Ed)=E x=e.qdA2 x b2,?73 0,?u?(0,kA2(x+d)=A2 x+A2d b2.nL,3,?u?(0,kA(x+d)b,E(x+d)=e,=x+d1:,l?d:x?1.6/73JJIIJIBackClose75.?x1:,d:x?1.d1?,3,?u?(0,kA(x+d)b,E(x+d)=eA1(x+d)b1,A2(x+d)b2,E(x+d)=e.udA1(x+d)=A1 x+(A1d)b1,A1 x=b1,0A1d 0.qdE(x+d)=E x+(Ed)=e,E x=e,0Ed=0.y.?7/73JJIIJIB

4、ackCloseln11.1,K(11.1)?1:x?e1d,L)e55yK?:min f(x)Td,s.t.A1d 0,Ed=0,1 6 di6 1,i=1,2,n,(11.2):d=(d1,d2,dn)T.O?1 6 di6 1(i=1,2,n)?“kdk.5?d=0w,fK(11.2)?1),?8If(x)Td?7,?u?u0.e8I?z=f(x)Td b2,A=A1A2,b=b1b2.K xK(11.1)?KKT:?fK(11.2)?0.dun11.1?yI?Farkasn(n8.1),?B,eFarkasn?d.n11.2(Farkasn)?Am n?,cn,KATy=c(y 0)k

5、)?7Ax 6 0,cTx 0),x,yOn,m.9/73JJIIJIBackClosen11.1?y 5?xKKT:?3 0,?f(x)AT1 ET=0.(11.3)-=1 2(1,2 0),r(11.3)?AT1 ETET?12=f(x),12 0.(11.4)Farkasn(n11.2),(11.4)k)?A1EEd 6 0,f(x)Td 0(11.5),=A1d 0,Ed=0,f(x)Td 0f(xk+dk),s.t.A(xk+dk)b,E(xk+dk)=e.(11.8)?K(11.8)?z:dk1,kEdk=0;?xk1:,ukExk=e.d,K(11.8)?g,72.d?,3xk?

6、,?k?k?,?A1xk=b1,A2xk b2,(11.9)12/73JJIIJIBackCloseA=A1A2,b=b1b2.KK(11.8)?A1xk+A1dk b1,(11.10)A2xk+A2dk b2.(11.11)qdk1,dn11.1A1dk 0.5?A1xk=b19 0,d,(11.10)g,.u,K(11.8)?e(11.11),?K(11.8)zmin0f(xk+dk),s.t.A2(xk+dk)b2.(11.12)13/73JJIIJIBackClosee?K(11.12)?.K(11.12)?1?U?A2dk b2 A2xk.ePb=b2 A2xk,d=A2dk,(11

7、.13)Kkd b,0.5?(11.9),ukb 0.dd?.O =minnbidi=(b2 A2xk)i(A2dk)i?di 0,+,d 0,(11.14)14/73JJIIJIBackClose:bi,diOb,d?1i.d,)K(11.12)?du)min f(xk+dk),s.t.0 6 6 ,(11.15)d(11.14)O.d,?)K(11.1)?1?O.11.1(5?1)0,1:x0 Rn,?0 b2,15/73JJIIJIBackCloseA=A1A2,b=b1b2.2,exk1?S:(dK(11.1)vk?,=E=0A1=0)kf(xk)k 1,K?|dk=f(xk),=5(

8、=8I?KF|2,daqu?zK).exk1?S:,K=3.16/73JJIIJIBackClose3,)55yKmin z=f(xk)Td,s.t.A1d 0,Ed=0,1 6 di6 1,i=1,2,n,(11.16):d=(d1,d2,dn)T,?)Odkzk.4,e|zk|0,i=1,2,m,(11.17):f(x)gi(x)(i=1,2,m)Y?.e?n?K(11.17)?e1dAv?.n11.3?xK(11.17)?1:,I8I(x)=i|gi(x)=0,f(x)gi(x)(i I(x)3 x?,gi(x)(i 6 I(x)3 x?Y.ef(x)Td 0,i I(x),18/73J

