Một vài kết quả về cận sai số trong tối ưu Vectơ

f : X → R ∪ {+∞}
X f α
τ
τd(x, [f ≤ α]) ≤ (f(x) − α)
+
, ∀x ∈ X
[f ≤ α] := {x ∈ X : f(x) ≤ α}, d(x, [f ≤ α])
x [f ≤ α] t
+
= max{0, t}.

min
C
F (x)
x ∈ A
F X
Y ; A X C Y
F (A)
A X \ (A + int(F
−1
(C)))
(Z, d) f : Z → R ∪ {+∞}
Z
(f) = {x ∈ Z : f(x) < +∞} f.
C
α
(f) = {x ∈ X : f(x) ≤ α} f.

(f) = {(x, α) ∈ Z × R : f(x) ≤ α} f.
f x
0
∈ Z
lim inf
x→x
0
f(x) ≥ f(x
0
),
lim inf
x→x
0
f(x) = sup
η>0
inf{f(x) : x ∈ Z, d(x − x
0
) ≤ η}. f
Z f
Z.
f(x) =



1 x < 0
−1 x ≥ 0,
R. 0
Ω ⊂ Z Ω
I
Ω

(x) =



0 x ∈ Ω
+∞ x /∈ Ω,
Ω.
f Z.
(f) Z × R
C
α
(f) Z
f(x) U
f(x) U
U f
U U
f : R
2
→ R f(x) = x
2
1
+ (x
1
x
2
−
1)
2
, ∀x = (x
1

, x
2
) ∈ R
2
f ≥ 0 ∀x ∈ R
2
inf
R
2
f = 0. f
R
2
.
f
 > 0 x

∈ Z  f Z
inf
x∈Z
f(x) ≤ f(x

) ≤ inf
x∈Z
f(x) + .
x

∈ Z 
 x
∗
f.

(Z, d) f : Z → R ∪ {+∞}
Z  > 0 x

∈ Z
f(x

) ≤ inf
x∈Z
f(x) + .
λ > 0 x
∗
∈ Z
d(x
∗
, x

) ≤ λ.
f(x
∗
) +

λ
d(x
∗
, x

) ≤ f(x

).
f(x

∗
) < f(x) +

λ
d(x
∗
, x

) ∀x ∈ Z \ {x
∗
}.
Z
Z
∗
Z x
∗
∈ Z
∗
, x ∈ Z
x
∗
, x = x
∗
(x)
C Z.
C x
1
∈ Z, x
2
∈ C

∀t ∈ [0, 1] tx
1
+ (1 − t)x
2
∈ C
C Z C x ∈ C, λ ≥
0 λx ∈ C C C
0.
R
n
+
= {x = (x
1
, x
2
, , x
n
) ∈ R
n
: x
i
≥ 0, i = 1, 2, , n}.
M = {x = (x
1
, x
2
) ∈ R
2
: x
2

≥ 0}.
C int(C)
int(C) := {x
0
∈ C| ∃ V x
0
: V ⊂ C}
C rint(C)
rint(C) := {x
0
∈ C| ∃ V x
0
: V ∩ aff(C) ⊂ C},
C.
C
C
+
:= {x
∗
∈ Z
∗
: x
∗
, x ≥ 0, ∀x ∈ C}.
C Z C
+
Z
∗
.
C

C = {x ∈ Z : x
∗
, x ≥ 0, ∀x
∗
∈ C
+
}.
C Z x
0
∈ C.
C x
0
T
C
(x
0
) := cl(∪
t>0
(C − z))
C x
0
N(x
0
, C) := {x
∗
∈ Z
∗
: x
∗
, x − x

0
 ≤ 0 ∀x ∈ C}.
T
C
(x
0
) N(x
0
, C)
x
0
∈ C T
C
(x
0
) = Z N(x
0
, C) = {0}.
f : Z → R ∪ {+∞} Z
f(tx
1
+ (1 − t)x
2
) ≤ tf(x
1
) + (1 − t)f(x
2
), ∀x
1
, x

2
∈ Z, ∀t ∈ [0, 1].
f : Z → R ∪ {+∞}
f (f) = ∅
f(x) = a, x + b a ∈ R
n
, b ∈ R
I
C
(x) =



