Luận văn: Bài toán tối ưu với hàm thuần nhất dương pptx

đại học Thái Nguyên
Tr-ờng đại học khoa học


0






Nguyễn Xuân Huy



Bài toán tối -u
với hàm thuần nhất d-ơng





Luận văn thạc sĩ toán học

Thái Nguyên - 2009
www.VNMATH.com
S húa bi Trung tõm Hc liu i hc Thỏi Nguyờn

đại học Thái Nguyên
Tr-ờng đại học khoa học


0




Nguyễn Xuân Huy




Bài toán tối -u
với hàm thuần nhất d-ơng


Chuyên ngành: Toán ứng dụng
Mã số: 60.46.36


Luận văn thạc sĩ toán học

Ng-ời h-ớng dẫn khoa học
GS-TS Trần Vũ Thiệu






Thái Nguyên - 2009
www.VNMATH.com
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min-max max
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x
1
x
2
R
n
x
1
x
2
x = λx
1
+ (1 − λ) x
2

= x
2
+ λ(x
1
− x
2
) λ ∈ R
M ⊆ R
n
M
M ∀x
1
, x
2
∈ M λ ∈ R ⇒ λx
1
+ (1 −λ)x
2
∈ M
M
M 1 x ∈ R
x =
k

i=1
λ
i
x
i
λ

1
, λ
2
, ··· , λ
k
∈ R
k

i=1
λ
i
= 1
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λ
1
, λ
2
, ··· , λ
k
∈ R
n
M ⊆ R
n
x
0
∈ M L = M −x
0
=

x − x

0
| x ∈ M

a, b ∈ L c = λa + µb λ, µ ∈ R L
L
M = x
0
+ L =

x
0
+ v | v ∈ L

x
0
∈ M L L
M x
0
x
0
M L L
M
M
E ⊆ R
n
E E aff E
M Ax = b
A m × n b ∈ R
m
x

1
, x
2
∈ M ∀λ ∈ R
A

λx
1
+ (1 − λ) x
2

= λAx
1
+ (1 − λ) Ax
2
= λb + (1 − λ)b = b
⇒ λx
1
+ (1 − λ) x
2
∈ M
E =

x ∈ R
3
| 0 ≤ x
1
≤ 1, 0 ≤ x
2
≤ 1, x

3
= 0

E aff E =

x ∈ R
3
| x
3
= 0

M ⊆ R
n
dim M M ⊆ R
n
dim M < n a ∈ M
M
B(a, ǫ)

B(a, ǫ) ∩ aff M

⊂ M
M ri M
M M ⊆ R
n
dim M = n M int M = ∅
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E =

x ∈ R

3
| 0 ≤ x
1
≤ 1, 0 ≤ x
2
≤ 1, x
3
= 0

int E = ∅ ri E =

x ∈ R
3
| 0 < x
1
< 1, 0 < x
2
< 1, x
3
= 0

dim E = 2
x
1
, x
2
∈ R
n
x = λx
1

+ (1 − λ) x
2
= x
2
+ λ(x
1
− x
2
) 0 ≤ λ ≤ 1
x
1
x
2

x
1
, x
2

M ⊆ R
n
∀x
1
, x
2
∈ M 0 ≤ λ ≤ 1
λx
1
+ (1 − λ) x
2

∈ M
x ∈ R
n
x =
k

i=1
λ
i
x
i
λ
1
, λ
2
, ··· , λ
k
≥ 0
k

i=1
λ
i
= 1
x
1
, x
2
, ··· , x
k

∈ R
n
λ
i
≥ 0 ∀i = 1, 2, ··· , k
x x
1
, x
2
, ··· , x
k
∈ R
n
M ⊂ R
n
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M ⊂ R
n
α ∈ R
n
αM = { y | y = αx, x ∈ M}
M
1
, M
2
⊂ R
n
M
1
+ M