9、JIIJIBackCloseodK(11.17)3 x?e1.y dn8.3?e1?“(8.10),d7K(11.17)3 x?e1.y.?dn11.3,K(11.17)31:x?e1dAvf(x)Td 0,i I(x).(11.18)?3(11.18)?9Cz?,?due?5?|dz:f(x)Td 6 z,gi(x)Td 6 z,i I(x),z 6 0.(11.19)19/73JJIIJIBackClose5?v(11.19)?e1d9zk,g,FU8Ie?d,?(11.19)=zz8I?55yKmin z,s.t.f(x)Td 6 z,gi(x)Td 6 z,i I(x),1 6 di6

10、 1,i=1,2,n,(11.20):d=(d1,d2,dn)T.?K(11.20)?)d,z.o,e z 0,KdK(11.17)3 x?e1;K,e z=0,Ke?ny:A?x7K(11.17)?Fritz-John:.n11.2?xK(11.17)?1:,I8I(x)=i|gi(x)=20/73JJIIJIBackClose0,K xK(11.17)?Fritz-John:?fK(11.20)?0.y ufK(11.20),0?|f(x)Td 0,i I(x),=f(x)Td 0,gi(x)Td 0,i=1,2,m.(11.23)e)K(11.17)?1?O.11.2(5?1)0,1:x

11、0 Rn,?0 1,2?1.-k:=0.22/73JJIIJIBackClose1,(xk?k?I8I(xk),=I(xk)=i|gi(xk)=0.eI(xk)=kf(xk)k 1,K?|dk=f(xk),=4.,eI(xk)6=,=2.2,)55yKmin z,s.t.f(xk)Td 6 z,gi(xk)Td 6 z,i I(xk),1 6 di6 1,i=1,2,n,:d=(d1,d2,dn)T,?)Odkzk.23/73JJIIJIBackClose3,e|zk|.?.(2)3ezk 0,3xk?e1,xk?K?Fritz-John:.(3)11.2e2?5?zK,S“Ly?(J.u?c

12、1S“:xk,J24/73JJIIJIBackCloseu?1.vaf,vaN?n?.11.1Zoutendijk1)e?Kmin f(x)=x21+x22 x1x2 2x1+3x2,s.t.x1 x2+3 0,x1 5x2+6 0,x1,x2 0,?1:x0=(0,0)T.)8I?Ff(x)=2x1 x2 2x1+2x2+3.25/73JJIIJIBackClose11gS“.3x0?Ff(x0)=(2,3)T,k?k?X?mOA1=1 00 1,A2=1 11 5,b1=00,b2=36.k3x0?e1d0.)55yKmin f(x0)Td,s.t.A1d 0,1 6 di6 1,i=1,

13、2,=min 2d1+3d2,s.t.d1 0,d2 0,1 6 d16 1,1 6 d26 1.26/73JJIIJIBackClosedX/?)d0=(1,0)T,d8I?2(0,1 6 d16 1,1 6 d26 1.dX/?)d1=(d1,0)T,d1?.d,8I?0.dn11.1,x1=(1,0)T?K28/73JJIIJIBackClose?KKT:.?11.2FKu?zK,?:eF0,KKFc?,o?#?8Ie?:,F.u?zK,XJ2KFc?,U1?,dIKFK?1?.,?S“:xk1D?S:,?d=f(xk)|;K,?xk1D?.:,?f(xk)3?.?8?K|.FK?g,/

14、FK0?d5.FKRosenu1961c?5?zKkJ5?z.gc,Rosenq?2?n29/73JJIIJIBackClose5?/.?5q?u,)55yK?a-?.11.2.1FK?n:5?zKminf(x),x Rn,s.t.Ax b,Ex=e,(11.24):fY?n?;A Rmn;E Rln;b Rm;e Rl.1D=x Rn|Ax b,Ex=e.3N0?FKc,kK?Vg9k5.n11.330/73JJIIJIBackClose11.1?P RnnK?,PvP=PT,P2=P.d11.1,?K?.K?kXe?5,yz12.n11.4?P Rnn.(1)ePK?,KP?;(2)PK?

