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1、1/47JJIIJIBackClosezO9MATLABSy1?g5yK2/47JJIIJIBackClose?g5y5z?A/,?8I?g,?5.du?g5y?,Bu)(=gu55y),?5zK=z)X?g5yK(=1130?/S?g5y0),d,?g5y?)?/0,d Rn,d 6=0,Ad=0.K5|(10.8)?X?,=|(10.8)k).y?dT,T?Te?g5|?):HATA0d=0,(10.9)=Hd AT=0,Ad=0.?dTHd=dTAT=0,Ad=0.9/47JJIIJIBackCloseud?57kd=0,l?AT=Hd=0.5?A1,?7k=0.dd,g5|(10.9)
2、k),d,X?7,.y.?e5?(10.8)?).n10.1,.KF?7,?,?_HATA01=GBTBC.d?HATA0GBTBC=In0nm0mnIm,10/47JJIIJIBackClose?HG+ATB=In,HBT ATC=0nm,AG=0mn,ABT=Im.ud4?G,B,C?LG=H1 H1AT(AH1AT)1AH1,(10.10)B=(AH1AT)1AH1,(10.11)C=(AH1AT)1.(10.12)d,d(10.8)?)?L x=GBTBCcb=Gc+BTbBc Cb,(10.13)G,B,COd(10.10)(10.12).11/47JJIIJIBackClosee
3、x?,?dL.?xkK(10.1)?1:,=xkvAxk=b.?3d:?8I?Fgk=f(xk)=Hxk+c.|xkgk,(10.13)U?x=xk GgkBgk.(10.14)e)?g5y.KFf?MATLABS.S10.1?S.KFf)?g5yK.function x,lambda,val=qlag(H,c,A,b)%U:.KFf)?g5y:%min f(x)=0.5*xHx+cx,s.t.Ax=b12/47JJIIJIBackClose%:H,cO8I?,%A,bO?%:(x,lambda)KT:,valIH=inv(H);AHA=A*IH*A;IAHA=inv(AHA);AIH=A*I
4、H;G=IH-AIH*IAHA*AIH;B=IAHA*AIH;C=-IAHA;x=B*b-G*c;lambda=B*c-C*b;val=0.5*x*H*x+c*x;13/47JJIIJIBackClose10.1|S10.1)e?g5yKminf(x)=32x21+x22+12x23 x1x2 x2x3+x1+x2+x3,s.t.x1+2x2+x3=4.)N?H=310121011,c=111,A=?1 2 1?,b=4.3MATLAB-IXe-:H=3-1 0;-1 2-1;0-1 1;c=1;1;1;A=1 2 1;b=4;14/47JJIIJIBackClose x,lambda,val
5、=qlag(H,c,A,b)?x=0.38891.22221.1667lambda=0.9444val=3.277815/47JJIIJIBackClose10.2?g5y?k?8?g5ymin12xTHx+cTx,s.t.aTix bi=0,i E=1,2,l,aTix bi 0,i I=l+1,l+2,m,(10.15):Hn?.PI(x)=i|aTix bi=0,i I,e?n?K(10.15)?5,yz1.n10.2 x?g5yK(10.15)?4?:?=?16/47JJIIJIBackClose(1)3 Rm,?Hx+c PiEiaiPiIiai=0,aTix bi=0,i E,aT
6、ix bi 0,i I,i 0,i I;i=0,i II(x);(2)PS=?d Rn0|dTai=0,i E;dTai 0,i I(x);dTai=0,i I(x)i 0?,Ku?d S,kdTHd 0.Nuy,K(10.15)?g5y?H?.d,n10.2(2)g,v.5?zK?4?:?4?:?5,uke?n.17/47JJIIJIBackClosen10.3 x?g5y?4?:?xvKKT,=3 Rm,?Hx+c PiEiaiPiIiai=0,aTix bi=0,i E,aTix bi 0,i I,i 0,i I;i=0,i II(x).e0?)?g5yK?k?89MATLABy.10.