2
=

x | x = x
1
+ x
2
, x
1
∈ M
1
, x
2
∈ M
2

E ⊂ R
n
E
conv E E
convE
convE
E ⊂ R
n
E
M ⊂ R
n
x ∈ M
M x
M

 ∃y, z ∈ M, y = z x = λy + (1 − λ)z, 0 < λ < 1
M ⊂ R
n
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R
n
a ∈ R
n
\ {0} α ∈ R
H := { x ∈ R
n
|< a, x > = α}
n − 1 a
x
0
x
a
< a, x >= α
R
2
{x ∈ R
n
|< a, x > ≤ α} {x ∈ R
n
|< a, x > ≥ α}
a ∈ R
n
\ {0} α ∈ R
n
{x ∈ R

n
|< a, x > < α} {x ∈ R
n
|< a, x > > α}
{x ∈ R
n
|< a, x > = α}
M ⊂ R
n
a ∈ R
n
\{0} α ∈ R
H := { x ∈ R
n
|< a, x > = α}
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M x
0
∈ M x
0
∈ H
M H
< a, x
0
>= α < a, x > ≤ α, ∀x ∈ M
M
a
x
0
M x

0
x
0
M ⊂ R
n
M x
0
M ⊂ R
n
M ⊂ R
n
x ∈ M, λ ≥ 0 ⇒ λx ∈ M
0 ∈ R
n
M ⊂ R
n
M x
1
, x
2
∈ M λ
1
, λ
2
≥ 0
λ
1
x
1
+ λ

2
x
2
∈ M
0
0
0 R
n
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• R
n
+
:= {x = (x
1
, x
2
, ··· , x
n
) : x
i
≥ 0, i = 1, 2, ··· , n}
• R
n
++
:= {x = (x
1
, x
2
, ··· , x
n

) : x
i
> 0, i = 1, 2, ··· , n}
M ⊂ R
n
k v
1
, v
2
, ··· , v
k
∈ R
n
cone

v
1
, v
2
, ··· , v
n

:=

v ∈ R
n
| v =
k

i=1

λ
i
v
i
, λ
i
≥ 0, i = 1, ··· , k

⊂ R
n

v
1
, v
2
, ··· , v
k

v
h
∈

v
1
, v
2
, ··· , v
k

cone


v
1
, ··· , v
h−1
, v
h+1
, ··· , v
k

= cone

v
1
, v
2
, ··· , v
k

0 0
v
1
v
2
v
3
v
1
v
2

v
3
v
2
v
2
v
3
D ⊆ R
n
d = 0
D
{x + λd | λ ≥ 0} ⊂ D x ∈ D
d
D D D
D
D ⊆ R
n
0
D rec D
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d
1
d
2
d
1
= αd
2
α > 0

d D
D d
1
d
2
D
d = λ
1
d
1
+ λ
2
d
2
λ
1
, λ
2
> 0
C, D ⊂ R
n
H := {x ∈ R
n
|< a, x > = α} a ∈ R
n
\ {0} α ∈ R
H C D
< a, x > ≤ α ≤ < a, y > ∀x ∈ C, ∀y ∈ D
H C, D
< a, x > < α < < a, y > ∀x ∈ C, ∀y ∈ D

C, D ⊂ R
n
C, D ⊂ R
n
a ∈ R
n
A m ×n
< a, x > ≥ 0 x Ax ≥ 0 ∃y ∈ R
n
, y ≥ 0
a = A
T
y
K = {x ∈ R
n
| Ax ≥ 0}
{x ∈ R
n
|< a, x > ≥ 0}
{x ∈ R
n
|< a, x > = 0} A
P ⊆ R
n
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< a
i
, x > ≥ b
i
, i = 1, 2, ··· , m.

(1.1)
k ∈ {i = 1, 2, ··· , m}

x |< a
i
, x > ≥ b
i
, i = 1, 2, ··· , m

=

x |< a
i
, x > ≥ b
i
, i ∈ {1, 2, ··· , m} \ {k}

A m×n a
i
= (a
1
, a
2
, ··· , a
n
), i = 1, 2, ··· , n
b = ( b
1
, b
2

, ··· , b
m
)
T
x = (x
1
, x
2
, ··· , x
n
)
T
(1.1)
Ax ≥ b



< a
i
, x > = b
i
, i = 1, 2, ··· , m
1
< a
i
, x > ≥ b
i
, i = m
1
+ 1, ··· , m

a
1
a
2
a
3
a
4
a
5
D (1.1) x
0
∈ D
< a
i
, x
0
>= b
i
x
0
i
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I(x
0
) :=

i ∈ {1, 2, ··· , m} < a
i
, x

0
>= b
i

x
0
∈ D
D D
F = ∅ D F
D F
y ∈ D, z ∈ D, x = λy + (1 −λ)z ∈ F 0 < λ < 1 ⇒ y ∈ F, z ∈ F
f D ⊆ R
n
x
1
, x
2
∈ D λ ∈ [0, 1]
f

