:
min F=Z()
.
1. , (.. ).
2. : S=0.
3. Z() grad Z().
grad Z()=0, 11, 4.
4. , ..
= + t grad Z() ( n ):
5. Z()=Z(t) ( Z).
6. t* minZ(t).
7. Z()=Z(t) Z().
Z()> Z(), , .8; , .. Z()<Z(t), , (). , .. , 9, 8.
8. S=S+1, , . S<M ( ), , ( 7).
S ≥ M, 10.
9. , : = ; Z()=Z() 3.
10. S M, ( ).
11. *,
grad Z( *) = 0, .
17:
minZ=
1. (0;0) : (0,0) ?
, .
2. S=0 ( =5).
I .
3. Z()=
grad Z() ={2x1-3; 2x1-2}={-3; -2}; grad Z() = - 5 0, . 4.
4. :
= + t∙ grad Z()
(-3t; -2t).
5. Z():
Z() = Z(t) = (-3t)2+(-2t)2 -3(-3t)-2(-2)t = 9t2+4t2+9t+4t = 13t2+13.
6. t* min Z(t):
∂Z/∂t = 0; 26t+13 = 0; t*= - 13/26; t* = - 0,5.
∂2Z/∂t2 = 26 > 0, t* - .
7. Z(t*) = 13 (- 0,5)2 + 13 (-0,5) =3,25 6,5 = - 3,25,
Z(t*)< Z() (-3,25<0), , S=0.
= (-3t; -2t) = (-3(-0,5); - 2 (-0,5)); (1,5; 1).
Z() = Z(t) = - 3,25.
: (1,5; 1) ?
21,5 + 1≤ 5? →4<5
1,5 + 31≤ 4? →4,5>4
1,5≥0; 1≥0.
, (1,5; 1) , .. , . 8.
8. S=S+1; S=0+1=1. S<M (1<4), t*= t*/2 = - 0,5 / 2; t*= - 0,25, . 7.
7. Z(t*) = 13 (- 0,25)2 + 13 (-0,25) =0,8125 3,25 =- 2,4375,
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Z(t*)< Z() (-2,4375<0), , S=1.
II .
1. :
(-3t; -2t) = (-3(-0,25); -2 (-0,25)) = (0,75; 0,5);
(0,75; 0,5).
2. : (0,75; 0,5) ?
20,75 + 0,5≤ 5? →2<5
0,75 + 30,5≤ 4? →2,25<4
0,75≥0; 0,5≥0
(0,75; 0,5) , .. (S=S+0; S=1). S=1+0;
S=1. S<M (1<4), = 0,75; 0,5) . 3.
3. Z() = Z() = Z(t) = -2,4375.
III .
3.
grad Z() ={2x1-3; 2x1-2} = {20,75-3; 20,5-2} = {-1,5; -1} = - 2,5 0, . 4.
4. :
= + t* grad Z(); 1=0,75+t(-1,5); 2=0,5+t(-1)
(0,75 - 1,5t; 0,5 - t).
5. Z():
Z()=Z(t)=(0,75- 1,5t)2+(0,5- t)2 3(0,75- 1,5t)-2(0,5 - t)= 2,25t2+t2-2,25t -t+4,5t+2t+0,5625+0,25-2,25-1=3,25t2+3,25t - 2,4375.
6. t* min Z(t):
∂Z/∂t = 0; 6,5t+3,25 = 0; t*=-3,25/6,25; t* = - 0,5.
∂2Z/∂t2 = 6,5 > 0, t* - .
7. Z(t*) = 3,25t2+3,25t - 2,4375 = 3,25 (- 0,5)2 +3,25(-0,5) - 2,4375; 3,250,25 +3,25(-0,5) - 2,4375 = - 0,8125 2,4375 = -3,25,
Z(t*) < Z() (-3,25<-2,4375), ,
(S=S+0; S=1. S=1+0=1);
IV .
1. :
(0,75 - 1,5t; 0,5 - t). (0,75 - 1,5(-0,5); 0,5 (-0,5)).
(1,5; 1)
2. : (1,5; 1) ?
