1. ?
1 , ( ) ;
2 , ( ), - ;
3 , ( ), - .
2. ?
1 n ;
2 3 ;
3 2 .
3. () ?
1 , ;
2 ;
3 .
4*. ?
1 ;
2 n ;
3 3 ;
4 2 .
5. ?
1 ;
2 ;
3 ;
4 , , , .
6. Z(X1,X2) = =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11=C22>0, ?
1 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2+C0;
2 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2+C0;
3 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2.
7. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11=C22<0, ?
1 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2+C0;
2 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2+C0;
3 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2.
8. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22>0, ?
1 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2+C0;
2 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2;
3 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2 +C0.
9. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22 >0, ?
1 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2;
2 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2+C0;
3 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2.
10. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22, C11C22 <0, ?
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1 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2+C0;
2 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2+C0;
3 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2.
11. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C22 =0, C2 , ?
1 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2+C0;
2 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2+C0;
3 Z(X1,X2)= aX1 2 +bX1+c - X2.
12. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C 11 =0, C1 , ?
1 Z(X1,X2)= aX2 2 +bX1+c - X1;
2 Z(X1,X2)= C11(X1 -a)2 -C22 (X2-b)2+C0;
3 Z(X1,X2)= C11(X1 -a)2 +C22 (X2-b)2+C0.
13. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C2= C22 =0, Z(X1,X2)= C11X12 +C1X1+C0 ?
1 Z(X1,X2)= aX2 2 +bX1+c - X1;
2 Z(X1,X2)= aX1 2 +bX1+c;
3 Z(X1,X2)= aX2 2 +bX2+c.
14. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C1=C11=0, Z(X1,X2)= C22X22 +C2X2 +C0 ?
1 Z(X1,X2)= aX2 2 +bX1+c - X1;
2 Z(X1,X2)= aX1 2 +bX1+c;
3 Z(X1,X2)= aX2 2 +bX2+c.
15. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11=C22>0, ?
1 O`(a,b);
2 O`(C11,C22);
3 O`(-C11,-C22).
16. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11=C22<0, ?
1 O`(C11,C22);
2 O`(a,b);
3 O`(-C11,-C22).
17. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22>0, ?
1 O`(C11,C22);
2 O`(-C11,-C22);
3 O`(a,b).
18. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22 >0, ?
1 O`(C11,C22);
2 O`(a,b);
3 O`(-C11,-C22).
19. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22, C11C22 <0, ?
1 O`(a,b);
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2 O`(C11,C22);.
3 O`(-C11,-C22).
20. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C22 =0, C2 , ?
1 (-b/2a; X2);
2 (X1;-b/2a);
3 (a,b).
21. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C 11 =0, C1 , ?
1 (-b/2a; X2);
2 (X1;-b/2a);
3 (a,b).
22. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C2= C22 =0,
Z(X1,X2)= C11X12 +C1X1+C0 ?
1 , X2, (-b/2a; X2);
2 , X1, (X1;-b/2a);
3 , X2, b2-4ac >0 , b2-4ac <0;
4 , X1, b2-4ac >0 , b2-4ac <0.
23. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C1=C11=0,
Z(X1,X2)= C11X12 +C1X1+C0 ?
1 , X2, (-b/2a; X2);
2 , X1, (X1;-b/2a);
3 , X2, b2-4ac >0 , b2-4ac <0;
4 , X1, b2-4ac >0 , b2-4ac <0.
24. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C22 =0, C2 , Z(X1,X2)= =C22X12 +C2X2+C0 ?
1 , X2, (b/2a; X2);
2 , X1, (X1;b/2a);
3 , X2, b24ac >0 , b24ac <0;
4 , X1, b2-4ac >0 , b24ac <0.
25. Z(X1,X2) = =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C22 =0, C2 , Z(X1,X2)= =aX1 2 +bX1+c - X2?
1 , X2, (-b/2a; X2);
2 , X1, (X1;-b/2a);
3 , X2, b2-4ac >0 , b2-4ac <0;
4 , X1, b2-4ac >0 , b2-4ac <0.
26. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11C22 =0, C11 C22 :
C 11 =0, C1 , Z(X1,X2)= aX2 2 +bX2+c - X1.
