Luận văn: Dưới vi phân của hàm lồi và một số ứng dụng trong tối ưu doc

R
n
R
n
+
R
n
R (R = R
1
)
R (R = R ∪ {−∞, +∞})
N
2
R
n
R
n
x, y ∈ R
n
x
i
x
T
x, y = x
T
y = xy :=

n
j=1

x
j
y
j
||x|| =


n
j=1
x
2
j
[x, y]
(x, y)
A
A A
coA A
aff A A
intA A
ri A A
f
f f
dom f f
f
∗
f
epi f f
∂f(x) f
∂


f(x)  f
f(x) f

(x) f
f

(x, d) d f
R R
n
R
n
{x ∈ R
n
| x = αa + βb , α  0 , β  0 , α + β = 1}.
C ⊆ R
n
C
C ∀x, y ∈ C, λ ∈ [0, 1] =⇒ λx + (1 −λ)y ∈ C.
C = R
2
+
C = [−2; 3)
C ≡ oxy R
3
C = (−2; 0) ∪ (0; 3)
C = {(x, y) ∈ R
2
| xy = 0}
x

1
, , x
k
x =
k

j=1
λ
j
x
j
, λ
j
 0 , ∀j = 1, , k ,
k

j=1
λ
j
= 1.
R
n
{x ∈ R
n
| a
T
x = α},
a ∈ R
n
α ∈ R

{x | a
T
x  α},
a = 0 α ∈ R
C ⊆ R
n
x ∈ C
N
C
(x) := {ω | ω, y − x  0 , ∀y ∈ C},
C
N
C
(x)
R
2
C = R
2
+
N
C
(0) = {ω | ω, y − 0  0 , ∀y ∈ C}
= {ω |
2

i=1
ω
i
y
i

 0}
= {ω | ω
i
 0}.
a ∈ C C
C aff C
C ri C
ri C := {a ∈ C | ∃B : (a + B) ∩ aff C ⊂ C},
B
ri C := {a ∈ aff C | ∃B : (a + B) ∩aff C ⊂ C}.
C C C \ri C
C
C ⊆ R
n
x ∈ ri C
y ∈ C
ri C 0  λ < 1 (1 −λ) ri C + λC ⊂ ri C
R
n
x ∈ R
n
{x ∈ R
n
| x = αa + βb , α , β ∈ R , α + β = 1}.
C
∀x, y ∈ C , ∀λ ∈ R =⇒ λx + (1 −λ)y ∈ C.
C = R
2
E
E E coE

E E
E coE
E E
E aff E
E ⊆ R
n
E U(a)
U(a) ⊂ E
E intE B
intE = {x | ∃r > 0 : x + rB ⊂ E}.
E
E E
E E E
E E
E E
R
n
E E
F ⊂ C C
F ∀x, y ∈ C , tx + (1 − t)y ∈ F , 0 < t < 1 =⇒ [x, y] ⊂ F.
C := {(x, y, z) ∈ R
3
| x, y, z ∈ [0, 1]}
F
1
:= {(x, y, z) ∈ R
3
| x, y ∈ [0, 1], z = 0} C
F
2

:= {(x, y, z) ∈ R
3
| y ∈ [0, 1], x = 1, z = 0}
C
x
0
∈ C a
T
x = α C
x
0
a
T
x
0
= α , a
T
x  α ∀x ∈ C.
C x
0
∈ C x
0
C a
T
x  α
C x
0
C D
a
T

x = α C D
a
T
x  α  a
T
y , ∀x ∈ C , ∀y ∈ D.
a
T
x = α C D
a
T
x < α < a
T
y , ∀x ∈ C , ∀y ∈ D.
a
T
x = α C D
Sup
x∈C
a
T
x < α < inf
y∈D
a
T
y.
C = {(x, y) ∈ R
2
| x
2

+ y
2
 1},
D = {(x, y) ∈ R
2
| − 1  x  1, 1  y  3}.
C D
C, D (0, 1)(x, y) = 1
(0, 1)(x, y)  1  (0, 1)(x