21,5 + 1≤ 5? →4<5
1,5 + 31≤ 4? →4,5>4
1,5≥0; 1≥0
(1,5; 1) , .. (S=S+1; S=1; S=1+1 S=2; S<M; 2<4).
t*= t*/2 = - 0,5 / 2; t*= - 0,25, . 7.
(0,75 - 1,5t; 0,5 - t). =(0,75 1,5×(-0,25); 0,5 + 0,25).
(0,75 + 0,375; 0,75); (1,125; 0,75).
: (1,125; 0,75) ?
21,125 + 0,75≤ 5? → 3 < 5
1,125 + 30,75≤ 4? →3,375 < 4
1,125≥0; 0,75≥0.
(1,125; 0,75) Î .
Z()=(1,125)2+(0,75)2 3 × 1,125 - 2× 0,75 = 1,265625 + 0,5625 3,375 1,5 = = - 3,609375.
Z() < Z() (- 3,609375 < -2,4375), , S=2; S<M; 2<4.
() = (1,125; 0,75); Z() = Z() = - 3,609375.
IV : (1,125; 0,75);
Z() = -3,609375.
. 3, grad Z() ..
( )
, .
1. ( ). t*, .
2. ∆t, z( 0 ) ( ).
:
= 0 + ∆t ∙ z 0 ,
1 = 10 + ∆t∙ ( 0 );
2 = 20 + ∆t∙ ( 0 );
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n = n0 + ∆t∙ ( 0 ); ∆t .
3. z, , . z; .
6
1 12 ().
1. min z = (1 -1)2 + (2 -1)2; 0 (2,2)
2. max z = x12 + 2x1ּx2 + 2x22; 0 (1,0)
3. max z = x12 + 4x1ּx2 + x22; 0 (1,0)
4. min z = x12 - 2x1ּx2 + 3x22; 0 (2,1)
5. max z = x12 - 2x1ּx2 + 2x22; 0 (2,1)
6. min z = 2x1 + x2 - x22; 0 (1,-1)
7. min z = -x12 + 2x1ּx2 - 4x22; 0 (1,1)
8. max z = 3x12 + 4x1ּx2 + 6x22; 0 (-1,-1)
9. min z = 2x1 + 3x2 - x12 - 2x22; 0 (1,2)
10. max z = (1 -2) 2 + 2x22; 0 (2,1)
11. max z = 9x1 - 8x2 0,5 x12 - 2x1ּx2; 0 (1,2)
12. min z = -10x1 - x12 + 2x1ּx2 + x22; 0 (1,1)
7
1 30 (). = 5. ∆ = 0,1.
1. max z = 2x1 + 3x2 - 2x22
: x1 + 4x2 ≤ 4
x1 + x2 ≤ 2
x1 ≥ 0; x2 ≥ 0
2. max z = 2x1 + 3x2 x12 - 2x22
: x1 - x2 ≥ 0
- x1 + 2x2 ≤ 2
x1 + x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
3. max z = x1 + 5x2 2x12 + 2x1ּx2 - 2x22
: 2x1 + x2 ≥ 2,0
3x1 + x2 ≤ 4
x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
4. min z = x1 - 3x1 + x12 - 2x22
: x1 - 2x2 ≤ 8
x2 ≤ 3
x1 ≥ 0; x2 ≥ 0
5. max z = x12 + 4x2 - x22
: x1 + x2 ≤ 6
x1 - x2 ≤ 1
x1 ≥ 0; x2 ≥ 0
6. min z = x12 - 4x2 + x22 - 3x2
: x1 - x2 ≤ 3
x1 ≤ 5
x1 + 2x2 ≥ 1
x1 ≥ 0; x2 ≥ 0
7. min z = x12 - 2x1 + 2x22 - x2
: 2x1 - 3x2 ≤ 0
x2 ≤ 5
x1 ≥ 0; x2 ≥ 0
8. max z = x12 + x22 - 2x1 - 2x2 + 2
: x1 + x2 ≥ 1
2x1 + x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
