Phép vô hướng hóa và điều kiện tối ưu cho bài toán cân bằng véc tơ

. . .
. . .
X f : X → R
X f : X → R
f x ∈ X,
x U x K > 0
(∀x, x

∈ U) |f(x) − f(x

)|  Kx − x

.
f Y ⊂ X f
x ∈ Y
f K Y ⊂ X
x, x

∈ Y
F : X → Y X Y
F x γ > 0
K > 0
F (x

) − F(x


)
Y
 Kx

− x


X
(∀x

, x

∈ x + γB),
B
f x ∈ X
f v(∈ X) x f
◦
(¯x; v)
f
◦
(¯x; v) = lim sup
x→¯x;t↓0
f(x + tv) − f(x)
t
,
x ∈ X, t > 0
f K x
v → f
◦
(x; v)

X
|f
◦
(x; v)| ≤ Kv;
f
◦
(x; v) (x, v); f
◦
(x; .)
K X
f
◦
(x; −v) = (−f)
◦
(x; v).
f : X → R
X X
∗
X
f ¯x ∂f(¯x)
X
∗
∂f(¯x) := {ξ ∈ X
∗
: f
◦
(¯x; u) ≥ ξ, u, ∀u ∈ X}.
f K ¯x
∂f(¯x) = ∅, compact
∗

X
∗
ξ
∗
≤ K (∀ξ ∈ ∂f(¯x)),
.
∗
X
∗
v ∈ X
f
◦
(¯x; v) = max{ξ, v : ξ ∈ ∂f(¯x)}.
a) f
◦
(¯x; .)
X ζ : X → R
f
◦
(¯x; v) ≥ ζ; v (∀v ∈ X)
ζ ∈ ∂f(¯x) ∂f(¯x) = ∅
∂f(¯x) ξ
1
, ξ
2
∈ ∂f(¯x), 0 ≤ α ≤ 1
f
◦
(¯x; u) ≥ ξ
i

, u (∀u ∈ X, i = 1, 2)
⇒ f
◦
(¯x; u) = αf
◦
(¯x; u) + (1 − α)f
◦
(¯x; u)
≥ αξ
1
, u + (1 − α)ξ
2
, u
= αξ
1
+ (1 − α)ξ
2
, u
⇒ αξ
1
+ (1 − α)ξ
2
∈ ∂f(¯x) ⇒ f(¯x) .
∂f(¯x)
∗
ξ ∈ ∂f(¯x), ξ
∗
≤ K ⇒
∂f(¯x) ⊂
¯

B
∗
(0, K)
¯
B
∗
(0, K)
K
¯
B
∗
(0, K)
∗
X
∗
∂f(¯x)
∗
⇒ ∂f(¯x)
∗
b)
max{ξ, v : ξ ∈ ∂f(¯x)} ≤ f
◦
(¯x, v).
v
0
max{ξ, v
0
 : ξ ∈ ∂f(¯x)} ≤ f
◦
(¯x, v

0
).
ζ
ζ, v ≤ f
◦
(¯x, v) (∀v ∈ X),
ζ, v
0
 = f
◦
(¯x, v
0
)
⇒ ζ ∈ ∂f(¯x) ⇒ f
◦
(¯x, v
0
) > ζ, v
0
 = f
◦
(¯x, v
0
).
f ¯x, s ∈ R
∂(sf)(¯x) = s∂f(¯x).
f ¯x sf
¯x
s ≥ 0, (sf)
◦

= sf
◦
⇒ ∂(sf)(¯x) = s∂f(¯x)
s < 0 s = −1
ζ ∈ ∂(−f)(¯x) ⇔ (−f)
◦
(¯x; v) ≥ ζ; v (∀v)
⇔ f
◦
(¯x, −v) ≥ ζ; v (∀v)
⇔ −ζ ∈ ∂f(¯x)
⇔ ζ ∈ −∂f(¯x).
∂(−f)(¯x) = −∂f(¯x)
∂

n

i=1
f
i

(¯x) ⊂
n

i=1
∂f
i
(¯x).
∗
n = 2

(f
1
+ f
2
)
◦
(¯x; v) f
◦
1
(¯x; v) + f
◦
2
(¯x; v).
(f
1
+ f
2
)
◦
(¯x; v) ≤ f
◦
1
(¯x; v) + f
◦
2
(¯x; v).
∂(f
1
+ f
2