, y

) ∀(x, y) ∈ C, ∀(x

, y

) ∈ D.
y  1  y

∀(x, y) ∈ C, ∀(x

, y

) ∈ D.
C, D
(a
1
, a
2
)(x, y) = α

(a
1
, a
2
)(x, y) < α < (a
1
, a
2
)(x

, y

) ∀(x, y) ∈ C, ∀(x

, y

) ∈ D.
C = {(x, y) ∈ R
2
| x  0, y = 0},
D = {(x, y) ∈ R
2
| y 
1
x
, y > 0, x > 0}.
C D
C, D (0, 1)(x, y) = 0
(0, 1)(x, y) = 0  (0, 1)(x


, y

) ∀(x, y) ∈ C, ∀(x

, y

) ∈ D.
y = 0  y

∀(x, y) ∈ C, ∀(x

, y

) ∈ D.
C, D
Sup
(x,y)∈C
(0, 1)(x, y) = 0,
inf
(x

,y

)∈D
(0, 1)(x

, y

) = 0.
C D R

n
C ∩ D = ∅
C D
C ⊂ R
n
x
0
∈ C
t ∈ R
n
, t = 0
t, x  t, x
0
 ∀x ∈ C.
C D C ∩ D = ∅
C ⊂ R
n
0 ∈ C
t ∈ R
n
, t = 0 α > 0
t, x  α > 0 , ∀x ∈ C.
C ⊆ R
n
f : C −→ R ∪ {−∞, +∞}
dom f := {x ∈ C | f (x) < +∞} dom f
f
epi f := {(x, µ) ∈ C × R | f(x)  µ} epi f
f
f(x) = +∞ x ∈ C f

dom f := {x ∈ R
n
| f(x) < +∞}.
epi f := {(x, µ) ∈ R
n
× R | f(x)  µ}.
∅ = C ⊆ R
n
f : C −→ R ∪{−∞, +∞}
f C epi f R
n+1
f : R
n
−→ R ∪ {+∞}
f : R
n
−→ R ∪ {+∞} C
f[λx + (1 −λ)y]  λf(x) + (1 −λ)f(y) , ∀x, y ∈ C , ∀λ ∈ (0, 1)
f : R
n
−→ R ∪ {+∞} C
f[λx + (1 −λ)y] < λf(x) + (1 −λ)f(y) , ∀x, y ∈ C , ∀λ ∈ (0, 1)
f : R
n
−→ R ∪{+∞} C η > 0
f[λx + (1 −λ)y]  λf(x) + (1 −λ)f(y) −
1
2
ηλ(1 − λ)||x − y||
2

,
∀x, y ∈ C , ∀λ ∈ (0, 1).
f C −f C
f(x) = a
T
x + α, a ∈ R
n
, α ∈ R
∀x, y ∈ R
n
, ∀λ ∈ (0, 1)
f[λx + (1 −λ)y] = a
T
[λx + (1 − λ)y] + α
= λa
T
x + (1 − λ)a
T
y + α
= λa
T
x + λα + (1 − λ)a
T
y + (1 − λ)α
= λ(a
T
x + α) + (1 − λ)(a
T
y + α)
= λf(x) + (1 − λ)f(y).

f R
n
∀x, y ∈ R
n
, ∀λ ∈ (0, 1)
−f[λx + (1 −λ)y] = −a
T
[λx + (1 − λ)y] −α
= −λa
T
x −(1 − λ)a
T
y −α
= −λa
T
x −λα − (1 − λ)a
T
y −(1 − λ)α
= −λ(a
T
x + α) − (1 − λ)(a
T
y + α)
= −λf(x) −(1 − λ)f(y).
−f R
n
f R
n
C = ∅
δ

C
(x) :=

0 x ∈ C,
+∞ x ∈ C.
δ
C
C
∀x, y ∈ C, ∀λ ∈ (0, 1) δ
C
(x) = 0 , δ
C
(y) = 0
C λx + (1 − λ)y ∈ C
δ
C
[λx + (1 −λ)y] = 0 = λδ
C
(x) + (1 − λ)δ
C
(y)
∀x ∈ C, ∀y ∈ C, ∀λ ∈ (0, 1)
δ
C
(x) = 0 , δ
C
(y) = +∞ , δ
C
[λx + (1 − λ)y]  +∞
δ

C
[λx + (1 −λ)y]  λδ
C
(x) + (1 − λ)δ
C
(y)
∀x, y ∈ C, ∀λ ∈ (0, 1)
δ
C
(x) = +∞ , δ
C
(y) = +∞ , δ
C
[λx + (1 − λ)y]  +∞
δ
C
[λx + (1 −λ)y]  λδ
C
(x) + (1 − λ)δ
C
(y)
δ
C
R
n
S
C
(y) := Sup
x∈C
y, x S

