中职 优化计算方法及其MATLAB程序实现第3章电子课件 高教版 .pdf

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1、1/35JJIIJIBackClosezO9MATLABSy1n F2/35JJIIJIBackClose?5zKminxRnf(x)(3.1)?F9U?.F)?zK?P?,8F2k5,?%?z?:,Nk?:LU?.d?,;?z,9kgA5?:,8E?E?“.3.1F9MATLABy312u?zKea?eJ?,?(|,?.F3/35JJIIJIBackCloseKFdk=f(xk)(3.2)|?.?f(x)3xkNCY,dk|,gk=f(xk).d?Vm,?f(xk+dk)=f(xk)+gTkdk+o(),0.o8If(x)3xk?dke?Czlim0f(xk+dk)f(xk)=lim0gTk

2、dk+o()=gTkdk=kgkkkdkkcosk,:kgkdk?Y?.w,u?dk,Cz?ugkY?u.Cz?,kcosk=1,=k=U?,=4/35JJIIJIBackClosedkAT?(3.2)?KF,=KF8If(x)3?c:?e,dFe.eF?NO.3.1(F)0,?:x0 Rn,NN?0 6?1.-k:=0.1,Ogk=f(xk).ekgkk 6,xkCq4?:.2,?dk=gk.3,d|(fk.4,-xk+1:=xk+kdk,k:=k+1,=1.5 3fk?(Q(|,(|,3nU?y?5.e5/35JJIIJIBackClose(|,=f(xk+kdk)=min0f(xk+dk

3、),okAv0()=ddf(xk+dk)|=k=f(xk+kdk)Tdk=0.d(3.2),kg(xk+1)Tg(xk)=0,=#:xk+1?F:xk?F?,S“:?r?.?,?(5).ddk=gk9(2.17),=cosk=gTkdkkgkkkdkk=gTk(gk)kgkkk gkk=1 k=0,6/35JJIIJIBackClose?(2.16)7,v(0 6 k62,(0,2),l?An2.2n2.3=?F?5n.n3.1?8If(x)YFg(x)LipschitzY?,xkdF?),fkd(|,dWolfeOK,dArmijoOK?),Kklimkkg(xk)k=0.e?n?F)?g4

4、?K?O,y?kz,d?y.n3.2?A Rnn?,b Rn.P1nOA?A?,=1/n.Xe4?zKminf(x)=12xTAx bTx.7/35JJIIJIBackClose?xk(|?F)K?)?S“S?,Kuk?k,e?kxk+1 xkA6?1+1?kxk xkA,(3.3):xK?),kxkA=xTAx.dn3.2w,e?Cu1(=A?A?A?C),F?.?(=ACqu?),?.euArmijo(|?FMATLABS.S3.1(FS)function k,x,val=grad(fun,gfun,x0,epsilon)%U:F)?zK:min f(x)%:fun,gfunO8I9F,x

5、0:,8/35JJIIJIBackClose%epsilonNN?%:kS“g,x,valOCq:maxk=5000;%S“gbeta=0.5;sigma=0.4;k=0;while(kmaxk)gk=feval(gfun,x0);%OFdk=-gk;%O|if(norm(gk)epsilon),break;end%u?OKm=0;mk=0;while(m20)%Armijo|if(feval(fun,x0+betam*dk).x0=-1.2;1.0;k,x,val=grad(fun,gfun,x0,1e-5)fun,gfunO8I9F?m.3.29MATLAByF?,)?zK?;,?g:S“

6、:xk?(F)?(Hesse?)8I?1?gCq,?r?g?4?:#?S“:,-EL,?v?Cq4?:.12/35JJIIJIBackClosee5?S“.?f(x)?Hesse?G(x)=2f(x)Y,?3xk?Vm?cn,?qk(x)=fk+gTk(x xk)+12(x xk)TGk(x xk),:fk=f(xk);gk=f(xk);Gk=2f(xk).?gqk(x)?-:,?qk(x)=gk+Gk(x xk)=0.eGk,o)?5|(P)xk+1)=?S“xk+1=xk G1kgk.(3.4)3S“(3.4),zS“IHesse?_G1k,3SOLk)Gkd=gk?dk,?-xk+1=x

7、k+dk5;_.?,?Xe.13/35JJIIJIBackClose3.2(?)0,?:x0 Rn,NN?0 6?1.-k:=0.1,Ogk=f(xk).ekgkk 6,x xk.2,OGk=2f(xk),)5|Gkd=gk,?)dk.3,-xk+1:=xk+dk,k:=k+1,=1.?:,k?5.e?nL?5.n3.3?f(x)k?Y?,34?:x?,G(x)=2f(x)?G(x)3x?SLipschitzY?.XJ:x0Cx,ok,S“14/35JJIIJIBackClose(3.4)?,?xk:?,4x?.y dG(x)?59f?gY,3x?N1(x),?x N1(x),kG(x)?.A

