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FAQ

⇐ 3456789101112 ⇒



- [10]

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min Z = f(X) = f(x1, x2,...,xn),

f(X) .

I . , X0 (x01, x02,...,x0n) .

II .

I. Y

Y=(X - X0)= (x1-x01, x2-x02,...,xn-x0n), X (x1, x2,...,xn) Î w, w - (). Y = X - X0. X (x1, x2,...,xn) , : 1) grad Z × Y = grad Z ×(X - X0) < 0, Z ; 2) X (x1, x2,...,xn) Î w.

II. .

min Z = grad Z(X0) × Y = grad Z(X0) × (X - X0) =

= [Z(X0)/(x1)]× (x1-x01) + [Z(X0)/(x2)]× (x2-x02) + [Z(X0)/(x3)]× (x3-x03) + ∙∙∙ +[Z(X0) (xn)]× (xn-x0n)

, i = 1 ¸ m.

X (x1, x2,...,xn) ³ 0.

3. , .

min Z = grad Z(X0) × Y = grad Z(X0) × (X - X0), Z X0 f(X) Y = X - X0 . . , X X0 . . , X (x1, x2,...,xn). , (, , ).

III . , . X* c () X0 . = X0 +t ·Y = X0 +t · (X - X0), t Î[0, 1].

IV . - .

. grad f(X0) = 0. . - :

f(X0)/(x1) = 0;

f(X0)/(x2) = 0;

f(X0)/(x3) = 0;

.

f(X0)/ (xn) = 0.

(), () . , , . , |f(X0)/(xj)| ≤ ε (j = 1 ¸ n), ε , .

. ( ) ε: | X0 t - X0 t+1 | ≤ ε.

() , (), .

1. () .

2. . Y. Y = X - X0.

3. Y, .

4. () . , 2.

14. min Z = 12+22

1+2 ≤ 2

1 ≤ 2

2 ≤ 1

1 ³ 0, 2 ³ 0

.

I . X0(1;1). ε = 0,02.

II .

1) grad Z(X) : grad Z(X) =

={ Z(X)/(x1); Z(X)/(x2)} = {2x1; 2x2}.

grad Z X0 grad Z(X0)={2 · 1;2 · 1 }={2; 2}.

2) .

min Z = grad Z(X0) × Y = grad Z(X0) × (X - X0) = [Z(X0)/(x1)]× (x1-x01) + +[Z(X0)/(x2)]× (x2-x02) ={2; 2} · [(x1-1); (x2-1)] = 2x1 2+2x2 2 = 2x1 +2x2 4.

:

min Z = 2x1 +2x2 4.

1+2 ≤ 2

1 ≤ 2

2 ≤ 1

1 ³ 0, 2 ³ 0.

3) , .

:

1) 1+2 ≤ 2 ;

1+2 = 2 - . .

 

j
1    
2    

 

(0,0) .

2) 1 ≤ 2, (0,0) .

3) 2 ≤ 2, (0,0) .


. 8.

 

4) 1 ³ 0,

2 ³ 0 I .

5) - ABCD.

6) Z = 2x1 +2x2 4

antigrad Z =- grad Z ={(Z)/(x1); (Z)/(x2)} = {2; 2 }. , (0; 0). (2; 2). .

7) Z = 2x1 + 2x2 4 = const, . . . ABCD. - A(0;0) , : min Z (A)= Z (0;0)= 2 × 0 +2 × 0 4 = - 4. , , grad Z(X) × Y = grad Z(X) × (X - X0) = {2; 2} · [(0 - 1); (0 - 1)] = -4 . X*(0,0) .

)
III . , . X* c () .

. = X0 +t ·Y = X0 +t · (X - X0), t Î[0, 1], X0 = (1;1).

. = (1;1) + t · [(x1-1); (x2-1)], t Î[0, 1].

x1 = 1 + t · (x1-1);

x2 = 1 + t · (x2-1);

. = [x1; x2 ] = [1 + t · (x1-1); 1 + t · (x2-1) ].