C
C
∀x, y ∈ C, ∀λ ∈ (0, 1)
S
C
[λx + (1 − λ)y] = Sup
z∈C
λx + (1 − λ)y, z
= Sup
z∈C
{λx, z + (1 − λ)y, z}
 Sup
z∈C
λx, z + Sup
z∈C
(1 −λ)y, z
= λ Sup
z∈C
x, z + (1 − λ) Sup
z∈C
y, z
= λS
C
(x) + (1 − λ)S
C
(y).
S
C
C
f : R

n
−→ R ∪ {+∞}
C ⊆ R
n
η
η f C λ ∈ (0, 1)
x, y ∈ C
f[(1 −λ)x + λy]  (1 −λ)f(x) + λf(y) −
1
2
ηλ(1 − λ)||x − y||
2
.
η = 0 f C
f C η > 0 f C η
f : R
n
−→ R ∪{+∞}
dom f = ∅ f (x) > −∞
f : R
n
−→ R ∪ {+∞} epi f
R
n+1
f C
f
f
e
(x) =


f(x) x ∈ C,
+∞ x ∈ C.
f
e
(x) = f(x) x ∈ C f
e
R
n
f
e
f f
e
f
f R
n
dom f dom f
R
n
epi f
dom f = {x|∃µ ∈ R : (x, µ) ∈ epi f}.
f : R
n
−→ R ∪ {+∞}
f R
n
f(λx) = λf(x) ∀x ∈ R
n
, ∀λ > 0.
f f(x + y)  f(x) + f(y) ∀x, y
f f

f(x) = x
∀x ∈ R
n
, ∀λ > 0 f(λx) = λx = |λ|.x = λx = λf(x)
∀x, y ∈ R
n
f(x + y) = x + y  x + y = f(x) + f(y)
f : R
n
−→ R ∪ {+∞}
R
n
f : E −→ R ∪{−∞, +∞}
f x ∈ E
{x
k
} ⊂ E, x
k
→ x
lim inf f (x
k
)  f(x).
f x ∈ E −f
x ∈ E f x ∈ E
{x
k
} ⊂ E, x
k
→ x
lim sup f(x

k
)  f(x).
f x ∈ E
x ∈ E
f E
E
f E
E
f E
E
f g R
n
g f epi g = epi f f
f epi f = epi f
f epi f = epi f
{f
α
}
α∈I
R
n
E ⊆ R
n
coE V
α∈I
f
α
(V
α∈I
f

α
)(x) := Sup
α∈I
f
α
(x)
x ∈ coE
{f
α
}
α∈I
R
n
E ⊆ R
n
coE
D ⊆ R
n
f
1
, , f
m
R
n
x ∈ D, f
i
(x) <= 0, i ∈ I
<=
< 
f

1
, , f
m
D = ∅
k ×n b ∈ ri A(D)
x ∈ D, Ax = b, f
i
(x) < 0 i = 1, , m
t ∈ R
k
λ
i
 0, i = 1, , m

m
i=1
λ
i
= 1
t, Ax − b+
m

i=1
λ
i
f
i
(x)  0 ∀x ∈ D.
f : R
n

−→ [−∞, +∞]
f
∗
(x
∗
) := Sup{x
∗
, x − f(x) | x ∈ R
n
}
f
−∞ f ≡ +∞ f
∗
≡ −∞
f −∞ f
∗
≡ +∞
+∞
−∞
f ≡ +∞ .
δ
C
(x) =

0 x ∈ C,
+∞ x ∈ C.
δ
∗
C
(x

∗
) := Sup
x∈R
n
{x
∗
, x − δ
C
(x)}
= Sup
x∈C
{x
∗
, x − δ
C
(x)}
= Sup
x∈C
{x
∗
, x − 0}
= Sup
x∈C
x
∗
, x
= S
C
(x
∗

).
f f
∗
f
∗
(x
∗
)  x
∗
, x − f(x) ∀x, ∀x
∗
.
f
∗∗
(x) := (f
∗
)
∗
(x) = Sup{x, s −f
∗
(s) | s ∈ R
n
}.
f ≡ +∞ f
epi f
∗∗
= co(epi f).
f ≡ f
∗∗
f

l f R
n
l R
n
l(x)  f(x) ∀x ∈ R
n
.
f : R
n
−→ R ∪{+∞}
y = 0
x
0
x
x
0
y x = x
0
+ λy λ ∈ R
ξ(λ) = f (x
0
+ λy) ξ R f R
n
f : R
n
−→ R ∪ {+∞} x
0
∈ R
n
f(x