λx
1
+ (1 − λ) x
2

≤ λf(x
1
) + (1 − λ)f(x
2
)

f D
f

λx
1
+ (1 − λ) x
2

< λf(x
1
) + (1 − λ)f(x
2
)
x
1
, x
2
∈ D, x
1
= x
2
λ ∈ (0, 1)
f dom f = {x ∈ D | f(x) < +∞}
f
epi(f) := {(x, α) ∈ D × R | x ∈ D, α ≥ f(x)}
f : D −→ R ∪{+∞}
R
n
f(x) = +∞, ∀x /∈ dom f
f R

n
f D ⊆ R
n
epi(f)
g D ⊆ R
n
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hypo(g) : = {(x, α) ∈ D ×R | x ∈ D, α ≤ g(x)}
f
1
D
1
⊆ R
n
f
2
D
2
⊆ R
n
λ > 0 λf
1
f
1
+ f
2
max{f
1
, f
2

}
(λf
1
)(x) := λf
1
(x), ∀x ∈ D
1
(f
1
+ f
2
)(x) := f
1
(x) + f
2
(x), ∀x ∈ D
1
∩ D
2
max{f
1
, f
2
}(x) := max{f
1
(x), f
2
(x)}, ∀x ∈ D
1
∩ D

2
f
1
D
1
f
2
D
2
α > 0, β > 0 αf
1
+ βf
2
max{f
1
, f
2
} D
1
∩D
2
f D ⊆ R
n
f D ⊆ R
n
f
D
f : D −→ R D ⊆ R
n
d ∈ R \ {0} x

0
∈ dom f
f
′
(x
0
, d) ≤ f(x
0
+ d) − f(x
0
)
f D f
d ∈ R \ {0} x
0
∈ dom f
< ▽f(x
0
), d >= f
′
(x
0
, d) ≤ f(x
0
+ d) − f(x
0
)
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f D ⊆ R
n
f

D
f(y) − f(x) ≥ < ▽f(x), y − x >, ∀x, y ∈ D
f D = (a, b) ⊆ R
f D f
′′
(x) ≥ 0, ∀x ∈ D
f D ⊆ R
n
f
D ▽
2
f(x)
D ∀x ∈ D
y
T
▽
2
f(x)y ≥ 0, ∀y ∈ R
n
f D ▽
2
f(x) D
x ∈ D
y
T
▽
2
f(x)y > 0, ∀y ∈ R
n
\ {0}.

f(x) =
1
2
< x, Qx > + < c, x > + α
Q n c ∈ R
n
α ∈ R f
R
n
Q f
R
n
Q
f(x
1
, x
2
) = 2x
2
1
+ 3x
1
x
2
+ 4x
2
2
▽f(x) =



4x
1
3x
2
3x
1
8x
2


▽
2
f(x) =


4 3
3 8


▽
2
f(x) f
R
2
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min f(x) x ∈ D (P
1
)
max f(x) x ∈ D (P

2
)
D ⊆ R
n
f : D −→ R x ∈ D
x
∗
∈ D
−∞ < f(x
∗
) ≤ f(x) , ∀x ∈ D
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(P
1
)
x
∗
∈ D
f(x
∗
) < f(x), ∀x ∈ D x = x
∗
(P
1
)
(P
1
)
min
x∈D

f(x) min{f(x) | x ∈ D}
(P
1
) x
∗
f(x
∗
) = min{f(x) | x ∈ D}
Agrmi n{f(x) | x ∈ D}
(P
1
) x
∗
x
∗
= agrmin{f(x) | x ∈ D} x
∗
∈ Agrmin{f(x) | x ∈ D}
x
∗
∈ D
(P
1
) ǫ B(x
∗
, ǫ)
x
∗
∈ D
f(x

