class="bi x0 y0 w1 h1"
class="bi x0 y1 w1 h2"
class="bi x0 y2 w2 h3"
class="bi x0 y3 w2 h4"
class="bi x0 y4 w3 h5"
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
⊂ 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}

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 = 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
}


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, 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


: 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

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

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

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

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

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
) →

)+
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

, 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

∈ 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

) , 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

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

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

, 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
) 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

) + 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

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

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

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

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

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]

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

}
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

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)