Bài toán tựa cân bằng tổng quát hỗn hợp

x ∈ D
f (x) = min
x∈D
f (x) ,
D X f
D
X, Y
1
, Y
2
, Z
D ⊂ X, K ⊂ Z
S: D × K → 2
D
, T : D × K → 2
K
, P : D → 2
D
, Q : K × D → 2
K
F : K×K ×D×D → 2
Y
1
, G :K ×D×D → 2
Y
2

. (x, y) ∈ D×K
x ∈ S (x, y) ; y ∈ T (x, y) ;
0 ∈ F (y, y, x, t) t ∈ S (x, y) ;
0 ∈ G (y, x, t) t ∈ P (x) , y ∈ Q (x, t)
(x, y) ∈ D × K
x ∈ S (x, y) ; y ∈ T (x, y) ;
0 ∈ F (y, y, x, z) z ∈ S (x, y)
x ∈ D
x ∈ P
1
(x)
0 ∈ F (y, x, t) t ∈ P
2
(x) y ∈ Q
1
(x, t)
X τ X
X
∅ ∈ τ, X ∈ τ
τ τ
τ τ
τ X
X τ
(X, τ) X
X X
X X
x U
x
V

x
⊂ U
x
x U ∈ U
x
V ∈ V
x
V ⊂ U
X
x, y ∈ X, x = y U
x
x U
y
y
U
x
∩ U
y
= ∅
X
X
X
X
Y C
Y C Y tc ∈ C c ∈ C, t ≥ 0
C C Y
C Y clC, intC, convC
C l(C) = C ∩ (−C)
C C
C l(C) = {0}

C
C clC + C\l(C) ⊆ C
C C
{0} Y Y
Y = R
n
= {x = (x
1
, x
2
, , x
n
) |x
j
∈ R, j = 1, 2, , n}
C = R
n
+
= {x = (x
1
, x
2
, , x
n
) ∈ R
n
|x
j
≥ 0, j = 1, 2, , n}
R

n
C = {0} ∪ {x = (x
1
, x
2
, , x
n
) ∈ R
n
|x
1
> 0} C
C = {x = (x
1
, x
2
, , x
n
) ∈ R
n
|x
1
≥ 0} C
l (C) = {x = (0, x
2
, , x
n
) ∈ R
n
} = {0}

C =

(x, y, z) ∈ R
3
|x > 0, y > 0, z > 0

∪

(x, y, z) ∈ R
3
|x ≥ 0, y ≥ 0, z = 0

C
C Y
x, y ∈ Y, x  cy x − y ∈ C
x  y
x  y x − y ∈ C\l (C) x  y x − y ∈ intC
C
Y C
x  y, y  x
x = y
R
2
C = {(x
1
, 0) |x
1
∈ R} x, y ∈ C x ≥ y
R
n

C = R
n
+
x =
(x
1
, x
2
, , x
n
) , y = (y
1
, y
2
, , y
n
) x  y x
i
 y
i
i = 1, 2, , n
x  y x
i
 y
i
i = 1, 2, , n
Ω x = {x
n
}


p
=

x ∈ Ω| ||x|| = (

|x
n
|
p
)
1
/
p
< ∞

, 1 ≤ p < ∞
C 
p
C x, y ∈ 
p
x  y y  x
C Y B ⊆ Y
C C = cone(B) C = {tb|b ∈ B, t ≥ 0}
B c ∈ C, c = 0
b ∈ B, c = tb B C
B C = cone(convB)
R
2
C = {(x, y) | y − 2x = 0, x ≥ 0, y ≥ 0} ∪ {(x, y) | y − 3x = 0, x ≥ 0, y ≥ 0}
B

1
=

1
2
, 1

, (1, 2) , (1, 3)

B
2
= {(1, 2) , (1, 3)}
C
C = R
2
+
C = cone (conv {(1, 0) , (0, 1)})
C = R
2
+
Y
C A Y
x ∈ A A C
y − x ∈ C y ∈ A
A C
IMin(A\C) IMinA
x ∈ A A
C y ∈ A x − y ∈ C\l(C)
A C P Min(A\C)
Min(A\C) MinA

x ∈ A intC = ∅ C = Y A
C x ∈ Min (A\ {0} ∪ intC) x
C
0
= {0} ∪ intC
A C W Min(A\C) W MinA
x ∈ A A
C
∼
C
Y C\l(C)
x ∈ PMin