∗
) ≤ f(x) , ∀x ∈ B( x
∗
, ǫ) ∩ D
x
∗
∈ D
(P
1
) ǫ B(x
∗
, ǫ)
x
∗
∈ D
f(x
∗
) < f(x), ∀x ∈ B(x
∗
, ǫ) ∩ D x = x
∗
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(P
1
)
min{f(x) | x ∈ D} min
x∈D
f(x) f(x) −→ min x ∈ D
(P
2

)
max{f(x) | x ∈ D} max
x∈D
f(x) f(x) −→ max x ∈ D
(P
2
)
ǫ B(x
∗
, ǫ) x
∗
∈ D
f(x
∗
) ≥ f(x) , ∀x ∈ B( x
∗
, ǫ) ∩ D
x
∗
∈ D
(P
2
) ǫ B(x
∗
, ǫ)
x
∗
∈ D
f(x
∗

) > f(x), ∀x ∈ B(x
∗
, ǫ) ∩ D x = x
∗
x
∗
(P
2
)
x
∗
∈ D f(x
∗
) ≥ f(x ), ∀x ∈ D
(P
2
)
x
∗
∈ D f(x
∗
) > f(x), ∀x ∈ D x = x
∗
x
∗
(P
2
)
(P
2

)
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max
x∈D
f(x) max{f(x) | x ∈ D}
(P
1
) Agrma x{f(x) | x ∈ D}
P
2
x
∗
x
∗
= agrmax{f(x) | x ∈ D} x
∗
∈ Agrmax{f(x) | x ∈ D}
(P
1
)
max −f(x) x ∈ D
min{f(x) | x ∈ D} = −max{−f(x) | x ∈ D}.
(P
1
)
(P
2
)
D = R
n

(P
1
)
D ⊂ R
n
(P
1
)
D
D = {x ∈ R
n
| g
i
(x) ≤ 0, i = 1 , 2, ··· , m},
g
i
(x), i = 1, 2, ··· , m A ⊃ D
A = R
n
g
i
(x), i = 1, 2, ··· , m
g
i
(x) ≤ 0, ( i = 1, 2, ··· , m)
g
i
(x) ≥ 0 ⇔ −g
i
(x) ≤ 0

g
i
(x) =



g
i
(x) ≤ 0
−g
i
(x) ≤ 0
(2.1)
D f(x)
(P
1
)
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f : R
n
→ R D ⊂ R
n
min{f(x) | x ∈ D}
x
∗
x
∗
x
∗
f x

∗
(P
1
)
inf f(D)
f D t
0
= inf f(D) t
0
∈ R ∪ {−∞}
f(x) ≥ t
0
, ∀x ∈ D ∃{x
k
} ⊂ D
lim
k→∞
f(x
k
) = t
0
(P
2
)
sup f(D)
f D t
∗
= sup f(D) t
∗
∈ R ∪ {+∞}

f(x) ≤ t
∗
, ∀x ∈ D ∃{x
k
} ⊂ D
lim
k→∞
f(x
k
) = t
∗
f(x) = cosx x ∈ D = R (P
1
)
Argmi n{cos(x) | x ∈ D} = {x = (2k + 1)π, k = 0, ±1, ±2, ···}
min{cos(x) | x ∈ R} = −1
Argma x{cos(x) | x ∈ D} = { x = 2kπ, k = 0, ±1, ±2, ···}
max{cos(x) | x ∈ R} = 1
f(x) = x
1
D =

x ∈ R
2
| x
1
2
+ x
2
2

≤ 4, x
1
2
≥ 1

f
D x = (−2, 0)
T
x = (1,
√
3)
T
x = (1, −
√
3)
T
(P
1
) min
x∈D
f(x) = −2
x = (2, 0)
T
(P
2
)
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x = (−1,
√
3)

T
x = (−1, −
√
3)
T
(P
2
) max
x∈D
f(x) = 2
−2
O
2
x
1
x
2
min f(x) x ∈ R
n
(P
krb
)
f : R
n
→ R
f
R
n
x
∗

∈ R
n
(P
krb
)
▽f(x
∗
) = 0
f R
n
x
∗
∈ R
n
(P
krb
) ▽f(x
∗
) = 0
f
R
n
x
∗
∈ R
n
f R
n
▽f(x
∗

) = 0 ▽
2
f(x
∗
)
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