, X (x1; x2), (0; 0) . . X0 (1;1) (0; 0), x1 = 1 + t · (0 - 1);

x2 = 1 + t · (x2 - 1);

. = [x1; x2 ] = [1 + t · (0 - 1); 1 + t · (0 - 1) ]

x1 = 1 - t; x2 = 1 - t;

. = [x1; x2 ] = [1 - t; 1 - t ].

, t, t Î[0, 1].

4

tÎ[0, 1]. . Z = 12+22
x1 = 1 - t; x2 = 1 - t;
         
  0,5 0,5 0,5 0,5
  0,7 0,3 0,3 0,18
         

(0,0) t=1 , X0 (0; 0). () , Z(X)/(x1); Z(X)/(x2) X0 .

 

Z(X)/(x1) =2 · 1; Z(X0)/(x1) = 2 · 0 = 0;

Z(X)/(x2) =2 · 2; Z(X0)/(x2) = 2 · 0 = 0;

() X* (0; 0). ()

min Z (X0) = grad Z(X0) × Y = grad Z(X0) × (X - X0) = [ Z(X0)/(x1)]× (x1-x01) + +[ Z(X0)/(x2)]× (x2-x02) ={0; 0} · [(x1-0); (x2-0)] = 0

1+2 ≤ 2

1 ≤ 2

2 ≤ 1

1 ³ 0, 2 ³ 0.

 

, Z (X0) = 0, , . . , X = X* (2; 0).

. = X0 +t ·Y = X0 +t · (X - X0), t Î[0, 1], X0 = (0;0),

x1 = 0 + t · (1 - 0); x2 = 0 + t · (x2-0);

. = [x1; x2 ] = [0 + t · (1 - 0); 0 + t · (2 -0) ]

x1 = t · 1; x2 = t · 2;

. = [x1; x2 ] = [ t · 1; t · 2 ].

. = [x1; x2 ] = [ t · 1; t · 2 ] X* (2; 0), X . . (2 · t; 0 · t).

, t, t Î[0, 1] X0 = (0;0).

 

5

tÎ[0, 1]. . Z = 12+22
x1 = 2·t x2 = 0
         
  0,01 0,02   0,0004
  0,1 0,2   0,04

 

. t () : | X0 t - X0 t+1 | ≤ ε, ε = 0,02?

| X0 t - X0 t+1 | = | X0 t - X0 | = √[ (x1t-x10)2 + (x2t-x20)2 ].

| X0 1 - X0 | = √[ (0 - 0)2 + (0 - 0)2 ] = 0; 0 <0,02 = ε, .

| X0 2 - X0 | = √[ (0,02 - 0)2 + (0 - 0)2 ] = √ 0,0004 = 0,02; 0,02=0,02= ε, .

| X0 3 - X0 | = √[ (0,2 - 0)2 + (0 - 0)2 ] = √ 0,04 = 0,2; 0,2>0,02= ε, .

X = X* (0,02; 0). X0 1(0;0) (). , (), .

. : . .

15

min Z =912+922 +121 - 62

0 ≤ 1 ≤ 4

1 ≤ 2 ≤ 5

1 ³ 0, 2 ³ 0

.

I . X0(3;4). ε = 0,02.

II .

1) grad Z(X) :

grad Z(X) ={ Z(X)/(x1); Z(X)/(x2)} = {18x1+12; 18x2 - 6}.

grad Z X0:

grad Z(X0) = {18 · 3+12+18 · 4 - 6}=132.

2) .

min Z = grad Z(X0) × Y = grad Z(X0) × (X - X0) = [Z(X0)/(x1)]× (x1-x01) + +[Z(X0)/(x2)] × (x2-x02) ={18 ·3 +12; 18 ·4 - 6} · [(x1-3); (x2-4)] ={66; 66} · [(x1 -

-3); (x2-4)] = 66x1 198 + 66x2 264 = 66x1 +66x2 462.

:

min Z = 66x1 +66x2 462.

0 ≤ 1 ≤ 4

1 ≤ 2 ≤ 5

1 ³ 0, 2 ³ 0

3) , .

:

1. 1 ≤ 4 ;

1 = 4 - , (4;0), 2. (0,0) .

2. 2 ≤ 5 ;

2 = 5 - , (0;5), 1. (0,0) .

3. 2 ³ 1 ;

2 = 1 - , (0;1), 1. (0,0) .