8、O/,kG(x)1k3N1(x)k.,=3M 0,?kG(x)1k 6 M,x N1(x).qdG(x)?Y5,3?N(x),?kG(x)G(x)k 614M,x N(x)N1(x).d,?x0 N(x),kkx1 xk=kx0 x G10g0k6 kG10kkg(x0)g(x)G0(x0 x)k6 kG10k?Z10G(x+t(x x)(x x)dt G0(x0 x)?15/35JJIIJIBackClose6 MZ10kG(x+t(x0 x)G(x0)kkx0 xkdt6 M?Z10kG(x+t(x0 x)G(x)kdt+Z10kG(x0)G(x)kdt?kx0 xk612kx0 xk.(

9、3.5)(3.5)AO,x1 N(x).aq/,|8B?n,yuk?k 1kkxk+1 xk 612kxk xk.d,xk N(x)xk x(k ).?,aqu(3.5)?,?kxk+1 xk 6 MZ10kG(x+t(xk x)G(xk)kkxk xkdt16/35JJIIJIBackClose=o(kxk xk),=xk5ux.eG(x)3x?SLipschitzY,Kd?kxk+1 xk 6 M?Z10kG(x+t(xk x)G(x)kdt+Z10kG(x)G(xk)kdt?kxk xk6 LM?Z10tdt+1?kxk xk2=32LMkxk xk2,=xk?gux.y.?n3.3,:

10、Iv?/C04?:,KkU?17/35JJIIJIBackClose.duSK?(4?:?,d,:?S5?(J.?(J,|?,=?Z.euArmijo|?Z?N.3.3(uArmijo|?Z)0,?(0,1),(0,0.5),:x0 Rn,NN?0 6?1.-k:=0.1,Ogk=f(xk).ekgkk 6,x xk.2,OGk=2f(xk),)5|Gkd=gk,(3.6)?)dk.18/35JJIIJIBackClose3,Pmkve?K?m:f(xk+mdk)6 f(xk)+mgTkdk.(3.7)4,-k:=mk,xk+1:=xk+kdk,k:=k+1,=1.e3.3?5n.n3.4?f

11、(x)?gY3 0,?dTG(x)d kdk2,d Rn,x L(x0),(3.8):L(x0)=x|f(x)6 f(x0).?xkd3.3?)?:?,KT:?uf3Y8L(x0)?4?:.y d(3.8),fY8L(x0)?,d3?4?:x,xg(x)=0?).19/35JJIIJIBackCloseduf,?Y8L(x0)k.48.5?3.3?3,S?f(xk)Ne?.?w,kxk L(x0).d(3.6)(3.8)J?kgkk 6 kdkk,(3.9):0,.?xxk?4:,Kkfk f(x).PkKFgkdk=G1kgk?Y?.d(3.6)!(3.8)(3.9)?cosk=gTkdkk

12、gkkkdkk=dTkGkdkkgkkkdkkkdkk2kgkkkdkk=kdkkkgkk 0,20/35JJIIJIBackCloseKdn2.3,?limkgk=0,=xk?4:-:.df?5,-:=(?)4?:.?d4?:?5,xkuf3Y8L(x0)?4?:.y.?3.3?,I?e?n,?yLz5.n3.1?f:Rn R?gY,:?xkd3.3?).?xk xg(x)=0,G(x)?.o,elimkkG(xk)dk+gkkkdkk=0,(3.10)K(1)?k,fk 1;21/35JJIIJIBackClose(2):?xk5ux.n3.5?n3.4?,:?xkd3.3?),Kxk5u

13、f?4?:x.d?,eHesse?G(x)3x?LipschitzY,K?.y n3.4yxk xg(x)=f(x)=0.qd3.3?2w,k(3.10).?dn3.1(1),u?k,k=1v?|.d,dn3.3=?xk?ux.y.?.euArmijo(|?ZMATLABS.S3.2(ZS)function k,x,val=dampnm(fun,gfun,Hess,x0,epsilon)22/35JJIIJIBackClose%U:Z)?zK:min f(x)%:fun,gfun,HessO8I9FHesse?,%x0:,epsilonNN?%:kS“g,x,valOCq:maxk=5000;