0 x ∈ C
+∞ x /∈ C,
C ⊂ Z
Z f(x) = ||x||
f : Z → R ∪ {+∞}.
f Z.
(f)
f(

m
k=1
λ
k
x
k
) ≤


m
k=1
λ
k
f(x
k
) x
k
∈ Z,

m
k=1
λ
k
= 1 λ
k
≥ 0
k, m ∈ Z, m ≥ 2.
f Z α ∈ R ∪ {+∞}
{x ∈ Z : f(x) ≤ α}
f(x) = x
3
f(x) =

|x|
R
f : Z → R x
0
∈ Z
x

∗
∈ Z
∗
f x
0
x
∗
, x − x
0
 ≤ f(x) − f(x) ∀x ∈ Z.
∂f(x
0
) f x
0
∂f(x
0
)
f x
0
∂f(x
0
) f
x
0
.
f : R
n
→ R f(x) = a, x + b
a ∈ R
n

, b ∈ R ∂f(x) = {a} x ∈ R.
I
C
C x
0
∈ C
C x
0
.
f(x) = ||x||
∂f(x) =



{x
∗
∈ Z
∗
: ||x
∗
|| ≤ 1} x = 0
{x
∗
∈ Z
∗
: x
∗
, x = ||x||, ||x
∗
|| = 1} x = 0.

f Z
x
0
∈ f)
f : Z → R ∪ {+∞}.
x
∗
∈ ∂f(x
0
) ⇔ (x
∗
, −1) ∈ N
f
(x
0
, f(x
0
)).
∂f(x
0
)
f x
0
∂f(x
0
) = {∇f(x
0
)} ∇f(x
0
)

f x
0
∇f(x
0
)
∗
∇f(x
0
)
f Z C ⊂
int(domf) x
0
f C
0 ∈ ∂f(x
0
) + N(x
0
, Z).
D Z. d
D
(·)
d(·, D) D
d
D
(x) := inf
y∈D
{||x − y|},
d
D
(·) = +∞ D = ∅

D Z
D ∆
D
: Z → R ∪ {±∞}
∆
D
(x) := d
D
(x) − d
Z\D
(x)



d
D
(x) x ∈ Z \ D
−d
Z\D
(x) x ∈ D,
∆
D
≡ +∞ D = ∅ ∆
D
≡
−∞ D = Z
∆
D
= −∆
Z\D

.
D Z ∆
D
B
Z
, S
Z
S
D
Z
D
∆
D
A
Z x ∈ Z
d
A
(x) = d
cl(A)
(x) = sup
x
∗
∈B
Z
∗
inf
u∈cl(A)
x
∗
, x−u = sup

x
∗
∈B
Z
∗
inf
u∈A
x
∗
, x−
u
int(A) = ∅ x ∈ Z \ int(A)
d
A
(x) = sup
x
∗
∈S
Z
∗
inf
u∈A
x
∗
, x − u
A = D + K K int(A) = ∅ x ∈ X \ int(A)
d
A
(x) = sup
x

∗
∈S(K
+
)
inf
u∈A
−x
∗
, x − u = sup
x
∗
∈S(K
+
)
inf
x∈D
−x
∗
, x − x.
d
A
(x) = d
cl(A)
(x)
sup
x
∗
∈B
Z
∗

inf
u∈cl(A)
x
∗
, z − u = sup
x
∗
∈B
Z
∗
inf
u∈A
x
∗
, x − u.
d
cl(A)
(x) = sup
x
∗
∈B
Z
∗
inf
u∈cl(A)
x
∗
, z − u.
f(z) = ||z −x||
f

∗
f
d
cl(A)
(x) = inf
z∈Z
{f(z) + I
cl(A)
(z)}
= sup
x
∗
∈Z
∗
{−f
∗
(x
∗
) − I
∗
cl(A)
(x
∗
)}
= sup
x
∗
∈B
Z
∗

{x
∗
, x − sup
z∈cl(A)
x
∗
, z}
= sup
x
∗
∈B
Z
∗
inf
u∈cl(A)
x
∗
, x − u.
B
Z
∗
= ∪
0≤t≤1
tS
Z
∗
d
A
(x) = max{0, sup
x