4. 1 ³ 0,

2 ³ 0, I .

5. - ABCD.

6. Z = 66x1 +66x2 462

antigrad Z = - grad Z ={(Z)/(x1); (Z)/(x2)} = {66; 66}. , (0; 0). (66; 66). 11 , (6; 6). .

7. Z = 66x1 +66x2 462 = const, . . . ABCD. - A(0;1) , : min Z (A)= Z (0;1)= 66 × 0 +66 × 1 462 = - 396 < 0. , , grad Z(X) × Y = grad Z(X) × (X - X0) ={18x1+12; 18x2 - 6} · [(x1-3); (x2-4)]= 66x1 +66x2 462 = 66 × 0 +66 × 1 462 = - 396 . X*(0,1) - .

III . , . X* c () .

. = X0 +t ·Y = X0 +t · (X - X0), t Î[0, 1], X0 = (3;4).

. = [(3;4) + t · [(x1-3); (x2-4)], t Î[0, 1].

 

. 9.

 

x1 = 3 + t · (x1-3);

x2 = 4 + t · (x2-4);

. = [x1; x2 ] = [3 + t · (x1-3); 4 + t · (x2-4) ].

, X (x1; x2), (0; 1) . . X0 (3;4) (0; 1), x1 = 3 + t · (0-3);

x2 = 4 + t · (1-4);

. = [x1; x2 ] = [3 + t · (0-3); 4 + t · (1-4) ]

x1 = 3 - 3t; x2 = 4 - 3t;

. = [x1; x2 ] = [3 - 3t; 4 - 3t].

, t, t Î[0, 1].

6

tÎ[0, 1]. . Z = = 912+922+12 1-6 2
x1 = 3 - 3t; x2 = 4 -3t;
         
  0,4 1,8 2,8 118,12
  0,8 0,6 1,6 29,08
         

 

(0,1) t=1 , X0 (0; 1). , Z(X)/(x1); Z(X)/(x2) X0 .

Z(X)/(x1) =18 · 1 + 12; Z(X0)/(x1) = 18 · 0 + 12= 12;

Z(X0)/(x2) =18 · 2 - 6; Z(X0)/(x2) = 18 · 1 - 6 = 12.

() X* (0; 1). ():

min Z (X0) = grad Z(X0) × Y = grad Z(X0) × (X - X0) = [Z(X0)/(x1)]× (x1-x01) + [Z(X0)/(x2)]× (x2-x02) ={12; 12} · [(x1-0); (x2-1)] ={12 x1; 12 x2 -12}

min Z = 12 x1+ 12 x2 -12

min Z = 12 · 0+ 12 · 1 12 =0

, Z (X0) = 0, , . . , X = X* (2; 0).

. = X0 +t ·Y = X0 +t · (X - X0), t Î[0, 1], X0 = (0;1),

x1 = 0 + t · (1 - 0); x2 = 1 + t · (x2-1);

. = [x1; x2 ] = [0 + t · (1 -0); 1 + t · (2 -1) ]

x1 = t · 1; x2 = 1 + t · (2 -1);

. = [x1; x2 ] = [ t · 1; 1 + t · (2 -1)].

. = [x1; x2 ] = [ t · 1; 1 + t · (2 -1) ] X* (2; 0), X . . (2 · t; 1 + t · (0 -1)) =(2 · t; 1 - t).

, t, t Î[0, 1] X0 = (0;1).

7

tÎ[0, 1] . Z = 912+922+ +121 - 62
x1 =2·t x2 = 1 - t;
  -      
         
  0,01 0,02 0,99 3,2055
  0,1 0,2 0,9 4,65
         

 

(0,1) t=0 , X0 (0; 1).

.t .t+1 : | X0 t - X0 t+1 | ≤ ε, ε = 0,02?

| X0 t - X0 t+1 | = | X0 t - X0 | = √[ (x1t-x10)2 + (x2t-x20)2 ].

| X0 1 - X0 | = √[ (0 - 0)2 + (1 - 1)2 ] = 0; 0 <0,02 = ε, .

| X0 2 - X0 1 | = √[ (0 0,02)2 + (1 0,99)2 ] =√ 0,0004+0,0001=√ 0,0005; √ 0,0005 > 0,02 = ε, .

| X0 3 - X0 2 | = √[ (0,02 0,2)2 + (0,99 0,9)2 ] =√0,0324+0,0081= √ 0,0405 > 0,02= ε, .

| X0 4 - X0 3| = √[ (2 - 0,2)2 + (0 0,9)2 ] = =√3,24+0,81= √ 4,05 > 0,02= ε, .