15、=?I PK?,In?;(3)?PK?,Q=I P,KL=y=Px|x Rn,L=z=Qx|x Rnp?5fm,u?x Rn/Lx=y+z,y L,z L.31/73JJIIJIBackClose?xK(11.24)?1:,vA1 x=b1,A2 x b2,A=A1A2,b=b1b2.q?M=A1E1?,P=I MT(MMT)1M,Pf(x)6=0.e?d=Pf(x),Kd:x?e1.yJ?y,P=I MT(MMT)1MK?,?dPf(x)6=0,?f(x)Td=f(x)TPf(x)=kPf(x)k2 b2,A=A1A2,b=b1b2.q?M=A1E1?,-P=I MT(MMT)1M,=(MM

16、T)1Mf(x)=,OAuA1E.ePf(x)=0,K(1)XJ 0,K xK(11.24)?KKT:;34/73JJIIJIBackClose(2)XJ 6 0,”?j 0.5?Pf(x)=0,uk0=Pf(x)=I MT(MMT)1Mf(x)=f(x)MT(MMT)1Mf(x)=f(x)MT=f(x)?AT1,ET?=f(x)AT1 ET.(11.25)(11.25)TKKT.d,xKKT:.35/73JJIIJIBackClose(2)?j 0.kyPf(x)6=0.y.ePf(x)=0,KdP?,?0=Pf(x)=I MT(MMT)1Mf(x)=f(x)MT,(11.26):=(MMT

17、)1Mf(x).?A1Auj?1rj(1j1).duAT1+ET=AT1+jrTj+ET=MT +jrTj,(11.27):?Kj?.(11.27)“(11.25),?0=f(x)MT jrTj.(11.28)(11.26)?(11.28),?MT()+jrTj=0.(11.29)36/73JJIIJIBackClose(11.29)?M?1?5|,?kXj6=0.dd?M?15,M1g.d,7kPf(x)6=0.duPK?Pf(x)6=0,?f(x)Td=f(x)TPf(x)=kPf(x)k2rjP,?rjPf(x)rjPMT jrjPrTj=0.5?PMT=09d=Pf(x),=rjd+j

18、rjPrTj=0.(11.31)P?(rjPrTj 0)9j 0.(11.32)d(11.30)(11.32),?A1d 0,Ed=0.?,n11.1,d:x?1.y.?38/73JJIIJIBackClose11.2.2FK?Ou?,eRosenFK?O.11.3(RosenFK)0,1:x0 Rn.-k:=0.1,3xk?(k?A1xk=b1k?A2xkb2,A=A1A2,b=b1b2.2,-M=A1E.39/73JJIIJIBackCloseeM?,K-P=I(?);K,-P=IMT(MMT)1M.3,Odk=Pf(xk).ekdkk 6=0,=5;K,=4.4,O=(MMT)1Mf(x

19、k)=.e 0,xkKKT:;K,?,K,Xj 0,?A1,=?KA1Auj?1,=2.5,)e|K,(fk:min f(xk+dk),s.t.0 6 6 ,40/73JJIIJIBackClose de(:=minn(b2 A2xk)i(A2dk)i?(A2dk)i 0,+,A2dk 0.6,-xk+1:=xk+kdk,k:=k+1,=1.11.2RosenFK)e?zKmin f(x)=x21+x22+6x1+9,s.t.2x1+x2 4,x1,x2 0,?1:x0=(2,0)T.)8I?Ff(x)=2x1+62x2.41/73JJIIJIBackClose11gS“.3x0?Ff(x0)

20、=(10,0)T,k?k?X?mOA1=2 10 1,b1=40,A2=?1 0?,b2=0.E=,?M=A1.5?A1_?,l?K?P=I AT1(A1AT1)1A1=I AT1AT1A11A1=0.-d0=Pf(x0)=00.O=(MMT)1Mf(x0)42/73JJIIJIBackClose=MTf(x0)=120121100=55=12.?A1.?KA1A2=5?1,?A1=?2 1?.qM=A1,K?P=I A1T(A1A1T)1A1=1 00 121?2 1?21!1?2 1?=151224.43/73JJIIJIBackClose-d0=Pf(x0)=151224100=24.0

