Báo cáo nghiên cứu khoa học: "Kết hợp phương pháp chiếu và hàm phạt giải bài toán bất đẳng thức biến phân đơn điệu" pptx

(D, F )
D R
n
F : K → R
n
K D
K D K
K
D
D ⊂ R
n
F : R
n
→ R
n
x
∗
∈ D, F (x
∗
), x − x
∗
  0 ; ∀x ∈ D. (V IP (D, F ))
D, F D, F
F f (D, F )
f D
D, F )
F
D
D
K
D, F D

K D K K
K
K
K
D F D D, F
F lim
x∈D;||x||→∞
F (x) − F (x), x − x
||x − x||
= ∞
x ∈ D D, F
F : D → R
n
F D
F (x) − F (y), x − y  0 ; ∀x, y ∈ D.
F D
F (x) − F (y), x − y > 0 ; ∀x, y ∈ D; x = y.
F D α > 0
F (x) − F (y), x − y  α||x − y||
2
; ∀x, y ∈ D.
F D L > 0
||F (x) − F (y)||  L.||x − y|| ; ∀x, y ∈ D.
F D D, F (D, F ) = ∅
∀x
∗
∈ SOL − V IP (D, F ) F (x), x − x
∗
  0; ∀x ∈ D.
x

∗
D, F )
F D D, F
F D D, F
D R
n
x ∈ R
n
d(D, x) = inf
y∈D
||x − y||. d(D, .)
R
n
D y ∈ D d(D, x) = ||y − x||
D
P
D
(x).
D
K
K K
K = {x = (x
1
, x
2
, , x
n
)
T
∈ R

n
|a
i
 x
i
 b
i
; i = 1, 2, , n},
a = (a
1
, a
2
, , a
n
)
T
; b = (b
1
, b
2
, , b
n
)
T
∈ R
n
.
x K
(P
K

(x))
i
=





a
i
, x
i
< a
i
x
i
, x
i
∈ [a
i
; b
i
]
b
i
, x
i
> b
i
.

C I(a
1
, a
2
, , a
n
) ∈ R
n
R
C = {x ∈ R
n
|
n

i=1
(x
i
− a
i
)
2
 R
2
}
b = (b
1
, b
2
, , b
n

)
T
∈ R
n
P
C
(b) C b ∈ C
P
C
(b) ≡ b C C
I C S = {x ∈ R
n
|
n

i=1
(x
i
− a
i
)
2
= R
2
}
∆ = {x ∈ R
n
|x
i
= a

i
+ t(b
i
− a
i
) ; i = 1, 2, , n ; t ∈ R}.
x
i
= a
i
+ t(b
i
− a
i
) S t
2
n

i=1
(b
i
− a
i
)
2
= R
2
t =
R


n

i=1
(b
i
− a
i
)
2
.
P
C
(b)
x
i
= a
i
+ (b
i
− a
i
)
R

n

i=1
(b
i
− a

i
)
2
.
L ⊂ R
n
B = {η
1
, η
2
, , η
k
} b ∈ R
n
a =
k

j=1
c
j
η
j
∈ L ; c
j
f = b −a f, η
j
 = 0
j = 1, 2, , k L
c ∈ L f L
L

||b − c||
2
= b − a + a − c, b − a + a − c
= b − a, b − a + a − c, a − c + 2b − a, a − c
= ||b − a||
2
+ ||a − c||
2
 ||b − a||
2
.
L
i = 1, . . . , k f, η
i
 = 0,
b −
k

j=1
c
j
η
j
, η
i
 = 0,
k

j=1
η

i
, η
j
c
j
= b, η
i
.
A
ij
= η
i
, η
j
 ; D
i
= b, η
i
 ; i = 1, 2, , k ; j = 1, 2, , k
Ac = D , A = (A
ij
),
c = (c
1
, c
2
, , c
k
)
T

; D = (D
1
, D
2
, , D
k
)
T
.
A det(A) = 0
a =
k

j=1
c
j
η
j
c
i
B η
i
, η
j
 = 0 i = j i = j
A c
i
= D
i
= b, η

i

D ⊂ R
n
F : R
n
→ R
n
x
∗
∈ SOL −V IP (D, F ) ⇔ F
nat
D
(x
∗
) = 0, F
nat
D
(x
∗
) ≡ x −P
D
(x −ξF (x))
ξ > 0
D, F F
F
(D, F ) F
α F t
k
α/L

