Phương pháp điểm trong và ứng dụng9

Luận văn thạc sĩ toán học: Phương pháp điểm trong và ứng dụng

-57-

Phu Luc Một số thuật toán điểm trong giải bài toán quy hoạch tuyến tính ở dạng chính tắc

*%##%sk+okseksks+okkse£E+e£'Thuật toán tỷ lệ affine **** FFE EEREERRRAER ERE EERE

function[X,dem]=main(A,b,c)

x=timnghiemxuatphat(A,c,b);

[X,dem]=hamchinh2(A,c,x);

FCIAC ICI I AOR ICR ICR I CR Cf aC fe ak Fe fe fe € << C

function[x,y]=kiemtra(s,x,y)

%Xx,s la vec to cot

epsi=0.000001;

a=length(x);

T=ones(1,a);

X=diag(x);

hl=I*X*s;

h2=-X*X*s;

J=0;

for i=1:length(s)

if s(i)>=0

J=i*l;

bQ)=s();

end

end

bị

k1=length(b);

k2=length(s);

if(kl == k2)&(hl<epsi)

fprintf{(Nghiem hien hanh da toi uu.\n')

fprinf(x= %g',X);

fprinf(%g',y);

elseif (h2>=0)

fprinfCMuc tieu toi uu la am vo cuc)

else

return

end

SOIC đc Ác do CAC ACCA ECACC ACA CACC A A CACC OE

function[x,T]=kiemtra2(x)

j=0;

b=0;

k=0.01;

:length(x)

J=l+l;

if (x(i)==0)I(x(i)<k)

Tels

for

break

else

b0)=x@);

end

end

b2=length(x);

if(length(b)==b2)

T=0;

end

2 Re c dị eof fe fe 2 fe 2k a Re 2 28 ok eo oe oc os ac os ae ok ae ok ake akc ae ae

function[y,s]=nghiemdoingau(A,x,c)

%x.¢ la vecto cot

X=diag(x);

yl=A*X*X*A';

y2=inv(y);

y3=A*X*X*%c;

y=y2"y3;

s=c-A'*y;

33K oe eo He A fe oR RC AC Rk ae ake 2 fe di k dể c eo ak a ae a ae

function[x2]=timnghiemxuatphat(A,c,b) M=1000;

[m,n]=size(A);

x=ones(n+1,1);

e=ones(n,1);

Al=b-A*e;

for i=1:(n+1)

if (i<=n)

cl(,1)=cŒ,1);

else

c1(,1)=M;

end

- 59 -

end

c1;%vec to c khi them thanh phan

for i=1:m

for j=1:(n+1)

if j<=n

A2(,=AG,)s

else

A2(i,J=ALG,1)5

end

end

end

A2;%Matran A khi them thanh phan

(x1,d]=hamchinh2(A2,cl,x);

for i=1:(n+1)

if (i<=n)

x2(,1)=x1(,1);

end

end

x2;

3X đc đc 4444444444444 "huật toán Gidm the SERRE RR RAR AAA AAA ARR

function[x2,y2,s2]=nghiemtoiuu(A,c,b)

MI=100;

M2=200;

[m,n]=size(A);

e=ones(n,1);

for i=1:(n+2)

if (i<=n)

el (i,1)=c(i,1);

end

if i==n+1

cl(,1)=MI;

end

ifi==n+2

cl(i,1)=0;

end

end

cl%vec to c khi them thanh phan

%tim b gach

A2=b-A*e;

:(m+l)

for j=1:(n+2)

if (j<=n)&&(i<=m)

A30j)=AG¡j);

end

if G==n+1)&&(i<=m)

A3(i,jJ=A2(i, 1)s

end

if G==n+2)&&(i<=m)

if ==m+1)&&(j<=n)

A3G,)=VG,D;

end

m+1)&&(j==n+1)

A3(i,j)=0;

end

ifG==m+1)&&(j==n+2)

A3G,)=1;

end

end

end

A3;%Matran A khi them thanh phan

for i=1:(n+2)

if (i<=n)

x(i,1)=1;

end

ifi==n+l

x(Œ,I)=l;

end

ifi==n+2

x(,1)=M2-(e-c)'*e;

yG,1)=0;

end

we§T x

if i==m+1

yd,1)=-1;

end

end

if i==n+1

s(i,1)=M1;

end

ifi==n+2

sq,1)=l;

end

end

§S;