14、%S“gbeta=0.5;sigma=0.4;k=0;while(kmaxk)gk=feval(gfun,x0);%OFGk=feval(Hess,x0);%OHesse?dk=-Gkgk;%)|Gk*dk=-gk,O|if(norm(gk)epsilon),break;end%u?OKm=0;mk=0;while(m20)%Armijo|23/35JJIIJIBackCloseif(feval(fun,x0+betam*dk).=feval(fun,x0)+sigma*betam*gk*dk)mk=m;break;endm=m+1;endx0=x0+betam*dk;k=k+1;endx=x

15、0;val=feval(fun,x);3.2|Z(S3.2)?zKminxR2f(x)=4(x21 x2)2+3(x1 1)2.24/35JJIIJIBackCloseTKk()x=(1,1)T,f(x)=0.)?3.1?O8IF?m?,IHesse?m:%Hesse?function He=Hess(x)He=48*x(1)2-16*x(2)+6,-16*x(1);-16*x(1),8;|S3.2,OK?kf(xk)k 6 105.?:,(JXL3.2?.L3.2Z?(J25/35JJIIJIBackClose:(x0)S“g(k)8I(f(xk)(0.0,0.0)T68.7342 1020

16、(2.0,1.0)T75.8973 1022(1.2,1.0)T81.6636 1020(2.0,2.0)T75.9579 1021(20.0,20.0)T223.8193 1024dL3.2w,Z?-x0=-1.2;1.0;k,x,val=dampnm(fun,gfun,Hess,x0,1e-5)fun,gfun,HessO8I9FHesse?m.26/35JJIIJIBackClose3.3?9MATLAByl3.2!?,k$u?,?:.?T8I?Hesse?G(x)=2f(x)3zS“:xk?,KJ?ydk=G1kgkf3xk?e.,?1?.?F(5,?E?/-F0,?g:?2f(xk)

17、?,|;K,e2f(xk),?e,KKF|.e-F?O.3.4(-F)0,?:x0 Rn,NN?0 6?1.-k:=0.1,Ogk=f(xk).ekgkk 6,xkC27/35JJIIJIBackCloseq4?:.2,OGk=2f(xk),)5|Gkd=gk.(3.11)e|(3.11)k)dkvgTkdk 0,=3zS“?/?k?Ak=G(xk)+kI?.NXe.3.5(?)0,?(0,1),(0,0.5),:x0 Rn,NN?0 6?1,0,1.-k:=0.29/35JJIIJIBackClose1,Ogk=f(xk),k=kgkk1+.ekgkk 6,xkCq4?:.2,OHesse?

18、Gk=2f(xk),)5|(Gk+kI)d=gk,(3.12)?)dk.3,-mkve?K?m:f(xk+mdk)6 f(xk)+mgTkdk.(3.13)4,-k:=mk,xk+1:=xk+kdk,k:=k+1,=1.e?n?3.5?5,yz4.n3.7?f:Rn Rke.?gY,G(x)=2f(x)?LipschitzY,Kd3.5?)?S“S?xk?4:K(3.1)?).30/35JJIIJIBackCloseky,?8I?Hesse?G(x)=2f(x)34?:x?,?U$5.e)?Vg.3.1e3K(3.1)?4?:x?Hesse?G(x),KxK(3.1)?).?K(3.1)k),

19、)U,dXL)8,=X=x|f(x)=minf(x),x Rn.3.2?x X,:Rn R+.e3x?N(x)9 0,?(x)dist(x,X),x N(x),(3.14)K3?N(x)SK(3.1)8XJ?31/35JJIIJIBackClose?.,(3.14):dist(x,X):x?8X?l.e?n?3.5?O,yz5.n3.8?n3.7?.ed3.5?)?S“S?xkkf?xk:k Kux X,kg(x)k3x?,?SK(3.1)J?.,K?k K,k 1f?xk:k K?ux,=3 0?dist(xk+1,X)6 dist(xk,X)2.e?MATLABS.S3.3(?S)func

20、tion k,x,val=revisenm(fun,gfun,Hess,x0,epsilon)32/35JJIIJIBackClose%U:?)?zK:min f(x)%:fun,gfun,HessO8I9FHesse?,%x0:,epsilonNN?%:kS“g,x,valOCq:n=length(x0);maxk=5000;beta=0.5;sigma=0.4;tau=0.0;k=0;while(kmaxk)gk=feval(gfun,x0);%OFmuk=norm(gk)(1+tau);Gk=feval(Hess,x0);%OHesse?Ak=Gk+muk*eye(n);dk=-Akgk;%)|Ak*dk=-gk,O|33/35JJIIJIBackCloseif(norm(gk)epsilon),break;end%u?OKm=0;mk=0;while(m20)%Armijo|if(feval(fun,x0+betam*dk).x0=-1.2;1.0;k,x,val=revisenm(fun,gfun,Hess,x0,1e-5)fun,gfun,HessO8I9FHesse?m.