∗
∈S
Z
∗
inf
u∈A
x
∗
, x − u}.
A x ∈ Z \ int(A) z
∗
∈ S
Z
∗
inf
u∈A
z
∗
, x − u ≥ 0 sup
x
∗
∈S
Z
∗
inf
u∈A
x
∗
, x − u} ≥ 0
d

A
(x) = sup
x
∗
∈S
Z
∗
inf
u∈A
x
∗
, x − u.
inf
k∈K
x
∗
, k = −∞ x
∗
/∈ K
+
✷
A Z ri(A) = ∅
∆
A
(x) = sup
x
∗
∈S
Z
∗

inf
u∈A
x
∗
, x − u, ∀x ∈ Z.
D + A Z K int(K) = ∅,
∆
D+K
(x) = sup
x
∗
∈S(K
+
)
inf
u∈D+K
−x
∗
, x−u = sup
x
∗
∈S(K
+
)
inf
u∈D
−x
∗
, x−u, ∀x ∈ X.
d

Z\A
(x) = inf
x
∗
∈S
Z
∗
sup
u∈A
x
∗
, x − u, ∀x ∈ A.
x ∈ A r > d
Z\A
(x) x
1
∈ Z \ A
r > ||z
1
− x|| d
A
(z
1
) = sup
x
∗
∈S
Z
∗
inf

u∈A
x
∗
, z
1
− u
 > 0 x
∗

∈ S
Z
∗
d
A
(z
1
) −  = inf
u∈A
x
∗

, z
1
− u.
u ∈ A
−x
∗

, x − u = x
∗


, x − z
1
 + x
∗

, z
1
− u ≤ ||x − z
1
|| − (d
A
(z
1
) − )
< r − d
A
(z
1
) + .
d
Z\A
(x) = inf
x
∗
∈S
Z
∗
sup
u∈A

x
∗
, x − u < r − .
 → 0 r → d
Z\A
(x)
inf
x
∗
∈S
Z
∗
sup
u∈A
x
∗
, x − u ≤ d
Z\A
(x).
d
Z\A
(x) > 0.
r

, r

d
Z\A
(x) > r


> r

x
∗
∈ S
Z
∗
w ∈ B(x, r

) x
∗
, w > r

. B(0, r

) ⊂ A
B(0, r

) ⊂ A − x w ∈ B(0, r

) ⊂ A − x
r

< x
∗
, w < sup
u∈A
x
∗
, x − u

r

< inf
x
∗
∈S
Z
∗
sup
u∈A
x
∗
, x − u r

→ d
Z\A
(x)
d
Z\A
(x) ≤ inf
x
∗
∈S
Z
∗
sup
u∈A
x
∗
, x − u.

✷
cl(D + int(K)) = cl(int(D + K)) = cl(D + K),
D K
int(K) = ∅.
∆
D+int(K)
(z) = sup
x
∗
∈S(K
+
)
inf
a∈D
−z
∗
, z − a, ∀z ∈ Z,
∆
D+int(K)
(·) = ∆
D+K
(·) Z.
K int(K) = ∅.
∆
K
(z) = sup
x
∗
∈S(K
+

)
−z
∗
, z, ∀z ∈ Z.
K
Z int(K) = ∅ D ⊂ Z
D + K

D = Z \ (D + int(K)).
∆

D
∆

D
(z) = d

D
(z) − d
D+int(K)
(z).
≤
K
K
f : D → R ∪ {+∞}. C−
x
1
, x
2
∈ D, x

1
≤
C
x
2
f(x
1
) ≤ f(x
2
).
∆

D
(z) = d

D
(z) ∀z ∈ D
∆

D
(z) = inf
z
∗
∈S(K
+
)
sup
a∈D
z
∗

, z − a = inf
z
∗
∈S(K
+
)
{z
∗
, z − inf
a∈D
z
∗
, a}∀z ∈ Z
∆

D
K−
z ∈ D int(K) = ∅ e ∈ int(K)
lim
n→+∞
(z +
1
n
e) = z z ∈ cl(D + int(K)) d
D+int(K)
(z) = 0.
∆