{[x1,y1,s1]=main(A3,x,y,s,c 1);

for i=1:(n+2)

if (i<=n)

x2(,1)=x1(,1);

end

end

x2

for i=1:(m+1)

if (i<=m)

y24,1)=y1,1);

end

end

y2;

for k=1:(n+2)

if (k<=n)

s2(k,1)=s1(k,1);

end

end

s2;

FICCI ACI AICI ICICI a A I a AC a GIR I OR Ia a a

function[x,y3,s]=main(A,x,y,s,c)

K=kt(x,s);

ep=0.00000001;

while K>=ep

[D,U]=huong(A,x,s);

[x,y,s]=up(x,s,y,D,A,U),

K=kt(x,s);

end

yl=pinv(A’);

y2=c-s;

y3=y1*y2;

34OXEICOIECE XE CÁ 4X 4 đo ACE AR ACR SACI ACHR ACI ACHE function[D,U]=huong(A,x,s)

[m,n]=size(A);

q=n+sqrt(n);

Ro=0.28537;

X=diag(x);

A2=A*X;

A3=inv(A*X*X*A');

Al=A2*A3*A2;

UI=X%s;

U2=q/(s'*x);

e=ones(n,1);

T=eye(n);

U3=(U1*U2)-e;

U=(-A1)*(U3);

U4=norm(U);

D=-Ro*X*(U/U4);

Ee fe fe fee Re ake oe eo fe ake 2 2k eo a a a aa ak a aK a

function[x,y,s]=up(x,s,y,D,A,U)

[m,n]=size(A);

e=ones(n,1);

X=diag(x);

to=0.479056;

q=n+sqrt(n);

ul=norm(U);

K=inv(A*X*X*A‘);

if ul>=to

else

X=x;

s=((s'*x)/q)*(inv(X))*(U+e);

-63-

yayt(K*A*X*(X*s-(((s'*x)/q)*e)));

end

FCIAC CIC IGICICI AGI AIO GR ICI II sÉ S‡ Ác 2% a

function[K]=kt(x,s)

K=s*x;

3⁄4eletetetexexesete+etctct©+£Thuật toán theo đường trung tâm gỐC ***#**#*###########e4exexese+ function[x,y,s]=capnhat(A,x,d,y,c)

s=c-A'*y;

x=x+d;

y=y;

S4 đc đc các CACR ACC ACCA AC ACACIA A CAC A

function[d,y]=doanNT(A,x,c,muy)

%c.x la vec to cot

[m,n]=size(A);

X=diag(x);

T=eye(n);

e=ones(n,1);

AI=A*X*X*A;

A2=inv(AI);

d4=X*X*A'*A2*A;

dI=(-d4);

d3=(X*X*c)*(1/muy);

d2=(X*e-d3);

d=d1*d2;

y=A2*A*(X*X*c-muy*X*e);

SRC ACR A ACCRA ACCC RR CR A A AC RR A CCE đc

function[x,y,s]=hchinh(A,x,y,s,c,muy)

Epsi=0.000001;

%muy=2;

K=kt(s,x);

while K>=Epsi

muy=0.92* muy;

[d,y]=doanNT(A,x,c,muy);

[x,y,s]=capnhat(A,x,d,y,c);

K=kt(s,x);

end

JESSE SI CSIC IGIO SIO ISIS ICICI IIIA IR

function[K]=kt(x,s)

Kes'*x;

JSS CSI CASIO ISIC ISIC SIO ICR ICICI AAA AIR

function[x3,y2,s2]=nghiemxuatphat(A,c,b) M=100 ;

[m,n]=size(A);

% bai toan nhan tao goc

x=ones(n+2,1);

e=ones(n,1);

for i=1:(n+2)

if (i<=n)

el(i,1)=c(i,1);

end

ifi==n+l

cl(,1)=M;

end

if i==n+2

cl(i,1)=0;

end

end

cl;%vec to c khi them thanh phan

%im b gách

for i=l:m

for j=1:n

if A(i,j)<0

ALGj=-AGj)s

else

A1G,)=AG)s

end

end

end

Al;

for i=l:m

for j=l:n

U2=max(A1(i,j));

end

end

U2;

bl=max(abs(b));

c2=max(abs(©));

Ul=max(bl,c2);

U7=max(U2,U1);

U=U7;

-65-

%U=100

rl=(m*U)^m ;

12=(n+2)/(n*r1);

r3=(n*r1)/(n+2);

b2=r2*b;

%b2 chinh la b gach

A2=b2-A*e;

for i=1:(m+1)

for j=1:(n+2)

if §<=n)&&(i<=m)