21、.dub=b2 A2x0=0?1 0?20=2.d=A2d0=?1 0?24=2.?=22=1.44/73JJIIJIBackCloseu)e?|K0:min 202 20+25,s.t.0 6 6 1.)?0=12.l?x1=x0+0d0=20+1224=12.12gS“.3x1?Ff(x1)=(8,4)T,k?k?X?mOA1=?2 1?,b1=?4?,A2=1 00 1,b2=00.45/73JJIIJIBackCloseE=,?M=A1.OK?P=I AT1(A1AT1)1A1=151224.-d2=Pf(x1)=15122484=00.O=(A1AT1)1A1f(x1)=15?2 1

22、?84=4 0,dn11.4,x1=(1,2)T?K?KKT:.46/73JJIIJIBackClose11.3?F11.3.1 Wolfe?FWolfeu1963c?5?5zKJ?#?1,?F.e0?.k5?5zKmin f(x),x Rn,s.t.Ax=b,x 0,(11.33):A Rmn(m 0KCxB=B1b B1NxN 0,xN 0.yb?xz?1),=xB 0.duxBxN5L,d,f(x)zuxN?,=f(x)=f(xB,xN)=f(B1b B1NxN,xN):=F(xN).n mxN?F(xN)?Ff(x)?F,Pr(xN),=r(xN)=xNF(xN)=xNf(B1b B1

23、NxN,xN)=Nf(xB,xN)(B1N)TBf(xB,xN),(11.34):N=xN;B=xB.49/73JJIIJIBackClose2(|,=(3xk?e1dk,?U:xk+1=xk+kdk1:8Ie.-dk=dBkdNk.dke1,Ivf(xk)Tdk 0,e(xk)j=0.d?Adk=0,?BdBk+NdNk=0,=dBk=B1NdNk,(11.35)50/73JJIIJIBackCloseLdBkddNk(?.2de5f(xk)Tdk 0,?f(xk)Tdk=Bf(xk)TdBk+Nf(xk)TdNk=Bf(xk)TB1NdNk+Nf(xk)TdNk=r(xNk)TdNk 0.

24、(11.37)Juy,v(11.35)(11.37)?dkkN?,?(dNk)j=(xNk)jrj(xNk),rj(xNk)0rj(xNk),.(11.38)51/73JJIIJIBackClosedk=B1NdNkdNk=B1NInmdNk.(11.39).e?K(k.?xk+1 0,=(xk+1)j=(xk)j+(dk)j 0,j=1,2,n,(11.40)I(?.5?(dk)j 0,?0,(11.40).?(dk)j 0,A?6(xk)j(dk)j.52/73JJIIJIBackClosed,-=minn(xk)j(dk)j?(dk)j 0,+,dk 0.(11.41)y,U?E?|dk

25、,edk6=0,K7e1,K,A?xk7KKT:.n11.5?A=?BN?m n?,Bm?,xk=xBkxNkK(11.33)?1:,xBk 0AuB?m.qbf3:xk?Y,dkd(11.38)(11.39)?.K(1)edk6=0,Kdke1;(2)dk=0?xkKKT:.53/73JJIIJIBackClosey(1)ddk?,kAdk=BdBk+NdNk=B(B1NdNk)+NdNk=0.qd(11.37),?(xNk)j=0,(dNk)j 0.5?xBk 0,d,n11.1,dk1.d?kf(xk)Tdk=Bf(xk)TdBk+Nf(xk)TdNk=Bf(xk)T(B1NdNk)+N

26、f(xk)TdNk=r(xNk)TdNk.5?dNk6=0,(11.38)r(xNk)TdNk 09,?f(xk)AT =0,0,xk 0,Txk=0,=Bf(xk)Nf(xk)BTNT BN=00,(11.42)TBxBk=0,TNxNk=0.(11.43)exkKKT:,K.duxBk 0B 0,Kd(11.43)?1fB=0,l?d(11.42)?1,?=(BT)1Bf(xk).(11.44)55/73JJIIJIBackClose(11.44)“(11.42)?1?,?N=Nf(xk)(B1N)TBf(xk)=r(xNk)0.(11.45)d(11.45)9(11.43)?1?,?r(