D
= −∆

D+int(K)
z, z

∈ Z z −z

∈ K x
∗
∈ S(K
+
) z
∗
, z

 ≤ z
∗
, z
z
∗
, z

 − inf
a∈D
z
∗
, a ≤ z
∗
, z − inf
a∈D
z
∗

, a.
✷
rext(K
+
) K
+
0
x
∗
∈ rext(K
+
) x
∗
= 0 R
+
x
∗
K
+
. w
∗
∗
Z
∗
co
w
∗
[rext(K
+
)] = w

∗
−cl{
n

i=1
x
∗
i
: x
∗
i
∈ rext(K
+
)}.
e ∈ int(K) δ > 0 e + δB
Z
⊂ K
z
∗
, e ≥ δ||z
∗
||, ∀z
∗
∈ K
+
.
K
+
= co
w

∗
[rext(K
+
)] = co
w
∗
{tx
∗
: t > 0, x
∗
∈ rext(K
+
) ||x
∗
|| = 1}.
K = {x ∈ X : x
∗
, x ≥ 0, ∀z
∗
∈ S(K
+
) ∩ rext(K
+
).
inf
x
∗
∈S(K
+
)

x
∗
, k = inf
z
∗
∈S(K
+
)∩rext(K
+
)
z
∗
, k, ∀k ∈ K.
H := {x
∗
∈ Z
∗
: x
∗
, e = 1}.
H ∩ K
+
w
∗
−
{0}.
inf
x
∗
∈S(K

+
)
x
∗
, k ≥ inf
z
∗
∈S(K
+
)∩rext(K
+
)
z
∗
, k, ∀k ∈ K.
x
∗
∈ S(K
+
) k ∈ K
x
∗
, k ≥ inf
z
∗
∈S(K
+
)∩rext(K
+
)

z
∗
, k.
{λ
i
}
n
i=1
⊂ (0, +∞) {x
∗
i
}
n
i=1
⊂ co[S(K
+
) ∩ rext(K
+
)]
λ
i
x
∗
i
w
∗
−→ x
∗
||x
∗

i
|| ≤ 1 i ∈ N Z
∗
w
∗
lim inf
i→+∞
λ
i
≥ lim inf
i→+∞
||λ
i
x
∗
i
|| ≥ ||x
∗
|| = 1.
i ∈ N
λ
i
x
∗
i
, k ≥ λ
i

inf
z

∗
∈co[S(K
+
)∩rext(K
+
)]
z
∗
, k

= λ
i

inf
z
∗
∈S(K
+
)∩rext(K
+
)
z
∗
, k

.
lim inf
i→+∞
λ
i

x
∗
i
, k ≥ inf
z
∗
∈S(K
+
)∩rext(K
+
)
z
∗
, k,
1.15 λ
i
x
∗
i
w
∗
−→ x
∗
✷
ξ
D,K
(z) := inf
a∈D
∆
K

(z − a), ∀z ∈ Z.
ξ
D,K
ξ
D,K
(z) = sup
a∈D
inf
z
∗
∈S(K
+
)
z
∗
, z − a ∀z ∈ Z.
ξ
D,K
(z) = sup
a∈D
inf
z
∗
∈S(K
+
)∩rext(K
+
)
z
∗

, z − a ∀z ∈ Z
✷
(X, d)
f : X → R ∪ {+∞}
U ⊂ X r ∈ (0, +∞] r ∈ [0, +∞)
B
r
(U) B
r
(U) U
B
r
(U) = {x ∈ X : d(x, U) < r};
B
r
(U) = {x ∈ X : d(x, U) ≤ r}.
U = {x} B
r
(x) B
r
(x) α ∈ R
[f ≤ α] = {x ∈ X : f(x) ≤ α},
[f < α] = {x ∈ X : f(x) < α},
[f > α] = {x ∈ X : f(x) > α},
[f ≥ α] = {x ∈ X : f(x) ≥ α}.
f α ∈ R
σ
f(x) − α ≥ σd(x, [f ≤ α]), f(x) ∈ (α, +∞).
σ
α

(f) σ ∈ [0, +∞)
σ
α
(f) = inf
x∈[f>α]
f(x) − α
d(x, [f ≤ α])
,
σ
α
(f) =



0 [f ≤ α] = ∅ [f > α] = ∅
+∞ [f > α] = ∅,
f α σ
α
(f) > 0.
f : X → R ∪ {+∞}
x ∈ dom(f)
|∇f|(x) :=