Dg,j)=Adj);

end

==n+l)&&(<=m)

Dúq,j)=A2(,1;

end

if G==n+2)&&(i<=m)

Dúj)=0;

end

1ƒ (==m+1)&&(j<=n)

Dúj)=1;

end

if (i==m+1)&&G==n+1)

DG,)=1;

end

if G==m+1)&&(j==n+2)

DGD=1;

end

end

end

D;%Matran A khi them thanh phan

h=norm(c);

muy=4*(sqrt(h^2+M^2));

for i=1:(m+1)

if (i<=m)

y(i,1)=0;

end

if i: +1

y(,1)=-muy;

end

end

ys

d=c+muy*e;

for i=1:(n+2)

if (i<=n)

s(i,1)=d(i, 1);

end

if i==n+1

s(i, 1)=M+muy;

end

ifi==n+2

s(,1)=muy;

end

end

§

[x1,y1,sI]=hchinh(D,x,y,s,c1,muy);

for i=1:(n+2)

if (i<=n)

x2(i, =x 1(i,1);

end

end

x25

X3=137x2;

for i=1:(m+1)

if (i<=m)

y2,1)=y1đ,1);

end

end

y2;

for k=1:(n+2)

if (k<=n)

s2(k,1)=s1(k,1);

end

end

sy

esto eka a oe

“#Thuật toán đường trung tâm gốc-đối ngẫu***##%##%%#%

function[x,y,s]=chinh(A,b,c,x,y,s)

K=ktra(s,x);

epsi=0.00001;

[m,n]=size(A);

teta=0.5;

E1

-67-

muy=(s'*x)/n;

while (K >=epsi)&&(t<=10)

[Dx,Dy,Ds]=doan_Newton(A,x,s,muy);

p=gtnn(x,Dx);

d=gtnn(s,Ds);

[xy,s]=up(x,y,s,Dx,Dy,Ds,p,d);

muy=(1-teta)*muy

K=ktra(s,X);

t=t+l;

end

đã đc đồ Xí 4cd CSIC ACR CAC AC ICR doc doc function[x,y,s,t]=chinh11(A,b,c,x,y,s)

K=ktra(s,x);

epsi=0.0000001;

to=3.186;

[m,n]=size(A);

teta=0.999;

muy=(s'*x)/n;

t=0;

while n*muy >=epsi

muy=(1-teta)*muy;

G=giatri(x,s,muy);

u=0;

while G>=to

[Dy,Dx,Ds]=NT(A,x,s,muy);

p=gtnn(x,Dx);

d=gtnn(s,Ds);

[x,y,s]=up(x,y,s,Dx,Dy,Ds,p,d);

G=giatri(x,s,muy);

u=u+l;

end

t=t+l;

end

EIS ISISISIOISSIO SIO ASIII ACR SACK SCA RICA AA A function[Gi]=giatri(x,s,muy)

%muy=(1-0.99)*muy;

for i=1:length(x)

w1(i)=sqrt(x(i)*s(i)/muy);

end

wl;

for i=1:length(x)

w2(i)=sqrt(muy/(x(i)*s(i)));

end

w2;

Gi=0.5*norm(w1-w2);

TR RC fe Ae Re EE SE dị eo AE FE AE 2 EO AE EAR EE € ĐC

function[P]=gtnn(a,k)

anpha=0.995;

(h,g]=gtnn I (a,k);

if(length(g)==length(k))

PI=I;

end

if (length(h)<=length(k))

if h==

PI=l;

else

hl=-h;

bl=anpha*hl;

b2=min(b1);

P1=min(1,b2);

end

end

P=anpha*P1 ;

FICCI GCC IC ICICI ICICI I CR CR IR I eH 2k a 2k a i ake

function[{h,g]=gtnn1(a,k)

Yanpha=0.5;

%j=0;

t=0;

n=length(k);

h=0;

g=0;

for i=1:

Jj

%j=j+l;

if (k(i)<0)

j=i+l;

h@)=a()/k@);

else

t=t+l;

g(0=kQ);

-69-

end

end

ke oe ko ae ok 2 2 2k ae ok 2 feo eof 2 ke a fe oe ae a ae a ae a ae ai a ak a

function[K]=ktra(s,x)

K=s*x;

S3 E đc Ác đ đc đe đo đe 4cđdcX đcđk

function[Dy,Dx,Ds]=NT(A,x,s,muy)

X=diag(x);

S=diag(s);

SI=inv(S);

[m,n]=size(A);

I=eye(n);