9. max z = x12 + x22 -2x1 + 2x2 + 3
: x1 + 2x2 ≥ 3
2x1 - x2 ≤ 1
x1 + x2 ≤ 2
x1 ≥ 0; x2 ≥ 0
10. min z = 2x12 + 2x22 - x1
: 2x1 + x2 ≤ 3
-x1 + 2x2 ≥ 1
x1 ≥ 0; x2 ≥ 0
11. min z = x12 + x22 - 4x1 + 1
: x1 + x2 ≤ 10
2x1 - x2 ≤ 10
x1 ≥ 0; x2 ≥ 0
12. max z = 4x12 + 4x22 - 8x1 - 2 x2 + 1
: 5x1 + x2 ≥ 6
3x1 - 2x2 ≤ 1
x1 + 2x2 ≥ 3
x1 ≥ 0; x2 ≥ 0
13. max z = 4x12 + x22 + 2x1
: x1 - x2 ≥ 0
x1 + x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
14. max z = 9x12 + 4x22 + 2x1 7
: x1 - x2 ≥ 0
x1 + x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
15. max z = x12 + x22 - 3x1 - 2 x2
: x1 - x2 ≥ 0
x1 + x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
16. max z = 4x1 + 5x2 - 2x22
: x1 + 3x2 ≤ 6
x1 + x2 ≤ 2
x1 ≥ 0; x2 ≥ 0
17. max z = 5x1 + 3x2 x12 - 2x22
: x1 - x2 ≥ 0
x1 + 2x2 ≤ 6
-x1 + x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
18. max z =3 x1 + 5x2 2x12 + 2x1ּx2 - 2x22
: 2x1 + x2 ≥ 2
2x1 + x2 ≤ 4
x2 ≤ 5
x1 ≥ 0; x2 ≥ 0
19. min z = 3x1 - 3x2 + x12 - 2x22
: x1 - 2x2 ≤ 8
x2 ≤ 4
x1 ≥ 0; x2 ≥ 0
20. max z = 2x12 + 4x2 -3x22
: x1 + x2 ≤ 5
x1 - x2 ≤ 1
x1 ≥ 0; x2 ≥ 0
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21. min z = 2x12 - 4x2 + x22 - 3x2
: x1 - x2 ≤ 3
x1 ≤ 4
x1 + 2x2 ≥ 1
x1 ≥ 0; x2 ≥ 0
22. min z = 4x12 - 2x1 + 2x22 - x2
: 2x1 - 3x2 ≤ 0
x2 ≤ 5
x1 ≥ 0; x2 ≥ 0
23. max z =3x12 + x22 - 2x1 - 2x2 + 2
: x1 + x2 ≥ 1
2x1 + x2 ≤ 6
x1 ≥ 0; x2 ≥ 0
24. max z = 4x12 + x22 - 2x1 + 2x2 + 3
: x1 + 2x2 ≥ 3
2x1 - x2 ≤ 1
x1 + x2 ≤ 3
x1 ≥ 0; x2 ≥ 0
25. min z = 2x12 + 2x22 -x1 +2x2
: 2x1 + x2 ≤ 4
-x1 + 2x2 ≥ 1
x2 ≤ 2
x1 ≥ 0; x2 ≥ 0
26. min z = x12 + x22 - 4x1 + 2x2+ 10
: x1 + x2 ≤ 6
2x1 - x2 ≤ 1
x1 ≥ 0; x2 ≥ 0
27. max z = 2x12 + 4x22 - 4x1 - 2 x2 + 3
: 5x1 + x2 ≥ 1
3x1 - 2x2 ≤ 2
x1 + 2x2 ≥ 3
x1 ≥ 0; x2 ≥ 0
28. max z = 4x12 + x22 + 2x1+ 3x2
: x1 - x2 ≥ 0
x1 + x2 ≤ 5
x1 ≥ 0; x2 ≥ 0
29. max z = 9x12 + 4x22 + 2x1 x2 + 5
: x1 - x2 ≥ 0
x1 + x2 ≤ 6
x1 ≥ 0; x2 ≥ 0
30. max z = x12 + 3x22 - 3x1 - 2 x2 + 4
: x1 - x2 ≥ 0
x1 + x2 ≤ 6
x1 ≥ 0; x2 ≥ 0
. - ().