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2、z8I?/.1?Lk,?5,=?x Rn|ci(x)=0,i E;ci(x)0,i I?,(1.2):ci(x)(i E I)3RnY?,?.ui E,ci(x)=0?,E?I8;ui I,ci(x)0?,I?I8.eI8E I=,K?zK,K?zK.AO/,rE 6=I=?zK?zK;4/37JJIIJIBackClose?rI 6=E=?zK?zK.d?,r8I?g?5?zK?g5y;?r8I?5?zK55y.1.2?3?5,I?Vg9kn.?RnLnm,RmnLm n?N|?5m.3m,O?.x Rn?k kK,7Lve:(1)kxk 0,kxk=0 x=0;(2)kxk=|kxk,R;
3、5/37JJIIJIBackClose(3)kx+yk 6 kxk+kyk.ak?p-,kxkp=?nXi=1|xi|p?1p,p 1.(1.3)?k:1-:kxk1=nPi=1|xi|;2-:kxk2=?nPi=1|xi|2?12;-:kxk=max16i6n|xi|.?A Rmn?,?U?.?/e,?Iv5:kABk 6 kAkkBk,A Rmn,B Rnq,6/37JJIIJIBackClose:kABk,kAk,kBkORmq,Rmn,Rnq?.XJ?k kuk kv?kAxk 6 kAkkxk,A Rmn,x Rn,K?kkkkN?.?,e3x 6=0kAk=maxx6=0kAxkk
4、xk=maxkxk=1kAxk,(1.4)K?k kdk kp?5?f,f,kluk k?.df?k k.J?y,lukxk1,kxk2,kxk?OkAk1=max16j6nmXi=1|aij|,7/37JJIIJIBackClosekAk2=max?|(ATA)?,kAk=max16i6mnXj=1|aij|,O?,1.?3?S“?5,Ue?F-:kAkF=?mXi=1nXj=1a2ij?1/2=qtr(ATA).(1.5)e?S?S?5.ex(k)k=1 Rn,Klimkx(k)=x limkx(k)i=xi,:i=1,2,n.aq/,eA(k)k=1 Rmn,KlimkA(k)=A li
5、mka(k)ij=aij,8/37JJIIJIBackClose:i=1,2,m;j=1,2,n.?|4,7L?dn9?dn.n1.1(1)?k kk k03Rn?,K3?c1c2,kx Rnc1kxk 6 kxk06 c2kxk.(2)?k kk k03Rmn?,K3?m1m2,kA Rmnm1kAk 6 kAk06 m2kAk.e|?Vg,?d/S?S?5.n1.2(1)?x(k)nS?,k k3Rn9/37JJIIJIBackClose?,Klimkx(k)=x limkkx(k)xk=0.(2)?A(k)m n?S?,k k3Rmn?,KlimkA(k)=A limkkA(k)Ak=0
6、.1.3?!0?I?n?9?Vm.1.1?kn?f(x),gCx=(x1,x2,xn)T10/37JJIIJIBackCloseRn.f(x)=?f(x)x1,f(x)x2,f(x)xn?T(1.6)f(x)3x?F.?2f(x)=2f(x)x212f(x)x1x22f(x)x1xn2f(x)x2x12f(x)x222f(x)x2xn.2f(x)xnx12f(x)xnx22f(x)x2n(1.7)f(x)3x?Hesse?.eFf(x)?z3x?Y,Kf3x?Y.eHesse?2f(x)?3x?Y,Kf3x?Y.11/37JJIIJIBackCloseef3m8D?z:?Y,Kf3D?Y.ef
7、3m8D?z:?Y,Kf3D?Y.d1.1Juy,ef3x?Y,K2f(x)xixj=2f(x)xjxi,i,j=1,2,n,=Hesse?2f(x)?.1.1?gf(x)=12xTAx bTx,:b RnA Rnn?.o,JOf(x)3x?F9Hesse?Of(x)=Ax b,2f(x)=A.12/37JJIIJIBackClose1.2(?Vm)?f:Rn RY,Kf(x+h)=f(x)+Z10f(x+th)Thdt=f(x)+f(x+h)Th?(0,1)?=f(x)+f(x)Th+o(khk).?,ef?gY?,Kkf(x+h)=f(x)+f(x)Th+Z10(1 t)hT2f(x+th
8、)hdt=f(x)+f(x)Th+12hT2f(x+h)h?(0,1)?=f(x)+f(x)Th+12hT2f(x)h+o(khk2)9f(x+h)=f(x)+Z102f(x+th)Thdt13/37JJIIJIBackClose=f(x)+2f(x+h)Th?(0,1)?=f(x)+2f(x)Th+o(khk).e0?59n.?kF(x)=(F1(x),F2(x),Fm(x)T:Rn Rm.ezFi(Y)?,KF(Y)?.F3x?F0 Rmn3x?Jacobi?,PF0(x)JF(x),=F0(x):=JF(x):=F1(x)x1F1(x)x2F1(x)xnF2(x)x1F2(x)x2F2(