[m,n]=size(A);

e=ones(n, 1);

Al=inv(A*SI*X*A);

Dy=A1*A*S1*(X*S*e-muy*e);

Dx=-S1*(I-X*A'*A1*A*S1)*(X*S*e-muy*e);

Ds=-A'*Dy;

SEAR A ACI AC ACR A CCR RICCO RR

function[x,y,s]=up(x,y,s,Dx,Dy,Ds,p,d)

x=x+p*Dx;

y=y+d*Dy;

s=s++d*Ds;

**Thuat toán theo đường trung tâm gốc-đối ngẫu với hàm chắn tự chính quy#** function[x1,s1,y1]=main(A,b,c,x,y,s,t,k,v)

[m,n]=size(A);

muy=l;

teta=0.999;

ep=0.0000001;

to=3.2;

h=n+1;

jsh*muy;

while j >=ep

muy=(1-teta)*muy;

j=h*muy;

Gi=giatrihamchan(x,s,t,k,muy)

i=l;

while i<=10

% while Gi>=to

[Dy,Dx,Dt,Dv,Ds,Dk]=huongNewton(A,b,c,x,y,s,t,k,v,muy);

[anpha]=dodaibuoc(x,Dx,s,Ds,y,Dy,t,Dt,k,Dk,v,Dv);

[x,y,s,k,v]=capnhat(x,y,s,t,k,v,anpha,Dx,Dy,Ds,Dt,Dk,Dv);

Gi=giatrihamchan(x,s,t,k,muy);

i=i+1; end

end

xl=x/t;

yl=y/t,

sl=s/t;

JOISISOISIS SSIES AISIOI SISO SISSIES AIK AIK

function[x1,y1,s1,t1,k1,v1]=capnhat(x,y,s,t,k,v,anpha,Dx,Dy,Ds,Dt,Dk,Dv) xl=x+anpha*Dx;

yl=y+anpha*Dy;

sI=s+anpha*Ds;

tl=t+anpha*Dt;

kl=k+anpha*Dk;

vl=v+anpha*Dvy;

FICCI CIC ICICIGI A I ICR ICICI HCH ACH Ci aki ak eae

function[anpha]=dodaibuoc(x,Dx,s,Ds,y,Dy,t,Dt, k,Dk,v,Dv)

T1=do(x,Dx);

MI=max(T1);

T2=do(s,Ds);

M2=max(T2);

T3=do(y,Dy);

M3=max(T3);

T4=do(t,Dt);

14=max(T4);

T5=do(v,Dv);

%MS5=mac(T5);

T6=do(k,Dk);

%M6=max(T6);

M7= max(M1,M2);

M8=max(M3,T4);

M9=max(T5,T6);

M10=max(M7,M8);

M11=max(M9,M10);

M12=-M11;

anpha=0.99995*M 12;

GN

DHE RA HCE A THC A ACHE RF TICE RR I ACCES dc

291s

function[b]=do(x,s)

for i=1:length(x)

a(i)=x(i)/s(i);

end

jJ=*l;

ifa@)<0

bQ)=aQ);

else

b(j)=-100;

end

end

Fee fee Re CRC c dị fe fe Re aR ek a fe ae ea ae ake ae ake a aie ake ake ake

function[S]=giatrihamchan(x,s,t,k,muy) for i=1:length(x)

Z@)=xŒ)*s()/muy;

UI@)=sart());

end

Ul;

for i=1:length(x)

Z)=muy/(x()*s());

UQ)=sqrt(z0));

end

U;

U2=sqrt((t*k)/muy);

U3=((U2-1/U2)^2)*0.5;

t=U1;

for i=1:length(t)

T@)=(t)-1/t))*@@)-1⁄t))*0.5;

end

T;

$1=0;

for i=1:length(T)

$1=S1+T(i);

end

S1;

S=S1+U3;

FEO CC CC đc 4E đc dd ICICI I I A ACCC EI RoR

function[Dy,Dx,Dt,Dv,Ds,Dk]=huongNewton(A,b,c,x,y,s,t,k,v,muy) [Rp.Rd,Rg,Beta]=saiso(A,b,c,x,y,s,L,k,V);

[Rtk,Rxs,RIxs]=saiso I(muy,x,s,t,k);

Dv=(R1xs+Rtk)/Beta;

Rp1l=Rp*Dv;

X=diag(x);

S=diag(s);

XI=inv(X);

Rd1=Rd*Dv+X1*Rxs;%sai so voi cong thuc la*Rxs

Rgl=Rg*Dv+Rtk/t;