7、2.1k?8?n?ke?n,k?8n:,yz3.18/47JJIIJIBackClosen10.4?x?g5yK(10.15)?4?:,3x?k?8S(x)=E I(x),Kx?g5ymin12xTHx+cTx,s.t.aTix bi=0,i S(x)(10.16)?4?:.ln10.4uy,k?8?J:k?k?8S(x),d,k?E8S?%C,=l:x0u,Ok?8S(x0),)A?fK.-E,?k?8S?S(xk)(k=0,1,),?S(xk)S(x),?K?).un10.4,e40?k?8?n.19/47JJIIJIBackClose11:/fK|dk.?xkK(10.15)?1:,d(
8、A?k?8Sk=E I(xk),I(xk)=i|aTixk bi=0,i I.)A?fKmin12xTHx+cTx,s.t.aTix bi=0,i Sk.(10.17)K(10.17)?dumin qk(d)=12dTHd+gTkd,s.t.aTid=0,i Sk,(10.18):x=xk+d;gk=Hxk+c.?K(10.18)?4?:dk,kA?.KFf.12:?1|(k.b?dk6=0,/?.20/47JJIIJIBackClose(1)exk+dkK(10.15)?1:,=aTi(xk+dk)bi=0,i E9aTi(xk+dk)bi 0,i I.K-k=1,xk+1=xk+dk.(2
9、)exk+dkK(10.15)?1:,KL|e?1:.5?8I?g,o:AT31?.?.d,v1?k=.?i Sk,u?k 0,kaTidk=0aTi(xk+kdk)=aTixk=bi,d,k 0.?i 6 Sk,=1i?,dkvaTi(xk+kdk)bi,=kaTidk bi aTixk,i 6 Sk.5?m?,?aTidk 0,.?aTidk 021/47JJIIJIBackClose,d)?k6bi aTixkaTidk.?kk=k=min?bi aTixkaTidk?aTidk 0?.(1)(2),?k=min1,k.(10.19)13:?Sk.?k=1,k?8C,=Sk+1:=Sk.
10、?k 1,k=k=bik aTikxkaTikdk,?aTik(xk+kdk)=bik,d3xk+1?O?k?,=Sk+1:=Sk ik.22/47JJIIJIBackClose14:dk=0?/.dxkK(10.17)?4?:.eA?.KFfK,KxkK(10.15)?4?:,S“;K,XJA?.KFfkK?,oI-#e1.?ik 0,ik I(xk).y3e1dk,vgTkdk 0(i I(xk).B,Ue?dk:aTik(xk+dk)bik,aTi(xk+dk)=bi,i Sk,i 6=ik,=aTikdk 0,aTidk=0,i Sk,i 6=ik.(10.20)23/47JJIIJI
11、BackClose,5?xkfK(10.17)?4?:,?kHxk+c XiSkkiai=0,=gk=Akk,Ak=(ai)iSk,k=(ki)iSk.l?,gTkdk=TkATkdk.d(10.20)ATkdk=XiSk(aTidk)ei=(aTikdk)eik,ukgTkdk=Tk(aTikdk)eik=kik(aTikdk)0,xk?K?4?:;K,e(k)t 0,K-Sk:=Skt,=1.3,(k.-k=min1,k,k=mini6Sk?bi aTixkaTidk?aTidk 0?.-xk+1:=xk+kdk.4,ek=1,K-Sk+1:=Sk;K,ek 1,K-Sk+1:=Sk ik
12、,ikv k=bik aTikxkaTikdk.5,-k:=k+1,=1.e10.2?5n.27/47JJIIJIBackClosen10.5b?K(10.15)?H?.e310.2zS“?Ak=?ai?iSk?k6=0,K10.23kS?K(10.15)?4?:.y 5?edk=0,KxkfK(10.17)?KKT:?4?:.edk6=0k=1,KSk+1=Sk,uxk+1?fKEK(10.17),xk+1K(10.17)?4?:.k?k 0,x1,x2 0.?1:x0=(0,0)T.29/47JJIIJIBackClose)k,(?HcH=2 00 2,c=24.3x0?,k?8S0=2,
13、3,KA0=(ai)iS0=1 00 1,g0=Hx0+c=24.)A?fKmin q0(d)=d21+d22 2d1 4d2,s.t.d1=0,d2=0,?)d0=(0,0)T.d,x0A?fK(10.17)?).O.30/47JJIIJIBackCloseKFf0=(A0H1AT0)1A0H1g0=24,dd,x0K?).(0)3=4A?,=?K?13?lk?8S0?K,S0=2,2)A?fKmin d21+d22 2d1 4d2,s.t.d1=0,?)d0=(0,2)T.dud06=0,IO0.5?0=min?bi aTix0aTid0?i 6 S0,aTid0 0?31/47JJIIJ
14、IBackClose=1(1,1)(0,2)T=12=12,?0=min1,0=12.-x1=x0+0d0=(0,0)T+12(0,2)T=(0,1)T.0=12 0?33/47JJIIJIBackClose?.?S“u?.d?,10.2I(1:.e:?O:x Rn,e?55ymin eTz,s.t.aTix+izi bi=0,i E=1,2,l,aTix+zi bi 0,i I=l+1,l+2,m,z1,z2,zm 0,(10.21):e=(1,1,1)T;i=sgn(aTi x bi)(i E).K(10.21)?1:x=x,zi=|aTi x bi|(i E),zi=maxbi aTi