A/
∼
C

A C
P rMin(A\C) P rMinA
A
C
x ∈ IMinA A ⊂ x + C
x ∈ MinA A ∩ (x − C) ⊂ x + l(C)
x ∈ W MinA A ∩ (x − intC) = ∅
IMinA ⊆ P rMinA ⊆ MinA ⊆ W MinA
R
2
A B
A =


(x, y) ∈ R
2
: x
2
+ y
2
≤ 9, y ≤ 0

∪

(x, y) ∈ R
2
: x ≥ 0, −3 ≤ y ≤ 0

B = A ∪ {(−4, −4)}
C = R
2
+
IMinB = P rMinB = MinB = W MinB = {(−4, −4)}
IMinA = ∅
Pr MinA =

(x, y) ∈ R
2
: x
2
+ y
2
= 9, x < 0, y < 0


MinA = Pr MinA ∪ {(0, −3) , (−3, 0)}
W MinA = MinA ∪

(x, y) ∈ R
2
: x ≥ 0, y = −3

X, Y D X 2
Y
Y
F X 2
Y
X Y F : X → 2
Y
F : X ⇒ Y
x ∈ X F (x) Y
x F (x) x ∈ X
F (x) Y F X Y
F : X → 2
Y
F : X → Y
F
domF = {x ∈ X | F (x) = ∅}
Gr (F ) = {(x, y) ∈ X × Y | y ∈ F (x)}
rgeF = {y ∈ Y | ∃x ∈ X sao cho y ∈ F (x)}
F : X → 2
Y
F
−
1

: Y → 2
X
F
−
1
(y) = {x ∈ X : y ∈ F (x)}
F
F
−
1
(y) = {x ∈ X : y ∈ F (x)}
F
x
n
+ a
1
x
n−1
+ + a
n−1
x + a
n
= 0.
n ∈ N, a
i
∈ R, i = 1, 2, . . . , n a =
(a
1
, , a
n

) F (a)
F : R
n
→ C.
F (a) = ∅ a ∈ R
n
|F (a)| ≤ n, ∀a ∈ R
n
Gr (F ) =

(a, x) ∈ R
n
× C | x
n
+ a
1
x
n−1
+ + a
n−1
x + a
n
= 0

,
domF = R
n
f : X → Y X Y
f x ∈ X V f (x)
U x f (x) ∈ V, ∀x ∈ U f

X X f X
f
−1
(V ) = {x ∈ X : f (x) ∈ V } X
V x f (x) ⊆ V
f (x) ∩ V = ∅
F : X → 2
Y
X Y
F
x ∈ domF V ⊂ Y
F (x) ⊂ V U x F (x) ⊂ V, ∀x ∈ U
F x ∈ domF V
F (x)∩V = ∅ U ⊃ x F (x)∩V = ∅, ∀x ∈ U
F x ∈ X
x
F X x ∈ X.
X = [−1, 1]
F (x) =

{0} x = 0,
[−1, 1] x = 0.
0
F (x) =

{0} x = 0,
[−1, 1] x = 0.
0
X, Y F : X → 2
Y

F Gr(F )
X × Y
F (X) Y F
F
{x
α
} , {y
α
} , x
α
→ x, y
α
→ y, y
α
∈ F(x
α
) y ∈ F (x) F (x)
x ∈ X F
F : D → 2
Y
F
F F
Y F
F : D → 2
Y
F
x ∈ domF y ∈ F (x)
{x
α
} ∈ D, x

α
→ x {y
α
}
α∈∧
, y
α
∈ F (x
α
) sao cho y
α
→ y
∧
F coF
y ∈ D x ∈ (coF )
−1
(y) y ∈ co (F (x)) , y =
n

i=1
α
i
y
i
0 ≤ α
i
≤ 1,
n

i=1

α
i
= 1, y
i
∈ F (x) x ∈ F
−1
(y
i
) i = 1, . . . , n
F
−1
(y
i
) , i = 1, . . . , n U(x) x
U(x) ⊆ F
−1
(y
i
) i = 1, . . . , n y
i
∈ F (z)
z ∈ U(x) i = 1, . . . , n y =
n

i=1
α
i
y
i
∈ (coF ) (z)

z ∈ U(x) U (x) ⊆ (coF )
−1
(y) (coF )
−1
(y)
F : R → 2
R
F (x) = (−∞, −x] F
−1
(y) =
{x : y ∈ (−∞, −x]} = {x : y ≤ −x} = (−∞, −y]
V R ∪
y∈V
F
−1
(y) = {x : F (x) ∩ V = ∅}
b = inf {v : v ∈ V } . (−∞, −b) ⊆ ∪
y∈V
F
−1
(y)
x ∈ (−∞, −b) b < −x
b y ∈ V b < y ≤ −x
x ∈ (−∞, −y] ⊆ ∪
y∈V
F
−1
(y) (−∞, −b) ⊆
∪
y∈V