27、xNk)TxNk=0.(11.46)5?xNk 0,?drj(xNk)(xNk)j=0,j=1,2,n.(11.47)d,(11.45)!(11.47)!(11.35)(11.38)?dk=0.,edk=0,Kr(xNk)K.-N=r(xNk)=Nf(xk)(B1N)TBf(xk)0.?d(11.38),7kTNxNk=0.2-B=0,=(BT)1Bf(xk),56/73JJIIJIBackCloseKkTBxBk=09(11.42),?xkKKT:.y.?e?F?O.11.4(Wolfe?F)0,?.1:x0 Rn.-k:=0.1,(|.xk)xk=xBkxNk,:xBkC,dxk?m|,?

28、eI8PJk.A/,A)A=?B N?.UeOdk:r(xNk)=Nf(xBk,xNk)(B1N)TBf(xBk,xNk),57/73JJIIJIBackClose(dNk)j=(xNk)jrj(xNk),rj(xNk)0,rj(xNk),dk=dBkdNk=B1NInmdNk.2,u?OK.edk=0,KxkKKT:,;K,=3.3,?1|.)e|K,(k:minf(xk+dk),s.t.0 6 6 ,58/73JJIIJIBackClose de(:=minn(xk)j(dk)j?(dk)j 0,+,dk 0.-xk+1:=xk+kdk.4,?C.exBk+1 0,KCC;K,ekj?(x

29、Bk+1)j=0,K(xBk+1)j,?(xNk+1)j,?#?xBk+1xNk+1.5,-k:=k+1,=1.59/73JJIIJIBackClose11.3Wolfe?F-#)11.1,=min f(x)=x21+x22 x1x2 2x1+3x2,s.t.x1 x2+3 0,x1 5x2+6 0,x1,x2 0,?1:x0=(0,0)T.)k,tCx3,x4 0,?K=zXe?d?/IO/0:min f(x)=x21+x22 x1x2 2x1+3x2,s.t.x1+x2+x3=3,x1+5x2+x4=6,x1,x2,x3,x4 0,60/73JJIIJIBackCloseKA=1 1 1

30、01 5 0 1,b=36,f(x)=2x1 x2 2x1+2x2+300.?1:x0=(0,0,3,6)T.11gS“.J0=3,4f(x0)=(2,3,0,0)T.(B,N,?xB0=36,xN0=00,B=1 00 1,N=1 11 5.61/73JJIIJIBackCloseOB1N=1 11 5,Bf(x0)=00,Nf(x0)=23.O?Fr(xN0),?r(xN0)=Nf(x0)(B1N)TBf(x0)=231 11 5T00=23.?dN0=(2)0 3=20,dB0=B1NdN0=1 11 520=22.l?d0=(2,0,2,2)T.62/73JJIIJIBackClose

31、.=minn32,62o=32.lx0u,d0|:x0+d0=(2,0,3 2,6 2)T,f(x0+d0)=42 4.)|Kmin42 4,s.t.0 6 632,?0=12.u,kx1=x0+0d0=?1,0,2,5?T.63/73JJIIJIBackClose12gS“.J1=3,4f(x1)=(0,2,0,0)T.(B,N,?xB1=25,xN1=10,B=1 00 1,N=1 11 5.OB1N=1 11 5,Bf(x1)=00,Nf(x1)=02.O?Fr(xN1),?r(xN1)=Nf(x1)(B1N)TBf(x1)=021 11 5T00=02.64/73JJIIJIBackC

32、lose?dN1=1 00 2=00,dB1=(B1N)dN1=1 11 500=00.l?d1=(0,0,0,0)T.n11.5,x1KKT:,?x=(1,0)T?K?4?:.?11.3.22?FAbadieCarpentieru1969cWolfe?F2?5?/,J?2?F.?5?65/73JJIIJIBackClosezKmin f(x),x Rn,s.t.hi(x)=0,i E=1,2,l,gi(x)0,i I=1,2,m,:f,hi(i E),gi(i I)Y?.b?xk1kg1S“:,PIk=E i|gi(xk)=0,?c(xk)=?h1(xk),hl(xk),gi(xk)(i I

