Tính ổn định của bất đẳng thức biến phân minimax (LV01190)

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2



VŨ QUANG HUY


TÍNH ỔN ĐỊNH CỦA BẤT ĐẲNG THỨC
BIẾN PHÂN MINIMAX

Chuyên ngành: Toán giải tích
Mã số: 60 46 01 02

LUẬN VĂN THẠC SĨ TOÁN HỌC

Người hướng dẫn khoa học: PGS. TS. NGUYỄN QUANG HUY



HÀ NỘI, 2014


max
y∈L
min
x∈K
f(x, y),
L, K f : L × K → R

K, L
X Y f : Ω → R
Ω X × Y K × L ⊂ Ω
K, L f
max
y∈L
min
x∈K
f(x, y),
(¯x, ¯y) ∈ K ×L
f (¯x, y) ≤ f (¯x, ¯y) ≤ f (x, ¯y) ∀x ∈ K, ∀y ∈ L.
(¯x, ¯y) ∈ K × L
η = sup
y∈L
inf
x∈K
f(x, y), γ = inf
x∈K
sup
y∈L
f(x, y).
η ≤ γ (¯x, ¯y)
η ≥ f (¯x, ¯y) ≥ γ η = γ = f (¯x, ¯y)
sup
y∈L
inf
x∈K
f(x, y) = inf

x∈K
sup
y∈L
f(x, y).
η = γ = f (¯x, ¯y)
η = γ
(¯x, ¯y) ∈ K × L
F
2
(¯x, ¯y) , y − ¯y ≤ 0 ≤ F
1
(¯x, ¯y) , x − ¯x ∀x ∈ K, ∀y ∈ L,
F
1
(u, v) := ∇
x
f(u, v) F
2
(u, v) := ∇
y
f(u, v)
f(x, y) (u, v) x y
(¯x, ¯y) ∈ K × L
(x, y) ∈ K × L y
t
:= ¯y + t (y − ¯y) = (1 − t)¯y + ty
L t ∈ (0; 1)
y ∈ L
F
2

(¯x, ¯y) , y − ¯y = ∇
y
f (¯x, ¯y) (y − ¯y) = lim
t↓0
f (¯x, y
t
) − f (¯x, ¯y)
t
≤ 0,

(x, y) ∈ K × L, f(·, y)
K f(x, ·) L
(u, u

∈ K, ∇
x
f(u, y), u

− u ≥ 0) ⇒ f (u

, y) − f(u, y) ≥ 0
(v, v

∈ K, ∇
y
f(x, v), v

− v ≤ 0) ⇒ f (x, v

) − f(x, v) ≤ 0.

(¯x, ¯y) ∈ K × L F
1
(u, v) :=
∇
x
f(u, v) F
2
(u, v) := ∇
y
f(u, v) (¯x, ¯y) ∈ K × L
f(·, y) K f(x, ·)
L (x, y) ∈ K × L (¯x, ¯y) ∈ K × L
(¯x, ¯y) ∈ K × L
x ∈ K
∇
x
f(¯x, ¯y), x − ¯x = ∇
x
f(¯x, ¯y)(x − ¯x) = F
1
(¯x, ¯y), x − ¯x ≥ 0.
f(·.¯y) f(x, ¯y) −f(¯x, ¯y) ≥ 0

X, Y
X
∗
Y
∗
K ⊂ X, L ⊂ Y
F

1
: K ×L → X
∗
, F
2
: K ×L → Y
∗
{K, L, F
1
, F
2
} (¯x, ¯y) ∈ K × L
F
2
(¯x, ¯y) , y − ¯y ≤ 0 ≤ F
1
(¯x, ¯y) , x − ¯x ∀x ∈ K, ∀y ∈ L.
S S ⊂ (MV I) F
1
= ∇
x
f F
2
= ∇
y
f
f(·, y) K f(x, ·) L
(x, y) ∈ K ×L S = (MV I)
f(·, y) K K
f ((1 −t)x + tu, y) ≤ {f(x, y), f(u, y)} ∀x, u ∈ K; ∀t ∈ (0; 1) .

f(·, y) K
f ((1 −t)x + tu, y) < {f(x, y), f(u, y)}
∀x, u ∈ K; f(x, y) = f(u, y); ∀t ∈ (0; 1) .
G(x, y) = (F
1
(x, y), −F
2
(x, y))
(x, y) ∈ K × L G(x, y) ∈ X
∗
× Y
∗
(x, y) ∈
K × L
G(x, y), (u, v) = F
1
(x, y), u − F
2
(x, y), v.
X × Y
(x, y) = x + y.
K × L ⊂ X × Y G : K × L → X
∗
× Y
∗
(¯x, ¯y) ∈ K × L
G (¯x, ¯y) , (x, y) −(¯x, ¯y) ≥ 0, ∀(x, y) ∈ K × L.
(¯x, ¯y) ∈ (MV I)
(¯x, ¯y)
(¯x, ¯y) ∈ (MV I) (x, y) ∈ K ×L