F
−1
(y) ∪
y∈V
F
−1
(y) F
X Y D, K
X C Y F D
Y
F C− C
x
0
∈ D V 0 Y U
x
0
X
F (x) ⊂ F (x
0
) + V + C,
( F (x
0
) ⊂ F (x) + V − C) x ∈ U ∩ domF.
F C x
0
F C C
x
0
F C C C
D C C C

D x D
C = {0} F
{0}

{0}

F : K × D × D → 2
Y
, C : K × D → 2
Y
F C C (
y, x, z) ∈ domF
V 0 Y U (y, x, z)
F (y, x, z) ⊆ F (y, x, z) + V + C (y, x) ,

F (y, x, z) ⊆ F (y, x, z) + V − C (y, x)

(y, x, z) ∈ U ∩ domF
C
F : K × D × D → 2
Y
, C : K × D → 2
Y
C F C
(y
0
, x
0
, z
0

) ∈ domF F (y
0
, x
0
, z
0
) + C (y
0
, x
0
)
(y
β
, x
β
, z
β
) → (y
0
, x
0
, z
0
) , t
β
∈ F (y
β
, x
β
, z

β
) + C (y
β
, x
β
) , t
β
→
t
0
t
0
∈ F (y
0
, x
0
, z
0
) + C (y
0
, x
0
)
F (y
β
, x
β
, z
β
) →

(y
0
, x
0
, z
0
) , t
β
∈ F (y
β
, x
β
, z
β
)+C (y
β
, x
β
) , t
β
→ t
0
t
0
∈ F (y
0
, x
0
, z
0

)+
C (y
0
, x
0
) F C (y
0
, x
0
, z
0
)
F C (y
0
, x
0
, z
0
) ∈ domF
(y
β
, x
β
, z
β
) → (y
0
, x
0
, z

0
) t
0
∈ F (y
0
, x
0
, z
0
) + C (y
0
, x
0
)
{t
β
} , t
β
∈ F (y
β
, x
β
, z
β
)

t
β
γ


, t
β
γ
− t
0
→
c ∈ C (y
0
, x
0
)

t
β
γ
→ t
0
+ c ∈ t
0
+ C (y
0
, x
0
)

F (y
0
, x
0
, z

0
) (y
β
, x
β
, z
β
) →
(y
0
, x
0
, z
0
) t
0
∈ F (y
0
, x
0
, z
0
)+C (y
0
, x
0
) {t
β
} , t
β

∈ F (y
β
, x
β
, z
β
)

t
β
γ

, t
β
γ
− t
0
→ c ∈ C (y
0
, x
0
) F C
(y
0
, x
0
, z
0
)
F C (y

0
, x
0
, z
0
) ∈ domF (y
β
, x
β
, z
β
) →
(y
0
, x
0
, z
0
) , t
β
∈ F (y
β
, x
β
, z
β
) + C (y
β
, x
β

) , t
β
→ t
0
t
0
/∈
F (y
0
, x
0
, z
0
) + C (y
0
, x
0
) V
0
Y
(t
0
+ V
0
) ∩ (F (y
0
, x
0
, z
0

) + C (y
0
, x
0
)) = ∅.

t
0
+
V
0
/
2

∩

F (y
0
, x
0
, z
0
) + C (y
0
, x
0
) +
V
0
/

2

= ∅.
t
β
→ t
0
V 0 Y β
1
≥ 0
t
β
− t
0
∈ V /2 β ≥ β
1
t
β
∈ t
0
+ V /2 F C
(y
0
, x
0
, z
0
) U (y
0
, x

0
, z
0
)
F (y, x, z) ⊆ F (y
0
, x
0
, z
0
)+V
0
/4+C (y
0
, x
0
) (y, x, z) ∈ U ∩domF
(y
β
, x
β
, z
β
) → (y
0
, x
0
, z
0
) β

2
≥ 0 (y
β
, x
β
, z
β
) ∈ U
t
β
∈ F (y
β
, x
β
, z
β
) + C (y
β
, x
β
) ⊆ F (y
0
, x
0
, z
0
) + C (y
0
, x
0

