, . , , , , ., , .., .
: X=(x1,x2,..., xn), , /
ji(x1, x2,..., xn) £ bi, i=1,2,...,k, (1)
ji(x1, x2,..., xn) = bi, i=k+1,k+2,...,m
Z=f(x1, x2,..., xn); (2)
x1, x2,..., xn .
ji / f , (). . , , . , ( , ) .
, , () . , .
, . . , (1) , , ji(x1, x2,..., xn) f(x1, x2,..., xn) . .. : x1, x2,..., xn,
ji (x1, x2,..., xn) = bi, i=1,2,...,m, (3)
()
Z=f(x1, x2,..., xn). (4)
: - 1 2. , , , 1 2 . ( ) Z=f(x1, x2). . (x1 x2) (1 2). b=c1x1+c2x2. , Z.
|
|
: x1, x2, b=c1x1+c2x2, 1³0, 2³0, Z=f(x1, x2) . , Z .
, .
, , .
, Z=f(x1, x2,..., xn)= f() *=(x*1, x*2,..., x*n) . f(*)³ f() f(*)£ f(), , f() * ( ).
*, f() 0, .
: * f() , :
, ( ) .
Z=f() * D ( ) ( ), f(*)³ f() f(*)£ f()
Î D.
D , Z=f() ( ).
, () Z=f() D, :
1) D ;
2) D;
3) , .1 2: () () .
D () x1, x2,..., xn. , , .
Z=f(x1, x2,..., xn) , x1, x2,..., xn
ji(x1, x2,..., xn) = 0, i=1,2,...,m, m<n. (*)
, f ji . (*) .
Z=f() , (*), (), f()³ f() (f()£ f()) , .
, (3), (4).
, m n-m :
xi=yI(xm+1, ,xn), i=1,2,,m.
xi Z,
Z= f(y1(xm+1, ,xn),, ym(xm+1, ,xn), xm+1, ,xn), Z=F(xm+1, ,xn).
()
|
|
n-m . Z=F(xm+1, ,xn) , =(y1 ,,ym , Z=f(x1, x2,..., xn) ( ).
n=2, . ji(x1, x2,..., xn) £ bi, i=1,2,...,m, Z=f(x1, x2).
, () , ji(x1, x2,..., xn) £ bi. . , .
- , : f(x1, x2)=C, () , , , . , , , .
1. Z=(x1-4)2+(x2-6)2 :
x1³0, x2³0.
. ABCE (. 1). Z=Q (Q>0), - (x1-4)2+(x2-6)2=Q. () Q ( ) Z ().
(4, 6). M Q : Z(D)=196/13 D (24/13,36/13), . D , , MD CE.
Z : A(1;0)
.1 Z(A)=45, E(6;0) Z(e)=40.
Z()>Z(E), A .
2. ,
Z=(x1-4)2+(x2-1)2. .
. M(4;1) (.2): Z(M)=0. Z : E(6;0) Z(E)=5, (0;4) Z(C)=25, .
. 2
3. Z=x12+x22
x1³0, x2³0.
. (.3) . Z=17 A(1;4) L(4;1). Z
.3. :
D(2/3; 6) Z(D)=328/9 M(7; 4/7) Z(M)=2417/49. M .
(-) , , . , .
:
f(x1,x2,...,xn) (5)
jj(x1,x2,...,xn)=0, j=1,2,...,m, m<n. (6)
. , f ji . :
L(x1,x2,...,xn,l1,l2,...,lm)=f(x1,x2,...,xn)+ . (7)
, li - . , . f(x1,x2,...,xn) , =(x1,x2,...,xn), jj(x1,x2,...,xn) i- , , li () i- , i- ( ).
|
|
X xj (j=1,2,...,n), L - li (i=1,2,...,m). (j=1,2,...,n), (i=1,2,...,m) . :
(8)
, =jj(), .. (8) .
