(, ). . , . , , , , . , , , , , . 0,6% , 2% .
, . , , , , , , , . , - [30].
, . (, , , ..) (, , ..). aij i - j- , bi, i- (, ) cj, j - .
j - i, ( ):
(11.1)
(11.2)
(11.3)
( ), , , [2].
(11.1)(11.3), , f(x) = . (x1,...,xn), (11.2) (11.3). . ,
f() → min, Î . (11.4)
, x1Ox2 x1 x2. (11.3) , , (11.2) , (. 11.1), . . , , OABCDEF.
, . . , 1O2
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f()=1x1+2x2 (11.5)
, , a, Za. , Za ={(x1,x2):f(x)=a} ; grad f(x)= N, ( ) a. , OABCDE, Z . - ( ), - (, CD 1 2).
, , , . , , . , . . .. (1939 .). , .
. , . . , .
, , . :
(11.6)
, , . , (11.6). :
max f 2(x)= f 2(E), xÎP (. 11.2). (11.7)
. , : f2 (). ( ), .
1. , . . e, 0 < e < 1,
(11.8)
e=1. , e=0, (. 11.1 11.2). 0<e<1 , , , (. 11.3). , , .
, , . , , [39].
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1. Rn , , z , , {,z }= { Î Rn: =l +(1-l )z, 0 l 1, , .
, , , . . 11.4 , R2.
2. f(x), xÌ Rn, , z, X, lÎx[0,1] ( [ly+(1-l)z], 0 l 1, X)
, (11.9)
. (11.9) , f(x) .
( ) (. 11.5).
(11.9) , (y,f(y)) = = (z,f(z)), ( ) ( D). f(x) , () .
, (. 11.5) . Y = f(x) + f'(x)(x-x), , f(x)- f(x) f(x)(x-x) 0,
0<Q<1.
(- x)2 → x, ,
f (x) 0. (11.10)
x [, z] ( ). .
1. f(x) =ex, xÎ (-¥,+¥), f() = e > 0, , .
2. f(x) = sin x, xÎ[0,2p], f (x) = - sin x, , sin x [0, π] [π, 2 π].
, ,
f(x) → min, Î (f(x) → max, Î ), (11.11)
max f () , min , . , .
. , , 1,..., . (. . t>T ). , i - ti, ( , ) 1- , ai>0 ( , t i ). () , .
(11.12)
(11.13)
(11.14)
,
(11.15)
, , g(x)→ max (-g(x))→ min; , (11.13) (11.14) (. 11.6). , :
, (11.16)
Ж , (11.13) (11.14).
, f(t) . , .
, <0. , f(t) ti, i = 1, 2,..., n,
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t1 + t2 ++ tn = T( . 11.6). , .., , f(t) . f(t1,t2)= [0, T] 1, 2, α1, α2, (, ).
, . [18].
. , - . , , . . , , , 34:
, , , , . ( ). , , , 5 , . , , 2 , .
, ( ). , : , . . .
, . , , . , . , : , . 4, , , . . , . , 4- 4- ( 4 , ).
, - - , , . .
, , . ( ) . . , , (1 ). (1 ). , (, ),
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(,)=4(1-)+(1-)=+4-5. (11.17)
max min E(x,y).
x y
=4-5, (11.18)
>0 , Ey =0 = <0 . ,
( . 11.7). ,
(11.19)
. , F(x,y), (, ).
F(x,y)=5xy-x-4y. (11.20)
Fx=5y-1, Fx< 0 , Fx = 0 y = Fx>0 < ≤ 1. ,
( . 11.8). ,
(11.21)
.
.
,
; (11.22)
(,) , v = .
.
A: x1 ≥ 0,...,m ≥ 0, x1 +x2++xm=1 : y1 ≥ 0,...,ym ≥ 0, y1 +y2++ym=1
(11.23)
x = (x1,,xn) X, =(1,...,yn) Y.
[18] ( ); , , :
1. ;
(11.24)
.
2. ; .
3. , v. , * =(1*,...,*m)Î , , j =1, 2,...,n y(j) =
v ≤ E (x*, y(j)). (11.25),
* =(y*1,...,y*n)ÎY , , x(i) =
E (x(i), y*) ≤ v. (11.26)
, . 1.
1 [8], .
, , , , 1,2 m3 ; 15, 5 (. . 2 5 m3, 1 3 2).
, , . (. 11.1):
1,2,3, y1,y2,y3 , :
x1+x2+x3=l, y1+y2+y3=1, (11.27)
v ≤ -2x1, -2y1 ≤ v,
v ≤ -6x2, -62 ≤ v,
v ≤ -30x3, -30 3 ≤ v.
.
1=1= , x2=y2= , x3=3= . (11.28)
(11.27),
v = . (11.29)
. (11.30)
(11.31)
, . , . ( ) .
. , 4, , . . 11.2:
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1,2,3, y1,y2,y3,y4 v, :
x1+x2+x3=l, y1+y2+y3+y4=1, (11.27)
v ≤ -2x1, -y4 2y1 ≤ v,
v ≤ -6x2, -3y4 62 ≤ v,
v ≤ -30x3, -15y4 30 3 ≤ v.
v ≤ - x1 3x2 15 x3
, ( , , ). ,
-15y4 30 3 ≤ v. 3=0, 3>0. , 3 = 0,
-y4 2y1 = v, -3y4 62 = v, -15y4 30 3 = v
y1, 2, 4. , ,
(11.32), 3 =0, , . .
, (11.33)
, .
(11.34)
, 3, = 0, x3 = 0, . . m3 . , x3 = 0 ( ). , x1, 2 v
x1 + x2 + x3 = 1, v = -2x1, v = -6x2,
, x3 = 0,
(11.35)
,
(11.36)
(11.35) , . . (11.34), , , .
, , , . , , , .
1. (, , ) (, , ). = (1, 2, c3) (.) (10; 20; 50), b = (b1, b2, b3) () (1,2; 4; 1,5). , .
f1(x)= → min ≤ b, ≥ 0.
2. f2(x) = 2 → max.
3. f1(x)→ min, f2(x)→ max, f(x)= Θ f1(x)-(1- Θ )f2(x). .
4. 13.
5. = 1: 2: 3. p1, 2, p3, , .. t1 = , t2=, t3 = , , (t1 = t2 = t3 = T/3).
6. , , :
7. , , :