15、x,0(i I).Jy,XJ xK(10.15)?1:,o(x,0)fK(10.21)?).,XJK(10.15)k1:,KK(10.21)?34/47JJIIJIBackClose0,l?fK(10.21)?)?)K(10.15)?1:.ek?8)?g5yK?MATLABS,3,e,TS?.S10.2?Su)?e?g5yK.function x,lambda,exitflag,output=qpact(H,c,Ae,be,Ai,bi,x0)%U:k?8)?g5yK:%minf(x)=0.5*x*H*x+c*x,%s.t.a_i*x-b_i=0(i=1,2,.,l),%a_i*x-b_i=0(
16、i=l+1,l+2,.,m)%:H,cO8I?g.?,35/47JJIIJIBackClose%Ae=(a_1,.,a_l),be=(b_1,.,b_l),%Ai=(a_l+1,.,a_m),bi=(b_l+1,.,b_m),%x0:%:x),lambdaA?f,%exitflaga.,output(?C,%4?f(x),S“gk?&E%=Sm=%zepsilon=1.0e-9;err=1.0e-6;k=0;x=x0;n=length(x);maxk=1000;ne=length(be);ni=length(bi);index=ones(ni,1);for(i=1:ni)36/47JJIIJI
17、BackCloseif(Ai(i,:)*xbi(i)+epsilon),index(i)=0;endend%Swhile(k0),Aee=Ae;endfor(j=1:ni)if(index(j)0),Aee=Aee;Ai(j,:);endendgk=H*x+c;m1,n1=size(Aee);dk,lamk=qsubp(H,gk,Aee,zeros(m1,1);37/47JJIIJIBackCloseif(norm(dk)ne)y,jk=min(lamk(ne+1:length(lamk);endif(y=0)exitflag=0;elseexitflag=1;for(i=1:ni)if(in
18、dex(i)&(ne+sum(index(1:i)=jk)index(i)=0;break;end38/47JJIIJIBackCloseendendk=k+1;elseexitflag=1;%alpha=1.0;tm=1.0;for(i=1:ni)if(index(i)=0)&(Ai(i,:)*dk0)tm1=(bi(i)-Ai(i,:)*x)/(Ai(i,:)*dk);if(tm1tm)tm=tm1;ti=i;end39/47JJIIJIBackCloseendendalpha=min(alpha,tm);x=x+alpha*dk;%?k?8if(tm0);s=size(p,1);40/4
19、7JJIIJIBackClosefor i=1:slambda(ne+p(i)=lamk(ne+i);endoutput.fval=0.5*x*H*x+c*x;output.iter=k;%=)fK=%function x,lambda=qsubp(H,c,Ae,be)ginvH=pinv(H);m,n=size(Ae);if(m0)rb=Ae*ginvH*c+be;lambda=pinv(Ae*ginvH*Ae)*rb;x=ginvH*(Ae*lambda-c);41/47JJIIJIBackCloseelsex=-ginvH*c;lambda=0;end5(1)uS10.2,3fqsubp
20、,?2_(pinvMATLABS?2_?),?=OH?/,?y?O?-5.,?,Tf?fK?g5y?).(2)S10.2,IrJK?8I?k,L?m5).42/47JJIIJIBackClose10.3|S10.2-#)10.2,=min f(x)=x21+x22 2x1 4x2,s.t.x1 x2+1 0,x1,x2 0.?1:x0=(0,0)T.)k,(k:H=2 00 2,c=24,Ae=,be=,Ai=1 11001,bi=100.?|NS10.2?callqpact.m43/47JJIIJIBackClosefunction callqpactH=2 0;0 2;c=-2-4;Ae=
21、;be=;Ai=-1-1;1 0;0 1;bi=-1 0 0;x0=0 0;x,lambda,exitflag,output=qpact(H,c,Ae,be,Ai,bi,x0),?,3MATLAB-Icallqpact,=?(Jx=01lambda=44/47JJIIJIBackClose2.000000exitflag=0output=fval:-3iter:4w,(J10.2?.45/47JJIIJIBackClose10.4|S10.2)e?g5yKmin f(x)=32x21+x22+2x23 x1x2+2x1x3,s.t.3x1+2x2 5x3 4,2x1 3x2 2x3 3,x1,x2,x3 0.?1:x0=(0,0,0)T.)3MATLAB-IXe-:H=3-1 2;-1 2 0;2 0 4;c=1-3-2;Ai=-3 2-5;2-3-2;1 0 0;0 1 0;0 0 1;bi=-4-3 0 0 0;x0=0 0 0;46/47JJIIJIBackClose x,lambda,exitflag,output=qpact(H,c,Ai,bi,x0)?x=0.09520.90480.2381lambda=00.4286000exitflag=47/47JJIIJIBackClose0output=fval:-2.1905iter:10