)(¯x) ⊂ ∂f
1
(¯x) + f
2
(¯x).
s
i
∈ R, (i = 1, . . . , n),
∂

n

i=1
s
i
f
i

(¯x) ⊂
n

i=1
s
i
∂f
i
(¯x).
¯x.
∂


n

i=1
s
i
f
i

(¯x) ⊂
n

i=1
∂(s
i
∂f
i
)(¯x)
=
n

i=1
s
i
∂f
i
(¯x) .
x, y ∈ X, f [x, y]
u ∈ (x, y)
f(y) − f(x) ∈ ∂f(u), y − x.
g(t) := f(x

t
)
x
t
= x + t(y − x) x y ∈ X
g : [0, 1] → R (0, 1)
∂g(t) ⊂ ∂f(x
t
), y − x.
g (0, 1)
R v = ±1
max{∂g(t)v} ≤ max{∂f(x
t
), y − xv}.
max{∂g(t)v} = g
◦
(t; v) = lim sup
s→t;λ↓0
g(s + λv) − g(s)
λ
= lim sup
s→t;λ↓0
f(x + (s + λv)(y − x)) − f(x + s(y − x))
λ
≤ lim sup
y

→x
t
;λ↓0

f(y

+ λv(y − x)) − f(y

)
λ
= f
◦
(x
t
; v(y − x)) = max∂f(x
t
), v(y − x).
θ(t) = f(x
t
) + t[f(x) − f(y)], (0 ≤ t ≤ 1).
θ(0) = θ(1) = f(x), θ(.) [0, 1]
¯
t ∈ (0, 1)
θ(.)
¯
t
0 ∈ ∂θ(
¯
t)
⇒ 0 ∈ [f(x) − f(y)] + ∂g(
¯
t) ( )
⇒ 0 ∈ [f(x) − f(y)] + ∂f(x
¯

t
), y − x ( )
u = x
¯
t
h : X → R
n
, g : R
n
→ R, f = g ◦ h h = (h
1
, . . . , h
n
).
h
i
¯x, (i = 1, . . . , n), g
h(¯x)
∂f(¯x) ⊂ co{
n

i=1
α
i
ζ
i
: ζ
i
∈ ∂h
i

(¯x), α ∈ ∂g(h(¯x))},
co
∗
.
1) g h(¯x) h
i
¯x α ∈ ∂g(h(¯x))
α
i
≥ 0 f ¯x
2) g h(¯x) n = 1 co
3) g h(¯x) h ¯x f
¯x co
A ⊂ X, A = ∅ v ∈ X A
¯x ∈ A
d
◦
A
(¯x; v) = 0.
T
C
(A, ¯x) A ¯x ∈ A
T
C
(A, ¯x) := {v ∈ X : d
◦
A
(¯x; v) = 0}.
T
C

(A, ¯x)
A ¯x
A ¯x
N
C
(A, ¯x) := {ξ ∈ X
∗
: ξ, v ≤ 0, ∀v ∈ T
C
(A, ¯x)}.
A N
C
(A, ¯x)
A ¯x
K
C
(A, ¯x) := {v ∈ X : ∀ > 0, ∃t ∈ (0, ),
∃ω ∈ v + tB sao cho ¯x + tω ∈ A}.
A ¯x
T
C
(A, ¯x) = K
C
(A, ¯x).
G : X → Y x ∈ X
G