F
2
(¯x, ¯y) , y − ¯y ≤ 0 ≤ F
1
(¯x, ¯y) , x − ¯x.
G(¯x, ¯y), (x, y) −(¯x, ¯y) ≥ 0 (x, y) ∈
K × L (¯x, ¯y)
G(¯x, ¯y), (x, y) −(¯x, ¯y) ≥ 0 ∀(x, y) ∈ K × L.
x = ¯x F
2
(¯x, ¯y), y − ¯y ≤
0 y ∈ L y = ¯y 0 ≤ F
1
(¯x, ¯y), x − ¯x
x ∈ K (¯x, ¯y) ∈ (MV I) 
G = (F
1
, −F
2
) : K ×L → X
∗
×Y
∗
(x
0
, y
0
) ∈ K × L
lim
(x,y)→∞

(x,y)∈K×L
G (x, y) − G (x
0
, y
0
) , (x, y) − (x
0
, y
0
)
x − x
0
 + y −y
0

= +∞.
G(x, y) − G(u, v), (x − u, y −v) ≥ 0 ∀(x, y), (u, v) ∈ K × L.
G(x, y) − G(u, v), (x − u, y −v) > 0
∀(x, y), (u, v) ∈ K × L, (x, y) = (u, v).
((x, y), (u, v) ∈ K × L, G(u, v), (x − u, y −v) ≥ 0)
⇒ G(x, y), (x − u, y −v) ≥ 0.
((x, y), (u, v) ∈ K × L, (x, y) = (u, v), G(u, v), (x − u, y −v) ≥ 0)
⇒ G(x, y), (x − u, y −v) > 0.
α > 0
G(x, y) − G(u, v), (x − u, y −v) ≥ α

x − u
2
+ y −v
2


∀(x, y), (u, v) ∈ K × L.
⇒
⇒ ⇒ ⇒
⇒ G = (F
1
, −F
2
)
(x
0
, y
0
) ∈ K × L
lim
(x,y)→∞
(x,y)∈K×L
F
1
(x, y) − F
1
(x
0
, y
0
) , x − x
0
 − F
2
(x, y) − F

2
(x
0
, y
0
) , y −y
0

x − x
0
 + y −y
0

= +∞.
F
1
(x, y) − F
1
(u, v) , x − u − F
2
(x, y) − F
2
(u, v) , y − v ≥ 0
∀(x, y), (u, v) ∈ K × L.
F
1
(x, y) − F
1
(u, v) , x − u − F
2

(x, y) − F
2
(u, v) , y − v > 0
∀(x, y), (u, v) ∈ K × L, (x, y) = (u, v).
((x, y), (u, v) ∈ K × L, F
1
(u, v) , x − u − F
2
(u, v) , y − v ≥ 0)
⇒ F
1
(x, y) , x − u − F
2
(x, y) , y − v ≥ 0.
((x, y), (u, v) ∈ K × L, (x, y) = (u, v),
F
1
(u, v) , x − u − F
2
(u, v) , y − v ≥ 0)
⇒ F
1
(x, y) , x − u − F
2
(x, y) , y − v > 0.
α > 0
F
1
(x, y) − F
1

(u, v) , x − u − F
2
(x, y) − F
2
(u, v) , y − v
≥ α

x − u
2
+ y −v
2

∀(x, y), (u, v) ∈ K × L.
X = R
n
, Y = R
m
f(x, y) = x
T
By
B n × m
T
F
1
(x, y) = ∇
x
f(x, y) = By,
F
2
(x, y) = ∇

y
f(x, y) = B
T
x,
G(x, y) − G(u, v), (x − u, y −v)
= By −Bv, x − u +

−B
T
x + B
T
u, y −v

= B(y −v), x −u − x − u, B(y −v) = 0.
X = R
n
, Y = R
m
f(x, y) =
1
2
x
T
Ax + x
T
By −
1
2
y
T

Cy + a
T
x + b
T
y
A ∈ R
n×n
C ∈ R
m×m
B ∈ R
n×m
a ∈ R
n
, b ∈ R
m
F
1
(x, y) = ∇
x
f(x, y) = Ax + By + a,
F
2
(x, y) = ∇
y
f(x, y) = B
T
x − Cy + b,
G(x, y) = (Ax + By + a, −B
T
x + Cy −b).