) + C (y
β
, x
β
) +
V /4. C β
3
C (y
β
, x
β
) ⊆
C (y
0
, x
0
) + V /4 β ≥ β
3
t
β
∈ (t
0
+ V /2) ∩ (F (y
0
, x
0
, z
0
) + C (y
0

, x
0
) + V /2) = ∅ β ≥
max {β
1
, β
2
, β
3
} t
0
∈ F (y
0
, x
0
, z
0
)+C (y
0
, x
0
)
F (y
β
, x
β
, z
β
) →
(y

0
, x
0
, z
0
) , t
β
∈ F (y
β
, x
β
, z
β
) + C (y
β
, x
β
) , t
β
→ t
0
t
0
∈ F (y
0
, x
0
, z
0
) + C (y

0
, x
0
) F C
(y
0
, x
0
, z
0
) V Y
U
β
(y
0
, x
0
, z
0
) (y
β
, x
β
, z
β
) ∈ U
β
F (y
β
, x

β
, z
β
)  F (y
0
, x
0
, z
0
) + V + C (y
0
, x
0
) .
t
β
∈ F (y
β
, x
β
, z
β
) , t
β
/∈ F (y
0
, x
0
, z
0

) + V + C (y
0
, x
0
)
F (D) t
β
→ t
0
t
β
∈ F (y
β
, x
β
, z
β
)+C (y
β
, x
β
)
t
0
∈ F (y
0
, x
0
, z
0

) + C (y
0
, x
0
) t
β
→ t
0
β
0
≥ 0 t
β
− t
0
∈ V β ≥ β
0
t
β
∈ t
0
+ V ⊆
F (y
0
, x
0
, z
0
) + V + C (y
0
, x

0
) , β ≥ β
0
F C (y
0
, x
0
, z
0
) ∈
domF (y
β
, x
β
, z
β
) → (y
0
, x
0
, z
0
) t
0
∈ F (y
0
, x
0
, z
0

)
V Y U (y
0
, x
0
, z
0
)
F (y
0
, x
0
, z
0
) ⊆ F (y, x, z) + V − C (y
0
, x
0
) (y, x, z) ∈ U ∩ domF.
(y
β
, x
β
, z
β
) → (y
0
, x
0
, z

0
) β
0
≥ 0 (y
β
, x
β
, z
β
) ∈
U β ≥ β
0
F (y
0
, x
0
, z
0
) ⊆ F (y
β
, x
β
, z
β
) + V −
C (y
0
, x
0
) β ≥ β

0
. t
0
∈ F (y
0
, x
0
, z
0
) t
0
=
t
β
+ v
β
− c
β
t
β
∈ F (y
β
, x
β
, z
β
) ⊆ F (D),v
β
∈ V, c
β

∈ C (y
0
, x
0
) .
F (D) t
β
γ
→ t
∗
, v
β
γ
→ 0
c
β
γ
=t
β
γ
+ v
β
γ
− t
0
→ t
∗
− t
0
∈ C (y

0
, x
0
) t
β
γ
→ t
∗
∈ t
0
+ C (y
0
, x
0
)
(y
β
, x
β
, z
β
) → (y
0
, x
0
, z
0
) , t
0
∈ F (y

0
, x
0
, z
0
)
t
β
γ
∈ F

y
β
γ
, x
β
γ
, z
β
γ

t
β
γ
→ t
∗
∈ t
0
+ C (y
0

, x
0
)
F (y
0
, x
0
, z
0
) (y
β
, x
β
, z
β
) →
(y
0
, x
0
, z
0
) , t
0
∈ F (y
0
, x
0
, z
0

) + C (y
0
, x
0
) {t
β
} t
β
∈
F (y
β
, x
β
, z
β
)

t
β
γ

sao cho t
β
γ
−t
0
→ c ∈ C (y
0
, x
0

)
F C (y
0
, x
0
, z
0
) V
Y U
β
(y
0
, x
0
, z
0
) (y
β
, x
β
, z
β
) ∈ U
β
F (y
0
, x
0
, z
0

)  F (y
β
, x
β
, z
β
) + V − C (y
0
, x
0
) .
t
β
∈ F (y
0
, x
0
, z
0
) t
β
/∈ (F (y
β
, x
β
, z
β
) + V − C (y
0
, x

0
))
F (y
0
, x
0
, z
0
)
t
β
→ t
0
∈ F (y
0
, x
0
, z
0
) t
0
∈ F (y
0
, x
0
, z
0
) + C (y
0
, x