0 x f
lim sup
y→x

f(x)−f(y)
d(x,y)
.
x /∈ dom(f) |∇f|(x) := +∞. |∇f|(x) f
x.
f : R → R
f(x) :=



x x < 0
x
2
x ≥ 0.
|∇f|(0) = 1. 0 f
|∇f|(0) = lim sup
y→0
−f(y)
|y|
.
y < 0
−f(y)
|y|
=
−y
−y
= 1.
y ≥ 0
−f(y)
|y|

=
−y
2
y
= −y.
|∇f|(0) = 1.
X M X
z ∈ M d
z
: M → R
d
z
(x) = ||x − z|| ∀x ∈ M.
|∇d
z
|(x) = 1 ∀x = z
|∇d
z
|(x) = lim sup
y→x
d
z
(x) − d
z
(y)
||x − y||
= lim sup
y→x
||x − z|| − ||y − z||
||x − y||

≤ 1.
M y
n
= λ
n
z + (1 − λ
n
)x ∈ M, λ
n
∈
(0, 1], λ
n
→ 0.
lim
y
n
→x
||x − z|| − ||y
n
− z||)
||x − y
n
||
= lim
λ
n
→0
||x − z|| − ||λ
n
z + (1 − λ

n
)x − z||)
||x − λ
n
z + (1 − λ
n
)x||
= 1.
|∇d
z
|(x) = 1 ∀x = z.
f : X → R ∪{+∞} U ∈ X α ∈ R
inf
U∩[f>α]
|∇f| ≥ inf
γ≥α
inf
U∩[f>α]
f(x) − γ
d(x, [f ≤ γ])
.
inf
[f>α]
|∇f| ≥ inf
γ≥α
σ
α
(f).
inf
U∩[f>α]

|∇f| < +∞ U ∩ [f > α] = ∅
[f ≤ α] = ∅.
σ > 0
inf
U∩[f>α]
f(x) − γ
d(x, [f ≤ γ])
> σ, ∀γ > 0,
x ∈ U ∩ [f > α] γ
n
= f(x) −
1
n
n γ
n
≥ α
x
n
∈ [f ≤ γ
n
]
f(x) − γ
n
≥ σd(x, x
n
).
0 < d(x, x
n
) ≤
f(x) − γ

n
σ
→ 0 n → ∞.
f(x) − f(x
n
) ≥ σd(x, x
n
) + γ
n
− f(x
n
) > 0 x
n
= x x
f.
f(x) − f(x
n
)
d(x, x
n
)
≥
f(x) − γ
n
d(x, x
n
)
≥ σ.
|∇f|(x) = lim sup
x

n
→x
f(x) − f(x
n
)
d(x, x
n
)
≥ σ.
✷
X
X f : X → R ∪ {+∞}
x ∈ X σ > 0, r > 0
f(x) < inf
B
r
(x)
+σr.
x ∈ B
r
(x) |∇f|(x) < σ f(x) ≤ f(x)
X = B
r
(x),  = σ

r

, λ = r

x ∈ B

r
(x) ⊂ B
r
(x) f(x) ≤ f(x)
f(x) < f(y) + σ

d(x, y), ∀y ∈ B
r
(x) \ {x}.
f(x) − f(y)
d(x, y)
< σ

, ∀y ∈ B
r
(x) \ {x}.
lim sup
y→x
f(x) − f(y)
d(x, y)
≤ σ

.
|∇f|(x) = |∇f|
B
r
(x)
(x) ≤ σ

< σ. ✷

[f ≤ α] = ∅ [f > α] = ∅ inf
[f>α]
|∇f| = 0,
0 < inf
[f>α]
|∇f| < +∞ [f ≤ α] = ∅
[f ≤ α] = ∅ inf
X
f ≥ α f σ > 0 r > 0
x ∈ X f(x) < inf
B
r
(x)
f + σr.
x ∈ B
r
(x) |∇f|(x) < σ. σ > 0 |∇f|(x) ≤ 0.
inf
[f>α]
|∇f| ≥ 0. inf
[f>α]
|∇f| = 0.
0 < inf
[f>α]
|∇f|
X f : X → R ∪ {+∞}
U ∈ X α ∈ R σ > 0, ρ > 0 U ∩ [f < α + σρ] = ∅
inf
B
ρ