9、x)xn.Fm(x)x1Fm(x)x2Fm(x)xn.14/37JJIIJIBackClose?I?F,krF?Jacobi?=F3x?F?,PF(x)=JF(x)T=?F1(x),F2(x),Fm(x)?.Juy,1.2u?(2?/.X,eF:Rn RmY?,Ku?x,h Rn,kF(x+h)=F(x)+Z10F(x+th)Thdt=F(x)+F0(x)h+o(khk).uF,LipschitzY5?Vg.1.2?F:Rn Rm,x Rn,F3x?LipschitzY?,3L 0,?y Rn,vkF(x)F(y)k 6 Lkx yk,(1.8)15/37JJIIJIBackClose:LLi
10、pschitz.e(1.8)?x,y Rn,KF3RnSLipschitzY?.3S“?5,kI?/n0,yXe.n1.3?F:Rn RmY,o(1)?x,y Rn,kkF(x)F(y)k 6 sup06t61kF0(y+t(x y)kkx yk;(2)?x,y,z Rn,kkF(y)F(z)F0(x)(y z)k 6 sup06t61kF0(z+t(y z)F0(x)kky zk.dn1.3?(2)?e?(.16/37JJIIJIBackClose1.1?F:Rn RmY?,Jacobi?N?LipschitzY?,=3L 0?kF0(u)F0(v)k 6 Lku vk,u,v Rn,(1.
11、9)K?x,h Rn,kkF(x+h)F(x)F0(x)hk 612Lkhk2.(1.10)1.48?!0?8!I?kVg.1.3?8D Rn.8D8,?x,y D9?0,1,kx+(1 )y D.d1.3J?8?A,=8D Rn,e?:?EuT8,KT8D8.17/37JJIIJIBackCloseJy8?e?5.K1.1?D,D1,D28,o(1)D:=y|y=x,x D8;(2)?8D1 D28;(3)8D1+D2:=z|z=x+y,x D1,y D28.1.3 nAp?m?m:?|8,=8?x=mXi=1ixi?xi Rn,i 0,mXi=1i=1?8.1.4 nAp?m?H:=x?b
12、Tx=8,R,b Rn0?.d?,e?om8:18/37JJIIJIBackClose(1)?4mH+:=x|bTx ;(2)K?4mH:=x|bTx 6;(3)?mmH+:=x|bTx ;(4)K?mmH:=x|bTx 08.1.48D Rn?(convex hull)kD?8?8,Pconv(D):=TGDG,:G8.19/37JJIIJIBackCloseeII?.1.5?8G Rn.e?x G?0,kx G,KGI(cone).eG8,KGI(convex cone).d?,uIG,e0 G,KGkI(pointed cone).A/,0?IkI.1.6Nx Rn|Ax 0kI,I(p
13、olyhedral cone).1.78Rn+:=x Rn|xi 0,i=1,2,nkI,KI(nonnegative cone).A/,IRn+:=x Rn|xi 0,i=1,2,n20/37JJIIJIBackClose?I(positive cone).k?8?Vg?,8?.1.6?f:D Rn R,D8.(1)fD?,?x,y D9?0,1,kf(x+(1 )y)6 f(x)+(1 )f(y).(2)fD?,?x,y D,x 6=y9?(0,1),kf(x+(1 )y)0,?21/37JJIIJIBackClose?x,y D9?0,1,kf(x+(1 )y)+12(1 )kx yk2
14、6 f(x)+(1 )f(y).ke?5.K1.2?f,f1,f28D?,c1,c2 R+,R,Kk(1)c1f1(x)+c2f2(x)D?;(2)Y8L(f,)=x|x D,f(x)6 8.83znXv”-?,?|?5?k5N?.XJ22/37JJIIJIBackClose?Y?,K|?FHesse?5?O?y?5N?.eA?On.n1.4?f38D Rn?Y,K(1)f3D?f(x)f(y)+f(y)T(x y),x,y D;(1.11)(2)f3D?,?x 6=y,f(x)f(y)+f(y)T(x y),x,y D;(1.12)(3)f3D?,3c 0,?x,y D,f(x)f(y)+f
15、(y)T(x y)+ckx yk2.(1.13)23/37JJIIJIBackClose3?,ef(x)3m(a,b)?f00(x)0(0),Kf(x)3(a,b)S().u?Y?f:D Rn R,d?(Hesse?)5?C?L.1.7?n?f38D?Y?.eh Rn,khT2f(x)h 0,K2f3:x?.e0 6=h Rn,khT2f(x)h 0,K2f3:x?.?,e3c 0,?h Rn,x D,khT2f(x)h ckhk2,K2f3D?.k?1.7,r?u?L5?(J2?.n1.5?n?f38D Rn?Y,K24/37JJIIJIBackClose(1)f3D?2f(x)x D?;(