0
) < +∞
y ∈ R
n
lim
λ→0
f(x
0
+λy)−f(x
0
)
λ
f y x
0
f

(x
0
, y)
f
f(x) =





0 x < 0,
1 x = 0,
+∞ x > 0.
dom f = (−∞; 0] ⇒ dom f = ∅

f(x) > −∞, ∀x f
f

(0, −1) = lim
λ→0
f(0+λ(−1))−f(0)
λ
= lim
λ→0
0−1
λ
= −∞
f

(0, 0) = lim
λ→0
f(0+λ0)−f(0)
λ
= lim
λ→0
1−1
λ
= 0
f

(0, 1) = lim
λ→0
f(0+λ1)−f(0)
λ
= lim

λ→0
∞−1
λ
= +∞
f

(0, .)
f : R
n
−→ R ∪ {+∞} x ∈ dom f
y ∈ R
n
ϕ (0; +∞)
ϕ(λ) :=
f(x + λy) − f(x)
λ
,
f

(x, y) y ∈ R
n
f

(x, y) := inf
λ>0
f(x + λy) − f(x)
λ
.
f


(x, .)
f

(x, .) > −∞ f

(x, .) R
n
R
n
−f

(x, −y)  f

(x, y) ∀y ∈ R
n
f

(x, .) F x ∈ ri(dom f)
F dom f
ϕ
(0; +∞)
h : R −→ R ∪ {+∞}
h(λ) = f (x + λ.y) −f (x).
h(0) = 0
0 < λ

 λ f h
−∞
h(λ


) = h[
λ

λ
λ + (1 −
λ

λ
)0]

λ

λ
h(λ) + (1 −
λ

λ
)h(0)
=
λ

λ
h(λ).
ϕ(λ) =
f(x+λy)−f(x)
λ
=
h(λ)
λ
ϕ(λ


)  ϕ(λ)
ϕ (0; +∞)
f

(x, y) = lim
λ→0
ϕ(λ)
lim
λ→0
ϕ(λ) = inf
λ>0
ϕ(λ) = inf
λ>0
f(x + λ.y) − f(x)
λ
.
f

(x, 0) = lim
λ→0
f(x + λ0) −f (x)
λ
= 0.
t > 0
f

(x, ty) = lim
λ→0
f(x + λty) − f(x)

λ
.
λ

= λt
f

(x, ty) = t lim
λ→0
f(x + λ

y) − f(x)
λ

= tf

(x, y).
f

(x, .)
f

(x, .) > −∞ u v
f

(x, u + v) = inf
λ>0
f[x +
λ
2

(u + v)] − f(x)
λ
2
= inf
λ>0
f[(
x
2
+
λ
2
u) + (
x
2
+
λ
2
v)] −
1
2
f(x) −
1
2
f(x)
λ
2
.
f −∞
f[(
x

2
+
λ
2
u) + (
x
2
+
λ
2
v)] −
1
2
f(x) −
1
2
f(x)

1
2
[f(x + λu) −f (x)] +
1
2
[f(x + λv) − f(x)].
f

(x, u + v)  inf
λ>0
f(x + λu)
λ

+ inf
λ>0
f(x + λv)
λ
= f

(x, u) + f

(x, v).
f

(x, u) + f

(x, v) f

(x, .) > −∞
f

(x, .) f

(x, .)
R
n
f

(x, .) > −∞, f

(x, 0) = 0 f

(x, .) R

n
f

(x, 0) = 0
0 = f

(x, 0) = f

(x, y −y)  f

(x, y) + f

(x, −y) ∀y ∈ R
n
.
−f

(x, −y)  f

(x, y) y ∈ R
n
x ∈ ri(dom f) f

(x, .) F
f

(x, .) > −∞ f

(x, y) < +∞
y ∈ F

x ∈ ri(dom f) ∀y ∈ F , x + λ.y ∈ dom f ∀λ > 0
f

(x, y) = inf
λ>0
f(x+λ.y)−f(x)
λ
< +∞
f

(x, y) y ∈ F
x ∈ ri(dom f)
y ∈ F {λ
k
}
x + λ
k
.y ∈ dom f
f(x + λ
k
.y) − f(x) = +∞ .
f