D2=inv(X1*S);

a=-b-A*D2*c;

al=(b-c*D2*A;

G= k/t+(c *D2*c);

Rp12=Rp1-A*D2*Rd1;

Rg12=Rg1+(c'*D2*Rd1);

a3=a/G;

r=Rp12-Rg12*a3;

AI=A*D2*A;

11=pinv(A1)*r ;%gia nghịch dao

12=pinv(A1)*(-a3);

Dy=I1-12*((a1'*11)/(1+a1'*12));

Dt=(Rg12-al'*Dy)/G;

Dx=D2*Rd1+D2*A'*Dy-D2*c*Dt;

Ds=X1*(Rxs-S*Dx);

Dk=(Rtk-k*DUWt;

ĐI RR I CORR ARICA RCA RR a TM dd

functon[Rp, Rd,Rg,Beta]= saiso(A,b,c,x,y,s,t,k,v)

Rp=(A*x-b*0/V;

Rd=(c*t-A'*y-s)/v;

Rg=(-c'*x-k-b'*y)/v;

Beta=x'*s+t*k;

ACR RAH RF AR RA AA lO A OO

function[Rtk,Rxs,R1xs]=saiso1(muy,x,s,t,k)

%ham chan chon co dinh la ham T=((w-1/w)42)/2

%la ham chan tu-chinh quy

for i=1:length(x)

2(i)=(x(i)*s(i))/muy;

Ul(i=sqrt(a(i));

end

“¡no

Ul:

U2=sqrt((t*k)/muy);

W1=hamchan(U1);

W2=hamchan(U2);

Rtk=-muy*(U2*W2);

for i=1:length(UI)

Rxs2(i)=-U 1(i)*W 1(i)*muy;

end

Rxs2;

RXS=Rxs2'*

R1xs=-muy*U1*W1';

Ae He Re A CRC Fee fe Re Re c c dc sự c dị fe ak ae ake ae ake ae ake ak ae ake

function [T2] =hamchan (u)

teu;

for i=1:length(t)

T2 (i) =(t (i) -1/t (i) ) * (1+1/t (i) *2) 7

end

%format long

T2;

3K 3€ dk sc tự ậc c dc Í€d €3 2k csỆ c sự Re eae a ake ae ake akc aie ak ake ae

function{anpha]=CPDgiam(A,b,c,x,y,s,dx,dy,ds,t,m) for i=1:length(x)

k1()=dx()/x();

end

kl;

hl=norm(k1);

for :length(s)

1Q)=ds()/s();

end

1;

k=norm(1);

w=sqrt(h1^2+k^2);

anpha=1/w-1/(w+4*t*t);

HR Re ee ae ok a fee €s eRe 2 sự sự aR a ake ak afc ake ke ak ae ake ae aie a aie

function[T]=CPDchan(x,s,m)

for i=1:length(x)

ul (i)=sqrt((x(i)*s(i))/m);

end

nls

for i=1:length(x)

u2(i)=sqrt(m/(x(i)*s(i)));

end

u2;

T=0.5*norm(ul-u2);

Hee Re Re CR a Hee fe Re ak eR a ak a ae a ae a ea a ae 2K ae

function[Dx,Dy,Ds]=CPDbuoc(A,b,c,x,y,s,m) S=diag(s);

X=diag(x);

SI=inv(S);

for i=1:length(s)

s2(i,1)=1/s(i);

end

s2;

Dy=(nv(A*X*S1I*A'))*(b-m*A*s2);

Ds=-A'*Dy;

Dx=m*s2-x-x*s2'*Ds;

RS A eo Fe oe Ae € sự c dị Ee fe ae Ae fe € d3 s3 s3 s‡

function[x,y,s,dem1,x1,x2]=CPDchinh(A,b,c,x,y,s) [m,n]=size(A);

muy=x'*s/n;

ep=0.0001;

to=3.18671;

dem=0;

dem1=0;

while((n*muy)>ep)

muy=(1-0.999)*muy;

t=CPDchan(x,s,muy);

while (t>=to)

[dx,dy,ds]=CPDbuoc(A,b,c,x,y,s,muy); a=CPDgiam(A,b,c,x,y,s,dx,dy,ds,t,muy); x=x+a*dx;

s=s+a*ds;

y=yta*dy;

t=CPDchan(x,s,muy);

deml=dem1+1;

for i=dem1:dem1

x1(1,)=x(1,1);

x2(1,j)=x(2,1);

end

end

dem=dem+l;