Z=f(x1,x2,...,xn) X(o)=(x1(o), x2(o),..., xn(o)) , L(o)=(l1(o),l2(o),...,lm(o)),
(x1(o), x2(o),..., xn(o), l1(o), l2(o),..., lm(o)) (8). , Z=f() L(X,L). . , () . Z=f(x1,x2,...,xn) ji(x1,x2,...,xn) (i=1,2,...,m) , , (8). .
, (8), , .
4. Z=x1x2+x2x3 .
. L(x1,x2,x3,l1,l2)= x1x2+x2x3+l1(x1+x2-2)+ l2(x2+x3-2) x1, x2, x3, l1 l2. , :
, l1=l2=-x2;
, x1=x2=x3=1; Z=2.
9.3.
.
:
Z=f(x1,x2,...,xn) (5)
jj(x1,x2,...,xn)=0, j=1,2,...,m, m<n. (6)
, -, xj ³ 0, j=1, 2,...,n, .. .
f() xj ³ 0, .
X0=()
Z, .
f(x) , , 0, .. n n-1 m , x1=0, x2=0,..., xn=0 f(x1,x2,...,xn) ji(x1,x2,...,xn) (i=1, 2,..., m). n , 0 Cn2 n-2 m . .
0 n-m , m . Z. , , n-m , . . .
, f(x), , , f(x*). .
|
|
xj³0 X=(x1,x2,...,xn), , , .
, - , , .. :
ji(X)=ji(x1, x2,..., xn) £ bi i=1,2,...,p;
ji(X)=ji(x1,x2,..., xn)³bi i=p+1,p+2,..., q;
gi(X)=bi i=q+1, q+2,..., m.
. , , , .
, , . :
xn+i ³ 0, i = 1, 2,..., q.
z=f(X) - xn+i, i=1, 2,..., q, .
xn+i³0, i=1, 2,..., q.
-
. f() , , .
( ) =(x1(o), x2(o),..., xn(o), l1(o), l2(o),..., lm(o)), e- e- L(X, L(o)) £ L(X(o), L(o)) £ L(X(o), L).
L .
L(X, L(o)) £ L(X(o), L(o)) £ L(X(o), L) , .
-, , X0=(), f() .
: Z=f(x1,x2,...,xn)
ji (x1,x2,...,xn)=0, i=1,2,...,m,
x1, x2,...,xn ³ 0 (9)
:
L(X,L)=f(X) + (10)
( X, ji (X)>0 i), .
-. X(o) (9), L(o), X(o) ³ 0 L(o) ³ 0
X ³ 0 L ³ 0
L(X, L(o)) £ L(X(o), L(o)) £ L(X(o), L). (11)
.
(11). X(o) > 0 L(o) > 0 - L(X, L). (11) L(X, L) (10),
f(X)+ £f(X(o))+ £f(X(o))+
X ³ 0, L ³ 0.
L ³ 0,
ji (X) ³ 0, =0.
f(X(o)) ³ f(X)+ X ³ 0.
³ 0, f(X(o)) ³ f(X).
, X(o) (11) , X(o). L(X,L).
(11) .
f(X) ji(X) - , (11) -:
|
|
(12)
(13)
, (X(o), L(o)), X(o)=(x1(o), x2(o),..., xn(o)),
L(o)=(l1(o), l2(o),..., lm(o)).
. max Z = -x12 -x22
x1³0, x2³0.
. : x1=0,8, x2=0,4, Z= -0,8.
, L(o) ³ 0, - (12), (13) F(X, L):
F(X, L)=-x12-x22+l1(2x1+x2-2)+l2(8-2x1-x2)+l3(6-x1-x2).
:
= -2x1+2l1-l3; = -2x2+l1-l2-l3;
=2x1+x2-2; =8-2x1-x2; =6-x1-x2.
(13) l2 l3 , , , x1=0,8 x2=0,4 , , , , . l1 , . (12) (j=1,2) , X(o) . l1=0,8. , (X(o), L(o)) - .