(x) : X → Y h ∈ X
G


(x)(h) = lim
x

→x
λ↓0
1
λ
(G(x

+ λh) − G(x

)),
h G
x
A ⊂ X, g : X → R g
x A x
0 ∈ ∂g(x) + N
C
(A, x).
X, Y G X Y ϕ
Y G x ϕ
G(x) f = ϕ ◦ G x
∂f(x) ⊂ (G

(x))
∗
∂ϕ(G(x)).
f : X → R
epi f := {(x, r) ∈ X × R : f(x) ≤ r}.
f ¯x

T
epif
(¯x, f(¯x)) = epi f
◦
(¯x, .),
(v, r) ∈ T
epif
(¯x, f(¯x)) ⇔ f
◦
(¯x; v) ≤ r.
a) (v, r) ∈ T
epif
(¯x, f(¯x)) {x
i
} → ¯x
lim
i→∞
f(x
i
+ t
i
v) − f(x
i
)
t
i
= f
◦
(¯x; v).
(x

i
, f(x
i
)) ∈ f {(x
i
, f(x
i
))} → (¯x, f(¯x))
{(v
i
, r
i
)} → (v, r)
(x
i
, f(x
i
)) + t
i
(v
i
, r
i
) ∈ f.
f(x
i
) + t
i
r
i

≥ f(x
i
+ t
i
v
i
) ⇒
f(x
i
+ t
i
v
i
) − f(x
i
)
t
i
≤ r
i
i → ∞
f
◦
(¯x; v) ≤ r
⇒ (v, r) ∈ f
◦
(¯x, .).
b) v, δ ≥ 0
(v, f
◦

(¯x; v) + δ) ∈ T
epi f
(¯x, f(¯x)).
{(x
i
, r
i
)} ⊂ f (¯x, f(¯x)) t
i
↓ 0
{(v
i
, s
i
)} → (v, f
◦
(¯x; v) + δ)
(x
i
, r
i
) + t
i
(v
i
, s
i
) ∈ f (∀i),
f(x
i

+ t
i
v
i
) ≤ r
i
+ t
i
s
i
(∀i).
v
i
= v, s
i
= max{f
◦
(¯x; v) + δ,
f(x
i
+ t
i
v) − f(x
i
)
t
i
}.
lim sup
i→∞

f(x
i
+ t
i
v) − f(x
i
)
t
i
≤ f
◦
(¯x; v),
s
i
→ f
◦
(¯x; v) + δ
(v
i
, s
i
) → (v, f
◦
(¯x; v) + δ).
(x
i
, r
i
) ∈ f ⇒ f(x
i

) ≤ r
i
r
i
+ t
i
s
i
≥ r
i
+ [f(x
i
+ t
i
v) − f(x
i
)] ≥ f(x
i
+ t
i
v).
g : X → R
g x v ∈ X
g
−
(x, v) = lim inf
t↓0
g(x + tv) − g(x)
t
.

g x
∂
−
g(x) = {x
∗
∈ X
∗
: g
−
(x, v) ≥ x
∗
, v, v ∈ X}.
L X
∂
−
L
g(x) = {x
∗
∈ X
∗
: g
−
(x, v) ≥ x
∗
, v, v ∈ L}.
Φ : X ⇒ Y Φ(u) u → x
lim sup
u→x
Φ(u) = {y ∈ Y : ∃u
k

→ x, y
k
→ y y
k
∈ Φ(u
k
), ∀k = 1, 2, . . . }.
F X
∂
a
g(x) g x ∈ X
∂
a
g(x) =

L∈F
lim sup
u→x
∂
−
L
g(u).
D x ∈ D
N
a
(D, x) =

λ≥0
λ∂
a

d
D
(x),
d
D
D
d
D
X A ⊂ X, g ϕ
x ∈ X
0 ∈ ∂
a
g(x) g x
∂
a
(g + ϕ)(x) ⊂ ∂
a
g(x) + ∂
a
ϕ(x)
∂g(x) = cl
∗
co(∂
a
g(x)) cl
∗
co
∗
g(x) = λϕ(Ax + y) λ > 0 A
X Y ∂

a
g(x) = λA
∗
∂
a
ϕ(Ax + y)
N
C
(A, x) = cl
∗
coN
a
(A, x) x ∈ A.
X, Y Φ : X ⇒ Y y ∈ Φ(x)
D
∗
a
Φ(x, y) : Y
∗
⇒ X
∗
Φ (x, y)
D
∗
a
Φ(x, y)(y
∗
) = {x
∗
∈ X