2
L F
L L
2
t
k
x
0
∈ K η > 0
x
k
= P
D
(x
k
− ηF (x
k
)) x
k
∈ SOL − V IP (D, F )
x
k
x
k
= P
D
(x
k
− ηF (x
k

))
x
k+
1
2
= P
D
(x
k
− ηF (x
k
)),
x
k+1
= P
D
(x
k
− ηF (x
k+
1
2
)).
k := k + 1
D ⊂ R
n
F : D → R
n
D D, F L D x
∗

∈
SOL − V IP (D, F ) k ∈ N
||x
k+1
− x
∗
||
2
 ||x
k
− x
∗
||
2
− (1 − η
2
L
2
)||x
k+
1
2
− x
k
||
2
.
F : D → R
n
D D, F L

D 0 < η < 1/L {x
k
}
D, F
x
∗
∈ D
F (x
∗
), x − x
∗
  0 , ∀x ∈ D. (1)
D K D
P : K → R
P (x)  0 ⇔ x ∈ D. (2)
D = {x ∈ R
n
|g
i
(x)  0 , i = 1, 2, , m}, g
i
: R
n
→ R
P
P (x) :=
m

i=1
[max(0, g

i
(x))]
2
. (3)
P P
h
i
(x) = [max(0, g
i
(x))]
2
= (α ◦ g)(x),
α(x) := [max(0, x)]
2
=

x
2
, x > 0
0, x  0.
x > 0 α

(x) = x
2
x < 0 α

(x) = 0 x = 0
lim
∆x→0
+

α(0 + ∆x) − α(0)
∆x
= lim
∆x→0
+
(∆x)
2
∆x
= 0.
α(x) g
i
h
i
P (x)
P (x) = max
1im
(g
i
(x)). (4)
m = 1 P ≡ g
1
g
1
P
t > 0 (K, tF + ∇P )
x
t
∈ K
tF (x
t

) + ∇P (x
t
), x − x
t
  0 , ∀x ∈ K, (1
t
)
K ⊃ D
K
∇P (x
t
) P x
t
S(t) (1
t
)
t
∗
= {t  0 : S(t) ⊂ D}.
F P
S(t) ∩D = ∅ t > t
∗
S(t) ⊂ D 0 < t < t
∗
S(t
∗
) 0 < t
∗
< ∞
S(.) t

∗
t
∗
= ∞ {x
n
} x
n
∈ S(t
n
) t
n
→ t
∗
{x
n
}
P
P
t
0
a = 0; b = ∞
(1
t
k
) x
k
x
k
∈ D a := t
k

t
k+1
=



a + b
2
, b < ∞
2a, b = ∞.
k := k + 1
x
k
/∈ D b := t
k
t
k
:=
a + b
2
k := k + 1
{x
k
}
x
k
∈ D {x
k
}
{x

k
}
{x
k
}
P (x) ≡ 0 x ∈ D x
k
x
k
∈ D
D, F
(1
t
k
)
t
k
k
t
k
→ t
∗
D, F D R
n
F
K D K
D
K D
D, F
K D V IP (K, tF + ∇P ) P

K P (x)  0 x ∈ D.
(K, tF + ∇P )
P
F F
t
:= tF + ∇P
V IP (K, tF + ∇P )
K D P
t
0

k
> 0 
k
→ 0 k → ∞
a = 0 , b = ∞
k
0
λ > 0 , η ∈ (0; 1/L
t
k
) L
t
k
F
k
:= t
k
F + ∇P
j := 0 y

0
∈ K
k
1
||y
j
− P
K
(y
j
− λ.F
k
(y
j
))||  
k
x
k
:= y
j
k
k
||y
j
− P
K
(y
j
− λ.F
k

(y
j
))|| > 
k
k
2
k
2
y
j+
1
2
:= P
K
(y
j
− ηF
k
(y
j
)),
y
j+1
:= P
K
(y
j
− ηF
k
(y

j+
1
2
)),
j := j + 1 k
1
P
K
(x) x K
K
j x
k
(1
t
k
)
x
k
∈ D a := t
k
t
k+1
:=



a + b
2
, b < ∞
2a, b = ∞.