(x, y) = +∞ x ∈ ri(dom f)
f : R
n
−→ R ∪ {+∞} x
∗
∈ R
n

f x
x
∗
, z − x + f(x)  f(z) ∀z.
f x ∂f(x) ∂f(x)
∅ R
n
∂f(x) = ∅ f
x
x
∗
∈ ∂f(x)
∂f(x)
∂f(x)
dom(∂f) := {x|∂f (x) = ∅}
f(x) = x, x ∈ R
n
x = 0
∂f(0) = {x
∗
|x
∗
, x  x, ∀x}.
f(x)
lim
x→0
f(x) −f(0) −x
∗
, x − 0
x −0

= lim
x→0
x −x
∗
, x
x
= 1 = 0.
f(x) x = 0
f(x) = δ
C
(x) :=

0 x ∈ C,
+∞ x ∈ C.
C ∅
x
0
∈ C
∂f(x
0
) = ∂δ
C
(x
0
) = {x
∗
|x
∗
, x − x
0

  δ
C
(x), ∀x}.
x ∈ C δ
C
(x) = +∞
∂f(x
0
) = ∂δ
C
(x
0
) = {x
∗
|x
∗
, x − x
0
  0, ∀x ∈ C} = N
C
(x
0
)
C ∅
x
0
∈ C C x
0
x
∗

∈ ∂f(x) f

(x, y)  x
∗
, y , ∀y
f R
n
x ∈ dom(∂f)
f(x) = f(x) ∂f(x) = ∂f (x)
x
∗
∈ ∂f(x) ⇔ x
∗
, z − x + f(x)  f(z) ∀z.
y z = x + λ.y, λ > 0
x
∗
, λ.y + f(x)  f(x + λ.y).
x
∗
, y 
f(x + λ.y) − f(x)
λ
∀λ > 0.
f

(x, y) x
∗
, y  f


(x, y) ∀y
z y = z −x λ = 1
x
∗
, z − x  f(z) − f(x) ∀z.
x
∗
∈ ∂f(x)
x ∈ dom(∂f) ∂f(x) = ∅ x
∗
∈ ∂f(x)
f epi f = epi f
epi f ⊂ epi f epi f ⊂ epi f
f(x)  f(x).
f R
n
f
R
n
f(x) = f
∗∗
(x).
f
∗∗
(x) < x
∗
, x − f
∗
(x
∗

) = f (x).
f(x) = f(x)
y
∗
∈ ∂f(x) ∀z
y
∗
, z − x + f(x)  f(z).
f(z)  f(z)  y
∗
, z − x + f(x) = y
∗
, z − x + f(x).
y
∗
∈ ∂f(x)
∂f(x) ⊂ ∂f(x).
z
0
∈ ri(dom f) z
f(z) = f(z) = lim
t→0
f[(1 −t).z + t.z
0
].
x
∗
∈ ∂f(x) ⇔ x
∗
, (1 − t).z + t.z

0
− x+ f(x)  f[(1 − t).z + t.z
0
].
t → 0
x
∗
, z − x + f(x) 
f(z).
x
∗
, z − x + f(x)  f(z)
x
∗
∈ ∂f(x)
∂f(x) ⊂ ∂f(x).
∂f(x) = ∂f (x)
f : R
n
−→ R ∪ {+∞}
x ∈ dom f ∂f(x) = ∅
x ∈ ri(dom f) ∂f(x) = ∅
z ∈ dom f f(z) < +∞ x ∈ dom f
f(x) = +∞ x
∗
x
∗
, z − x + f(x)  f(z) < +∞.
∂f(x) = ∅
x ∈ ri(dom f) (x, f (x)) epi f

f epi f
(x, f (x))
p ∈ R
n
, t ∈ R 0
p, x + t.f(x)  p, y + t.µ , ∀(y, µ) ∈ epi f.
t = 0 t = 0 p, x  p, y , ∀y ∈ dom f
p, x − y  0 , ∀y ∈ dom f
x ∈ ri(dom f) p = 0
p, t t = 0
t > 0 t < 0 µ → ∞
t > 0

p
t
, x + f(x)  
p
t
, y + µ ∀y ∈ dom f.