∗
: (x
∗
, −y
∗
) ∈ N
a
(gph Φ, (x, y))},
gph Φ Φ Φ
D
∗
a
Φ(x)(y
∗
)
F : X → Y X, Y
f(x) = (g ◦ F)(x) g : Y → R
∂
a
f(x) ⊂

y
∗
∈∂
a
g(F (x))
D
∗
a
F (x)(y

∗
).
f : X → Y x ∈ X
Df(x) : X → Y h ∈ X
Df(x)(h) = lim
λ→0
1
λ
(f(x + λh) − f(x)),
f x
C Y f : X → Y
x, y ∈ X λ ∈ [0, 1]
λf(x) + (1 − λ)f(y) − f(λx + (1 − λ)y) ∈ C.
C Y, A
X f : X → Y A f
x
0
∈ A
f(x) − f(x
0
) − Df(x
0
)(x − x
0
) ∈ C, x ∈ A.
X Y
Y
∗
Y C Y

C
∗
= {y
∗
∈ Y
∗
: y
∗
(y) ≥ 0, ∀y ∈ C}
C
C
∗
C

C

:= {y
∗
∈ Y
∗
: y
∗
(y) > 0, ∀y ∈ C\{0}}.
D Y D
cone(D) = {td : t ≥ 0, d ∈ D}.
D cl(D) D intD
B C C
C = cone(B) 0 /∈ cl(B) C

= ∅ C

B C
C

(B) = {y
∗
∈ C

: t > 0 y
∗
(b) ≥ t, ∀b ∈ B}.
C

(B) = ∅
C

(B) ⊂ C

B C 0 /∈ cl(B)
y
∗
∈ Y
∗
\ {0}
r = inf{y
∗
(b) : b ∈ B} > y
∗
(0) = 0.
V
B

= {y ∈ Y : |y
∗
(y)| < r/2}.
V
B
0 Y V
B
inf{y
∗
(y) : y ∈ B + V
B
} ≥ r/2.
U 0 U ⊂ V
B
, B+U 0 /∈ cl(B +U)
C
U
(B) := cone(U+B) C\{0} ⊂ int C
U
(B).
A X, F : A × A → Y
x ∈ A
F (x, y) /∈ −K \ {0} (∀y ∈ A),
K Y
intC = ∅ x ∈ A
F (x, y) /∈ −int C (∀y ∈ A),
x ∈ A
F (x, A) =

y∈A

F (x, y).
x ∈ A
H ⊂ Y C \ {0} ⊂
intH
F (x, A) ∩ ((−H) \ {0}) = ∅.
x ∈ A
U 0 U ⊂ V
B
cone(F (x, A)) ∩ (−intC
U
(B)) = ∅.
x ∈ A
U 0 U ⊂ V
B
cone(F (x, A)) ∩ (U − B) = ∅.
x ∈ A
V 0 U 0
cone(F (x, A)) ∩ (U − C) ⊂ V.
L(X, Y )
X Y
F (x, y) = (Tx)(y − x), x, y ∈ A,
T A L(X, Y ).
F (x, y) = (Tx)(y −x), x, y ∈ A x ∈ A
x ∈ A
F (x, y) = f(y) − f(x), x, y ∈ A,
f : A → Y
F (x, y) = f(y)− f(x), x, y ∈ A x ∈ A
x ∈ A
Y K ⊂ Y
e ∈ intK

P (y) = inf{t ∈ R : y ∈ te − K}, y ∈ Y
Y
C Y intC = ∅ x ∈ A
P Y
y
2
− y
1
∈ intC P (y
1
) < P (y
2
)