X = X* (0; 1). X0 1(0;1) (); , (), .

min Z = 3, X*(0;1)

3

- .

1. max z = 2x1 + 3x2 +2x12 + 3x22

: x1 + 4x2 ≤ 4

x1 + x2 ≤ 2

x1 ≥ 0; x2 ≥ 0

2. max z =3x12 + 4x22 + 2x1 + 3x2

: x1 - x2 ≥ 0

- x1 + 2x2 ≤ 2

x1 + x2 ≤ 4

x1 ≥ 0; x2 ≥ 0

3. max z = x1 + 5x2 + 2x12 + 3x22

: 2x1 + x2 ≥ 2,0

3x1 + x2 ≤ 4

x2 ≤ 4

x1 ≥ 0; x2 ≥ 0

4. min z = 4x12 + 3x22 - x1 - 3x1

: x1 - 2x2 ≤ 8

x2 ≤ 3

x1 ≥ 0; x2 ≥ 0

5. max z = x1 + 4x2 - 612 + 3x22

: x1 + x2 ≤ 6

x1 - x2 ≤ 1

x1 ≥ 0; x2 ≥ 0

6. min z =2x12 + 5x22 + 2x1 - 4x2

: x1 - x2 ≤ 3

x1 ≤ 5

x1 + 2x2 ≥ 1

x1 ≥ 0; x2 ≥ 0

7. min z = - 2x1 - 4x2 + 2x12 + 3x22

: 2x1 - 3x2 ≤ 0

x2 ≤ 5

x1 ≥ 0; x2 ≥ 0

8. max z = - 5x1 - 2x2 +3x12 + 3x22 - 2

: x1 + x2 ≥ 1

2x1 + x2 ≤ 4

x1 ≥ 0; x2 ≥ 0

9. max z = 4x12 + 2x22 - 2x1 + 2x2 + 3

: x1 + 2x2 ≥ 3

2x1 - x2 ≤ 1

x1 + x2 ≤ 2

x1 ≥ 0; x2 ≥ 0

10. min z = 2x1 + 6x2 +2x12 + 3x22 - 1

: 2x1 + x2 ≤ 3

-x1 + 2x2 ≥ 1

x1 ≥ 0; x2 ≥ 0

11. min z = -2x12 + 3x22 + 4x1 + x2 + 1

: x1 + x2 ≤ 10

2x1 - x2 ≤ 10

x1 ≥ 0; x2 ≥ 0

12. max z = - 8x1 - 2 x2 +2x12 + 4x22 + 1

: 5x1 + x2 ≥ 6

3x1 - 2x2 ≤ 1

x1 + 2x2 ≥ 3

x1 ≥ 0; x2 ≥ 0

13. max z = 4x1 + x2 -2x12 + 4x22 - 2

: x1 - x2 ≥ 0

x1 + x2 ≤ 4

x1 ≥ 0; x2 ≥ 0

14. max z =3x12 + 2x22 + 9x1 + 4x2 + 2

: x1 - x2 ≥ 0

x1 + x2 ≤ 4

x1 ≥ 0; x2 ≥ 0

15. max z = 3x1 - 2 x2 - 2x12 + 4x22 - 5

: x1 - x2 ≥ 0

x1 + x2 ≤ 4

x1 ≥ 0; x2 ≥ 0

16. min z = -2x12 + 3x22 + 4x1 + 6 x2 + 1

: x1 + x2 ≤ 5

2x1 - x2 ≤ 6

x1 ≥ 0; x2 ≥ 0

17. min z = -2x12 - 3x22 + 4x1 +3x2 + 1

: x1 + x2 ≤ 7

2x1 - x2 ≤ 7

x1 ≥ 0; x2 ≥ 0

⇐ 3456789101112 ⇒


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: 2015-11-23; !; : 1102 |

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