1. .
.
: Z=CXmax(min) A×X = B, X ³ 0.
1=(11, 21,, n1) 2=(12, 22,, n2) , A×X1 = B, X1 ³ 0; A×X2 = B, X2 ³ 0.
, 1 2
=l11+l22, l1³ 0, l2³0, l1+l2=1 .
, =(l11+l22)= l11+l22=l1+l2=(l1+l2)=,
.. , AX = B, .
.. X1 ³ 0; X2 ³ 0, l1 ³ 0,l2 ³ 0, ³ 0, .. X ³ 0, .
¨
, , .
2. , . , , .
.
, , . . , .
2 2 3,
1, 2,,, 1
*. Xi i
Z(X*)£ Z(X) 0 1
. * - , .
, * . * 2 ( ) , ..
*=l11+l22++l, li ³ 0 (i=1,2,,p), =1.
Z (X) ,
Z (X*)= Z (l11+l22++l)= l1 Z (X1)+ l2 Z (X2)++ lp Z (Xp). (*)
Z(Xi) (i=1,2,..,p) ( (1££)) m, .. Z (X)=m. (*) Z (Xi) . , li ³ 0, =1,
Z(X*)³l1m+l2m++lm= m =m.
, * - , , , Z(X*)£m, , , , Z(X*)³m, , Z(X*)=m= Z(X), .
, , .
, Z (X) , 1, 2,,q,1<q£p;
Z(X1)= Z(X2)== Z(Xq)=m.
:
=l11+l22++lqq, li ³ 0 (i=1,2,,q), =1.
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, , Z (X) ,
Z(X)=Z(l11+l22++lqq)= l1Z(X1)+ l2Z(X2)++ lqZ(Xq)=l1m+l2m++lqm= =m =m,
.. Z , 1,2,,q. ¨
. , , 2 3.2., .
, .. . , .
.
3. ( ) , , .
.
= (x1, x2,..., xm,0,,0) - A1x1+A2x2++Anxn= B, m , n-m , ( , ).
, .
, .. . , , ,
1= (x1(1), x2(1),..., xn(1)), 2= (x1(2), x2(2),..., xn(2)),
, - 1 2 , .. (*) =l11+l22, l1> 0, l2>0, l1+l2=1.
(, l1¹0, l2¹0, 1 2).
(*) :
xj(1)³0, xj(2)³0 (j=1,2,,n), l1>0 l2>0, n-m , xm+1(1)=0, xm+1(2)=0,, xn(1)=0, xn(2)=0, .. 1, 2 A1x1+A2x2++Anxn= B n-m :
1= (x1(1), x2(1),..., xm(1),0,,0), 2= (x1(2), x2(2),..., xm(2),0,,0).
.
1 2 ,
A1x1(1)+A2x2(1)++Amxm(1)= B, A1x1(2)+A2x2(2)++Amxm(2)= B.
,
(x1(1)- x1(2))A1+(x2(1)- x2(2))A2++(xm(1)- xm(2))Am= 0.
, 1, 2,...,m , ,
x1(1)- x1(2)=0; x2(1)- x2(2)=0;;xm(1)- xm(2)= 0.
x1(1)=x1(2); x2(1)=x2(2);;xm(1)=xm(2).
, . , .
. = (x1, x2,..., xm,0,,0) m . , ( ).
, .. 1, 2,...,m . l1, l2,...,lk, ,
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l1 1+ l2 2+... + l mm=0. (*)
e>0, (*):
e l1 1+ e l2 2+... +e l mm=0. (**)
1x1+ 2x2+... +mxm=B. (***)
(**) (***):
(x1 + e l1 ) 1+ (x2 + e l2 ) 2+... +(xm + e lm) m=B,
(**) (***), :
(x1 - e l1 ) 1+ (x2 - e l2 ) 2+... +(xm- e lm) m=B.
(***), , e 1x1+ 2x2+... +nxn=B :
X1=(x1 + e l1; x2 + e l2; ; xm + e lm; 0;;0),
X2=(x1 - e l1; x2 - e l2; ; xm - e lm; 0;;0).
i³0 (j=1,2,,n), e , 1 2 , 1 2 .
, , ½ (1 + 2) =X ( ), .. ( ), , 1, 2. , , .
, , 1, 2,...,m , . , .
¨
1. 1, 2,..., n m, m
xi > 0 (i=1,2,..., m).
2. k£m 1, 2,..., n.
, 1, 2,..., n m . , m , .
3. , , , .
, , :
- , ;
- ( , m - , n 1, 2,...,n) .
1. (, , ) .
2. , , .