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1 , X2, (-b/2a; X2);
2 , X1, (X1;-b/2a);
3 , X2, b2-4ac >0 , b2-4ac <0;
4 , X1, b2-4ac >0 , b2-4ac <0.
27. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11=C22>0, ?
1 ;
2 ;
3 ;
4 ;
5 ;
6 .
28. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11=C22<0, ?
1 ;
2 ;
3 ;
4 ;
5 ;
6 .
29. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22>0, ?
1 ;
2 ;
3 ;
4 ;
5 ;
6 ;
30. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22 >0, ?
1 ;
2 ;
3 ;
4 ;
5 ;
6 .
31. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22, C11C22 <0, ?
1 ;
2 ;
3 ;
4 ;
5 ;
6 .
32. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22>0, ?
1 C22 C11;
2 C11 C22;
3 C22 C11;
4 C11 C22 .
33. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22 ; C11C22 >0, ?
1 C22 C11;
2 C11 C22;
3 C22 C11;
4 C11 C22 .
34. Z(X1,X2)= =C11X12+C22X22+C1X1+C2X2 +C0
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aI1 X1 +aI2X2.≤aI 0 , i=1÷m,
X1≥0, X2≥0,
C11 C22, C11C22 <0, ?
1 X2 = b + K (X1 -a) X2 =b -K (X1 -a). K C11 C22;
2 X2 = b + K (X1 -a) X2 =b -K (X1 -a). K C11 C22;
3 X1 = b + K (X2 -a) X1 =b -K (X2 -a). K C11 C22;
4 X1= b + K (X2 -a) X1 =b -K (X2 -a). K C11 C22.
1. .., .. . .: - , 1981.
2. .. . .: , 2007.
3. .., .. . .: , 2001.
4. .., .. . .: , 2005.
5. .. . . .: , 1998
6. .. . . .: , 1997, 2003, 2004.
7. .. : . .: , 2005.
8. .., .., .. . .: , 1980, 1986.
9. .., .., .., .. . : . .: - .. , 2011.
10. .., .., .., .., .. . . ( ). .: - .. , 2009.
11. : . / .. . 3- ., . ., , 2005.
12. / . .. . .: , 1995.
13. .., .. - . .: , 2005. 320 .
14. .. : : - . ., -, 2008.
15. .., .. : . .: ʰ, 2005.
16. - : . . . ...-2- ., . . .:-, 2005.
1. .. . .: , 1986.
2. .. . .: , 1982.
3. .. - : / .. . .: : , 2008.
4. ., . . .: , 1966.
5. .., .., .. . . .: - , 2007.
6. .., .. . .: , 1989.
7. .. . .: , 1983.
8. .. . .: , 1982.
9. .., .., .. . . 1. .: , 2001, 2007.
() , , - , . - () , , - , . , , . - Z = Z(X) = Z(x1, x2,...,xn) - . , Z, antigradZ= -i×Z/x1-j×Z/x2-...-k×Z/xn. Z : antigradZ= - gradZ= -{Z/x1;Z/x2;...;Z/xn } - gradZ= -Ñ Z (). | ||||||||
, . , , . | ||||||||
. , , , () . -Z = Z(X) = Z(x1, x2,...,xn) - . , Z, gradZ= i×Z/x1+j×Z/x2+...+k×Z/xn. Z : gradZ= {Z/x1;Z/x2;...;Z/xn } gradZ=Ñ Z (). . ( ) , 1 2, , f[(1 + 2)/2] £ [f(1) + f(2)] /2. ( ) , (). . f(X), ( ) , , (). w , , w. . - . , : , - , | ||||||||
. , , , . , . , - , . | ||||||||
. , ; ; - . , , , . f(X), , , . . ( ) , 1 2, , f[(1 + 2)/2] ³ [f(1) + f(2)] /2. ( ) , (). . f(X), ( ) , , (). | ||||||||