G(x, y) − G(u, v), (x − u, y −v)
= A(x − u) + B(y − v), x − u +

−B
T
(x − u) + C(y −v), y − v

= A(x − u), x −u + C(y −v), y − v + B(y −v), x − u
− (x −u), B(y −v)
= A(x − u), x −u + C(y −v), y − v.
u
T
Au > 0 u ∈
( K)\{0} v
T
Cv > 0 v ∈ ( L)\{0} M
M
u
T
Au ≥ 0 u ∈ K v
T
Cv ≥ 0
v ∈ L
u
T
Au ≥ 0 u ∈ K v
T
Cv ≥ 0
v ∈ L f(·, y) y ∈ L
f(x, ·) x ∈ K

f(x, y) K × L
X = R
n
, Y = R
m
X
∗
Y
∗
R
n
R
m
x
∗
∈ X
∗
x ∈ X
x
∗
, x R
n
K ⊂ R
n
F : K → R
n
¯x ∈ K F (¯x) , x − ¯x ≥ 0 ∀x ∈ K
K ⊂ R
n
, L ∈ R

m
F
1
: K × L → R
n
, F
2
: K × L → R
m
G : K×L → R
n
×R
m
K ×L ⊂ R
n
×R
m
G
(¯x, ¯y) ∈ K ×L (¯x, ¯y) ∈ (MV I) 
L
K ⊂ R
n
L ⊂ R
m
f(·, y)
K f(x, ·) L (x, y) ∈ K × L
f(·, y) K f(x, ·) L
(x, y) ∈ K × L
F
1

(u, v) = ∇
x
f(u, v) F
2
(u, v) = ∇
y
f(u, v)
(u, v) ∈ K ×L (¯x, ¯y) ∈ K ×L

K F
K ⊂ R
n
, L ∈ R
m
F
1
: K × L → R
n
, F
2
: K × L → R
m
G = (F
1
, −F
2
) :
K × L → R
n
× R

m
(x
0
, y
0
) ∈ K × L
(¯x, ¯y) ∈ K × L
(¯x, ¯y) ∈ (MV I) 
L =
K = R F
1
(x, y) = a, F
2
(x, y) = b a = 0 b ∈ R
K L
K ⊂ R
n
L ⊂ R
m
f(·, y) K
f(x, ·) L (x, y) ∈ K × L
(x
0
, y
0
) ∈ K × L F
1
(u, v) = ∇
x
f(u, v) F

2
(u, v) =
∇
y
f(u, v) f(·, y) K
f(x, ·) L (x, y) ∈ K × L
(x
0
, y
0
) ∈ K × L F
1
(u, v) = ∇
x
f(u, v)
F
2
(u, v) = ∇
y
f(u, v)
(x
0
, y
0
) ∈ K × L
(¯x, ¯y) ∈ K × L

(x
0
, y

0
) ∈ K × L
L = K = R f(x, y) = ax + by
a = 0 b ∈ R
X, Y
X × Y (x, y) =
x+ y X ×Y
(X × Y )
∗
≡ X
∗
×Y
∗
(x
∗
, y
∗
) ∈ X
∗
×Y
∗
(x, y) ∈ X ×Y
(x
∗
, y
∗
), (x, y) = x
∗
, x+ y
∗

, y
(x
∗
, y
∗
) = {x
∗
, y
∗
}
K ⊂ X
F : K → X
∗
M ⊂ X
K ∩ M = ∅ F : K ∩ M → X
∗
K ∩ M X
∗
F
K x, u ∈ K F (u), x − u ≥ 0
F (x), x − u ≥ 0
x
x ∈ K, F (x), u −x ≥ 0, ∀u ∈ K
x ∈ K, F (u), u −x ≥ 0, ∀u ∈ K.
K
x
0
∈ K
lim
x→∞

x∈K
F (x) − F (x
0
) , x − x
0

x − x
0

= +∞,
0 ∈ K
˜
K = K − x
0
K ⊂ X, L ⊂ Y
F
1
: K × L → X
∗
, F
2
: K × L → Y
∗
F
1
F
2
(¯x, ¯y)
F
1

(u), u − ¯x − F
2
(v), v − ¯y ≥ 0, ∀(u, v) ∈ K × L.
K L
G = (F
1
, −F
2
)
G : K × L → X
∗
× Y
∗
F
1
F
2

K ⊂ X
L ⊂ Y
((x, y), (u, v) ∈ K × L, ∇
x
f(u, v), x − u − ∇
y
f(u, v), y − v ≥ 0)
⇒ ∇
x
f(x, y), x − u − ∇
y
f(x, y), y − v ≥ 0,

(¯x, ¯y)
∇
x
f(u, v), u − ¯x − ∇
y
f(u, v), v − ¯y ≥ 0, ∀(u, v) ∈ K × L.
K L
(x
0
, y
0
) ∈ K × L
F
1
(u, v) = ∇
x
f(u, v) F
2
(u, v) = ∇
y
f(u, v)
f(·, y) K f(x, ·)
L (x, y) ∈ K × L
F
1
(x, y) = ∇
x
f(x, y), F
2
(x, y) = ∇

y
f(x, y)
G(x, y) = (F
1
(x, y), −F
2
(x, y)) ,

X, Y
X
∗
Y
∗
X Y
x
∗
∈ X
∗
x ∈ X x
∗
, x
X y
∗
∈ Y
∗
y ∈ Y
y
∗
, y Y (x, y), (u, v) = x, u+ y, v
(x, y), (u, v) ∈ X × Y X × Y