k := k + 1
x
k
/∈ D b := t
k
t
k
:=
a + b
2
k := k + 1

k
→ 0
F K ⊃ D P
K F ∇P K {x
k
}
{x
k
}
x
k
∗
K, t
k
+ ∇P )
y
j
→ x

k
∗
j → +∞ 
k
> 0 j k
k
1
j x
∗
D, F
||x
k
− x
∗
||  ||x
k
− x
k
∗
|| + ||x
k
∗
− x
∗
||.
x
k
= y
j
||x

k
−x
k
∗
||  
k

k
→ 0 x
k
∗
→ x
∗
||x
k
−x
∗
|| → 0
k → +∞
D D = {x ∈ R
n
|g(x)  0} ∩ K ,
K = {x = (x
1
, x
2
, , x
n
) ∈ R
n

|0  x
i
 a
i
, i = 1, 2, , n},
a
i
= 5 +
i
3i − 1
; g(x) :=
1
n
2

i
e
x
i
−
e
5
+ n − 1
n
2
.
g(x) R
n
P (x) ≡ g(x) R
n

P (x) =

 0, x ∈ D
> 0, x ∈ K \ D.
D K P D
∇P (x) =
1
n
2
· (e
x
1
, e
x
2
, , e
x
n
)
T
.
∇P (x) D e
6
/n
2
∇P (x) 1/n
2
tF + ∇P
F
F (x) = H(x) − p(σ

x
)e − p

(σ
x
)x,
e = (1, , 1)
T
∈ R
n
, σ
x
= x, e =

j
x
j
,
H(x) = (α
1
x
1
+ β
1
, . . . , α
n
x
n
+ β
n

)
T
,
α
i
, β
i
p(t) =
γ
t + 1
,
γ
F
F L
n
=

2(α
2
+ 4n
2
γ
2
)
ε = 10
−6
γ\n
γ\n
γ = 10
x

k
P
D = {x = (x
1
, x
2
)
T
∈ R
2
|g
1
(x)  0 , g
2
(x)  0},
g
1
(x) = x
2
− x
1
, g
2
(x) = −x
2
+ x
2
1
− 1 R
2

F
F (x) =

x
1
+ x
2
x
2
+ e
x
2
4
− 1

D x
2
= x
1
+1 x
2
= x
2
1
−1
F
P
P (x
1
, x

2
) = [max(0, x
2
− x
1
− 1)]
2
+ [max(0, −x
2
+ x
2
1
− 1)]
2
=









0, (x
1
, x
2
) ∈ D
(x

2
− x
1
− 1)
2
, (x
1
, x
2
) ∈ (I)
(x
2
− x
1
− 1)
2
+ (−x
2
+ x
2
1
− 1)
2
, (x
1
, x
2
) ∈ (II)
(−x
2

+ x
2
1
− 1)
2
, (x
1
, x
2
) ∈ (III).
D
K = {x = (x
1
; x
2
)| − 1  x
1
 2 ; −1  x
2
 3}.
F K l =
√
4.4 t > 0
l
t
= t.
√
4.4 F K
α =
1

2
.
tF (t > 0) α
t
=
t
2
.
∇P (x
1
, x
2
) =

































0
0

(x
1
, x
2
) ∈ D

−2(x
2
− x
1
− 1)
2(x

2
− x
1
− 1)

(x
1
, x
2
) ∈ (I)

−2(x
2
− x
1
− 1) + 4x
1
(−x
2
+ x
2
1
− 1)
2(x
2
− x
1
− 1) − 2(−x
2
+ x

2
1
− 1)

(x
1
, x
2
) ∈ (II)

4x
1
(−x
2
+ x
2
1
− 1)
−2(−x
2
+ x
2
1
− 1)

(x
1
, x
2
) ∈ (III)

K F
t
:= tF + ∆P
K L
t
= 69 + t
√
4.4
α
t
= t/2
(K, tF +
∆P )
t > 0 η
k
:= η
0 < η < 2 ·
α
t
L
2
t
=
t
(69 + t
√
4.4)
2
.
η

k
= η =
0.9t
(69 + t
√
4.4)
.
η
0 < η <
1
L
t
=
1
69 + t
√
4.4
.
η =
0.9t
69 + t
√
4.4
.
10
−12
, t
0
= 1
t

k
∞
∇P ≡ 0 D P (K, tF + ∆P )
D t
∗
= ∞
t
0
< t
∗
t
0
> 0 S(t
0
) ⊂ D
t
0
F F (x) =

x
1
+ x
2
+ 10
x
2
+ e
x
2
+ 10


.
F D F
10
−10
, t
0
= 1 , x
0
= (1; 1)
x
k
t
k
t
k
→ 0 t
∗
= 0 t
∗
= 0
x
k
t
k
D
{x
k
} x
k

∈ S(t
k
) t
k
→ t
∗
D 10
−8
D
(D, F )
D R
n
F : K → R
n
K ⊃ D D ⊂ R
n
K ⊃ D K
R
n
K