16、2)f3D?2f(x)x D?;(3)f3D?2f(x)x D?.52f?f?7.1.5?K?5?!?zKminxRnf(x)(1.14)25/37JJIIJIBackClose?5,?.k4?:?,?4?:4?:.1.8eu?x Rn,kf(x)6 f(x),Kxf?4?:.e?x 6=x,Kxf?4?:.1.9eu?x N(x,)=x Rn|kxxk 0,.e?x 6=x,Kxf?4?:.26/37JJIIJIBackClosed1.81.9,?4?:4?:,.5,?4?:?(J?,d,4?:(3SA,k4?:v?K?).?4?:?4?:.?Q?B,?e?P:g(x)=f(x),gk=f(
17、xk),G(x)=2f(x),Gk=2f(xk).n1.6(?7)?f(x)3m8D?Y.ex DK(1.14)?4?:,K7kg(x)=0.y?x=x g(x)D,0,Kkf(x)=f(x)+g(x)T(x x)+o(kx xk)=f(x)g(x)Tg(x)+o()=f(x)kg(x)k2+o().27/37JJIIJIBackClose5?f(x)f(x)9 0,uk0 6 kg(x)k26o().-0,?kg(x)k=0,=g(x)=0.y.?n1.7(?7)?f(x)3m8D?Y.ex DK(1.14)?4?:,K7kg(x)=0G(x)?.y?x4?:,odn1.6g(x)=0.eI
18、yG(x)?5.?x=x+d D,0d Rn.d?Vm,?0 6 f(x)f(x)=122dTG(x)d+o(2),=dTG(x)d+o(22)2 0.28/37JJIIJIBackClose-0,=?dTG(x)d 0,l?n.y.?n1.8(?)?f(x)3m8D?Y.ex Dvg(x)=09G(x)?,KxK(1.14)?4?:.y?x=x+d D,0d Rn.d?Vm,?f(x+d)=f(x)+g(x)Td+122dTG(x+d)d,(0,1).5?g(x)=0,G(x)?f?Y,?3 0,?G(x+d)3kdk f(x),l?n.y.?29/37JJIIJIBackClose5,8I
19、?-:4?:.?u8I?zK,-:!4?:?4?:n?d?.n1.9?f(x)3Rn?Y?,Kx RnK(1.14)?4?:?g(x)=0.y 75w,?,Iy5.?g(x)=0.d?On1.4(1),?f(x)f(x)+g(x)T(x x)=f(x),x Rn,Lx?4?:.y.?30/37JJIIJIBackClose1.6?zK?V3z,S“)?zKminxRnf(x)(1.15)?4?:.?g:k:x0,?U,S“5K?)S“S?xk,?eTS?k?,K?:K(1.15)?4?:;K,eS?xk:?,k4:4:=K(1.15)?4?:.?xk1kgS“:,dk1kg|,k1kgf,K1
20、kgS“?#(1k+1g)?S“:xk+1=xk+kdk.(1.16)d,?)?zK(1.15)?eXe.31/37JJIIJIBackClose1.1(?zK?e)0,z9S“:x0.-k:=0.1,exkv,OK,S“,xkCq4?:.2,L)xk?,fK(edk.3,L,|(fk,?f(xk+kdk)0,?(0,dk6=0,kf(xk+dk)f(xk),Kdkf(x)3xk?e.e8If?Y?,K?OdkekB?O.n1.1?f:D Rn R3m8D?Y,Kdkf(x)3xk?e?f(xk)Tdk 0,w,kf(xk+dk)f(xk),=dkf(x)3xk?e.,kf(xk)Tdk+o(
21、)0,=f(xk)Tdk+o()0.dulim0o()=0,?(1.17).y.?5?OS“?.u5,kXe?/50/?50Vg.1.11e,k?:x0?C4?:x,d?)?:?xkux,KTk5.e34/37JJIIJIBackCloseu?:x0,d?)?:?xkux,KTk?5.?-I,k?:Q-R-.eO?1.1.12?)?:?xku4?:x,limsupkkxk+1 xkkxk xkp6 c.(1)ep=10 c 1,KTkQ-5(Q-5?).(2)ep=1c=0,KTkQ-5(Q-5?).(3)ep=20 c 20 c 0q (0,1)?kxk xk 6 cqk,(1.19)KS?xk R-5?x.36/37JJIIJIBackClose(2)e3c 0u?qk?kxk xk 6 ckYi=0qi,(1.20)KS?xk R-5?x.3OyS“,IS“.?ke?n:(1)?,=kxk+1 xkk kxk+1 xkkkxkk,:?;(2)8I?,=|f(xk+1)f(xk)|f(xk+1)f(xk)|f(xk)|,37/37JJIIJIBackClose:?;(3)8I?F?,=kf(xk)k ,:?.