0
))
(y
β
, x
β
, z
β
) → (y
0
, x
0
, z
0
)

t

β

sao cho t

β
∈
F (y
β
, x
β
, z
β

)

t

β
γ

t

β
γ
− t
0
→ c ∈ C (y
0
, x
0
)
t

β
→ t
∗
∈ t
0
+ C (y
0
, x
0
)

β
1
≥ 0 t
β
∈ t
0
+ V /2, t

β
∈ t
∗
+ V /2 t
0
∈
t

β
+ V /2 − C (y
0
, x
0
) , β ≥ β
1
t
β
∈ t

β
+V /2+V /2−C (y
0

, x
0
) ⊆ F (y
β
, x
β
, z
β
)+V −C (y
0
, x
0
) , β ≥ β
1
X, Y D ⊂ X C
Y f : D → Y C D
x
1
, x
2
∈ D, α ∈ [0, 1]
f (αx
1
+ (1 − α) x
2
) ∈ αf(x
1
) + (1 − α) f (x
2
) − C.

f C D −f C D
Y = R, C = R
+
f
F : D → 2
Y
C Y
F C C
αF (x) + (1 − α) F (y) ⊂ F (αx + (1 − α) y) + C,
( F (αx + (1 − α) y) ⊂ αF (x) + (1 − α) F (y) − C)
x, y ∈ domF, α ∈ [0, 1] .
F C C
αF (x) + (1 − α) F (y) ⊂ F (αx + (1 − α) y) − C,
( F (αx + (1 − α) y) ⊂ αF (x) + (1 − α) F (y) + C)
x, y ∈ domF, α ∈ [0, 1] .
C = {0} {0} {0} F
F
F C C
C C C
C
F D ⊂ X 2
Y
Y
C
C D
x
1
, x
2
∈ D, α ∈ [0, 1]

F (x
1
) ⊆ F (αx
1
+ (1 − α) x
2
) + C,
F (x
2
) ⊆ F (αx
1
+ (1 − α) x
2
) + C.
C D
x
1
, x
2
∈ D, α ∈ [0, 1]
F (αx
1
+ (1 − α) x
2
) ⊆ F (x
1
) − C,
F (αx
1
+ (1 − α) x

2
) ⊆ F (x
2
) − C.
C C
X, Y K
X F : K → 2
Y
F imF Y
x ∈ K
F : K → 2
K
¯x ∈ F (¯x)
X K ⊂ X
F : K → 2
K
x ∈ K, F (x)
y ∈ K, F
−1
(y) K
x ∈ K ¯x ∈ F (¯x)
X K ⊂ X
F : K → 2
K
x ∈ K, x /∈ F (x) F (x)
y ∈ K, F
−1
(y) K
x ∈ K F(¯x) = ∅
F D → 2

X
{t
1
, . . . , t
n
} ⊂ D co {t
1
, , t
n
} ⊆
n

j=1
H(t
j
)
X, Z D ⊆ X, K ⊆ Z, F :
K × D × D → 2
X
, Q : D × D → 2
K
F
{t
1
, . . . , t
n
} ⊂ D x ∈ co {t
1
, , t
n

}
j ∈ {1, . . . , n} 0 ∈ F (y, x, t
j
) y ∈ Q (x, t
j
) .
X, Z, W D ⊆ X, K ⊆
Z, E ⊆ W F : K × D × E → 2
X
, Q : D × E → 2
K
F
{t
1
, . . . , t
n
} ⊂ E
{x
1
, , x
n
} ⊆ D x ∈ co {x
i
1
, , x
i
k
}
t
i

j
∈ {t
i
1
, , t
i
n
} 0 ∈ F

y, x, t
i
j

y ∈ Q

x, t
i
j

.
D
X F : D → 2
X
x ∈ D, F (x) x

∈ D
F (x

) ∩
x∈D

F (x) = ∅
X, Y
1
, Y
2
, Z
D ⊂ X, K ⊂ Z
S: D × K → 2
D
, T : D × K → 2
K
, P : D → 2
D
, Q : K × D → 2
K
F : K×K ×D×D → 2
Y
1
, G :K ×D×D → 2
Y
2
. (
x, y) ∈ D×K
x ∈ S (x, y) ; y ∈ T (x, y) ;
0 ∈ F (y, y, x, t) t ∈ S (x, y) ;
0 ∈ G (y, x, t) t ∈ P (x) , y ∈ Q (x, t)
(x, y) ∈ D × K
x ∈ S (x, y) ; y ∈ T (x, y) ;
0 ∈ F (y, y, x, z) z ∈ S (x, y)