33、kE)?T.(11.48)y3?X(?Fr(xNk).”5,?Ik=1,2,s(s l),(11.48)?Jacobi?h1(xk),hl(xk),gi(xk)(i IkE)?T1(”?cs?).Pxk?csC|?fxBk,n sC|?fPxNk.d66/73JJIIJIBackClose3n,3xk?,?S,uxNk?5|c(xk)=c(xBk,xNk)=0k)xBk=(xNk).xBk,xNk.B,?KeIk,Ps s?Bc(x)=?Bh1(x),Bhl(x),Bg1(x),Bgsl(x)?T,c(x)=?h1(x),hl(x),g1(x),gsl(x)?T:=?c1(x),cs(x)?T

34、9Bci(x)=ci(x)x1.ci(x)xs,i=1,2,s.67/73JJIIJIBackClosedb?,Bc(x).2P?Nc(x)=?Nc1(x),Nc2(x),Ncs(x)?T Rs(ns),Nci(x)=ci(x)xs+1.ci(x)xn,i=1,2,s.?c(x)=c(xB,xN)=0uxNF,?JBN(xN)Bc(x)+Nc(x)=0,68/73JJIIJIBackCloseJBN(xN)=?(x1,xs)(xs+1,xn)?T=x1xs+1x2xs+1xsxs+1.x1xnx2xnxsxn.l?kJBN(xN)=Nc(x)?Bc(x)?1.(11.49)5?f(x)f(xN

35、),xN),uxN?F(=?F),?r(xN)=Nf(x)+JBN(xN)Bf(x).(11.49)“,?r(xN)=Nf(x)Nc(x)?Bc(x)?1Bf(x).(11.50)69/73JJIIJIBackClosey3?e1d=?dTB,dTN?T,Kde1,dAvf(x)Td 0,c(x)Td=0.dc(x)Td=0,?Bc(x)TdB+Nc(x)TdN=0.ukdB=Bc(x)T1Nc(x)TdN=JBN(xN)TdN.(11.51)qdf(x)Td 0,?Bf(x)TdB+Nf(x)TdN 0.(11.51)“,?Nf(x)TdN Bf(x)TBc(x)T1Nc(x)TdN 0,7

36、0/73JJIIJIBackClose=r(xN)TdN 0,i I,(11.53)71/73JJIIJIBackClose?|k,?-xk+1:=xk+kdk=?U1S“:xk+1.?.KFf?O.dn8.4,34?:x?f(x)=XiEihi(x)+XiIigi(x),(11.54):I=i|gi(x)=0;i 0(i I).Pc(x)=?h1(x),hl(x),gi(x)(i I)?,=?()T,()T?T=?1,l,i(i I)?T.o(11.54)?f(x)=c(x).d2_?4?)=c(x)f(x),72/73JJIIJIBackClose:c(x)?c(x)?2_.d,Ue?OA

37、?f?O:k=?Tk,Tk?T=c(xk)f(xk).(11.55)e2?F?O.11.5(2?F)0,?.1:x0 Rn,?0 6?1.-k:=0.1,u?.(CxBkCxNk.d(11.50)O?Fr(xNk).ekr(xNk)k 6,xk?K?Cq4?:.2,(|.d(11.52)Oe1dk.3,?1|.)fK(11.53)?|k.-73/73JJIIJIBackClosexk+1:=xk+kdk.4,?k?8.kxk+1?k?8,?Ik+1.d(11.55)Ok+1.ek+1 0,KIk+1:=Ik+1;K,Ik+1Ik+1?k+1?A?I8.5,-k:=k+1,=1.5(1)311.5?1,?kr(xNk)k 6,SI?OAu?.KFf?K5,evI?1U?.(2)2?FL?,?C3m$,U?(),5).K,?8c)5zK?k?.