(). , , , . f(X), , , . 1. (-1), . 2. - . - , , , - - , . 3. - . , : , - , . | ||||||||
. . ( ) : , | ||||||||
. . | ||||||||
, : ei (). | ||||||||
. , (bx + a)′ = b, . . ; (xn)′ = nx n-1. , , , , . . , x 0, , . , , (). | ||||||||
, (, ) , , . , , ( , ), , . ( , -.) , ( ), : 1) , , ; 2) , ( ); 3) , , , . | ||||||||
: , () , . : min Z=f(X) = Z(x1, x2,..., xn) = C11 x12 + C12 x1x2 + C13 x1x3 +...+ C1nx1xn + C22 x22 + C23 x2x3 +...+ C2nx2xn +... + Cnn xn2 + C1 x1 + C2 x2 +...+ Cnxn + C0 . . , , - . , . . min Z=f(X) = Z(x1, x2,..., xn) = C11 x12 + C12 x1x2 + C13 x1x3 +...+ C1nx1xn + C22 x22 + C23 x2x3 +...+ C2nx2xn +... + Cnn xn2 + C1 x1 + C2 x2 +...+ Cnxn + C0 . , ( ) (). | ||||||||
- . ( ). , . , : XÎ , , ji(X) < 0 i, i = 1 ¸ m. X* , , m- λ*(λ*1,λ*2,,λ*m) , (X*, λ*) . , ≥0 , .. f(X)+ λ*iji(X) ≥ f(X*)+ λ*i ji(X*) ≥ f(X*)+ λi ji(X*), i = 1 ¸ m. λi≥0, i = 1 ¸ m. | ||||||||
- ( ) , . . - .. , . . - f (x) = f (x 1, x 2,..., xn), ( ) . , , - . | ||||||||
(x *, λ*) , xi λ i. , λi , i = 1¸ m. | ||||||||
, ( , ) ( ). , . | ||||||||
, . , , . . , , , , - , . . (). . , . , . | ||||||||
. min Z = f() j ≤ xj ≤ bj; j = 1 ÷ n, f() . 1. ω(j ≤ xj ≤ bj); j=1÷n. 2. (1, 2,..., n) = (x1 , x2 ,,xn ) = (x10, x20,..., xn0), ω. 3. Z = f(). 4. = , Z = f(). 5. S = 0, N = 1000 . 6. λ. 7. , Î w, : = xj0 +Pj , - , , =1, = + 1. , - , ≤ Pj ≤ . , . Pj =A+ λjּB, λj - λ. 8. , xj1 = xj0 + (A+ λjּB) . = - e. 9. = , Z = f(). f() f(). 10. , f() <f(), 11, 12. 11. = Z =f() = f(). 7. 12. f() ≥ f(), S = S + 1. 13. S N, N . 14. S ≤ N, 7, 15. 15. . Z = f() . | ||||||||
aij (n)- - ( n-1) , i- j- . | ||||||||
(), ( ) () . , , , . | ||||||||
min Z = f() j ≤ xj ≤ bj; j = 1 ÷ n, f() 1. ω(j ≤ xj ≤ bj); j=1÷n. 2. (1, 2,..., n) = = (x1 , x2 ,,xn ) = = (x10, x20,..., xn0), ω. 3. Z = f(). 4. = , Z = f(). 5. S = 0, N = 1000 . 6. λ. 7. , Î w, xj1 = xj +λj ּxj = j + λj ּbj. 8. = , Z = f(). 9. f() f(). , f() <f(), 11, 12. 10. = Z =f() = f(). 7. 11. f() ≥ f(), S = S + 1. 12. S N, N . S ≤ N, 7, 13 13. . Z =f() . | ||||||||
-, M 0, ε > 0 Sr (M 0) (S M 0) , M Î S | f (M) f (M 0) |< ε; -, f (M) V, . , . ( ), . ( ), . | ||||||||
, . min Z=f(X) φi(X) ≤ 0, i=1¸m, xj ≥0; j=1¸n, f(X) , φi(X), i=1¸m . , . . : (=), (£), (³). , , . ( A): det A. , ( ) : det A = a 11 a 22 a 12 a 21. , ( ) . | ||||||||
. , , . , - , , . .
. f (x) df / dx , . .
. , f (x 1,..., xn), ( ). df / dx. x 1,..., xn
; f , , , :
, , ′ ', , f(x) df / dx, f′(x).
: 2015-11-23; !; : 552 | : : , . |
: 0.126 .