(U)∩[α<f <α+σρ]
|∇f| ≥ σ.
[f ≤ α] = ∅
σd(x, [f ≤ α]) ≤ (f(x) − α)
+
, ∀x ∈ U ∩ [f < α + σρ].
[f ≤ α] = ∅ f(x) > α, ∀x ∈ X.
x ∈ U ∩ [f < α + σρ]
f(x) < α + σρ ≤ inf
X
+σρ.
x ∈ B
ρ
(x) ⊂ B
ρ
(U)
α < f(x) ≤ f(x) < α + σρ |∇f|(x) < σ.
x ∈ B
ρ
(U) ∩ [α < f < α + σρ] |∇f|(x) < σ.
✷
X f : X → R ∪ {+∞}
α ∈ R
inf
[f>α]
|∇f| = inf
γ≥α
σ
γ
(f) = inf

γ≥α
inf
x∈[f>γ]
f(x) − γ
d(x, [f ≤ γ])
.
σ
γ
(f) ≥ inf
[f>α]
|∇f|, ∀γ ≥ α.
γ ≥ α σ
γ
(f) < +∞
[f > α] = ∅ inf
[f>α]
|∇f| > 0 [f ≤ α] = ∅. σ > σ
γ
(f)
x ∈ [f ≤ γ]
f(x) − γ < σd(x, [f ≤ γ]).
r := d(x, [f ≤ γ]) > 0 [f ≤ γ] x /∈ [f ≤ γ] g := (f −γ)
+
≥
0 g := max{f(x) − γ, 0}
g(x) = max{f(x) − γ, 0} = f(x) − γ < σd(x, [f ≤ γ]) = σr ≤ inf
B(x)
+σr.
x ∈ B
r

(x) g(x) ≤ g(x) |∇g|(x) < σ.
r d(x, x) < r = d(x, [f ≤ γ]) f(x) > α
f(x) = g(x) + γ ≤ g(x) + γ = f(x) < +∞.
x ∈ [f > γ]
|∇f|(x) = |∇g|(x) < σ.
inf
[f>γ]
|∇f| < σ.
σ
γ
(f) inf
[f>γ]
|∇f| ≤ σ
γ
(f).
inf
[f>α]
|∇f| ≤ inf
[f>γ]
|∇f| ≤ σ
γ
(f).
✷
X
|| · || X
∗
X, d
∗
x ∈ X, x
∗

∈ X
∗
x
∗
, x = x
∗
(x)
f : X → R ∪ {+∞} f

(x, u)
u x
f

(x, u) = lim
t→0
+
f(x + tu) − f(x)
t
.
X f : X → R ∪ {+∞}
x ∈ X
f
|∇|f(x) = sup
f(z)<f (x)
f(x) − f(z)
||x − z||
= sup
f(z)<f (x)
−f


(x, z − x) − f(z)
||x − z||
= d
∗
(0, ∂f(x)).
x /∈ domf
∂f(x) = ∅ +∞.
x ∈ domf x f z ∈ X
f(z) < f(x) λ ∈ (0, 1]
f(x + λ(z − x)) = f((1 − λ)x + λz)
≤ (1 − λ)f(x) + λf(z).
λ(f(x) − f(z)) ≤ f(x) − f(x + λ(z − x)).
f(x) − f(z)
||x − z||
≤
f(x) − f(x + λ(z − x))
λ||x − z||
.
x
∗
∈ ∂f(x)
f(x + λ(z − x)) − f(x) ≤ x
∗
, λ(z − x).
f(x) − f(x + λ(z − x))
λ||x − z||
≤
−x
∗
, λ(z − x)

λ||x − z||
≤
λ||x
∗
||
∗
||z − x||
λ||x − z||
= ||x
∗
||
∗
.
|∇|f(x) ≤ sup
f(z)<f (x)
f(x) − f(z)
||x − z||
≤ sup
f(z)<f (x)
−f

(x, z − x) − f(z)
||x − z||
≤ d
∗
(0, ∂f(x)).
x f σ
0 < σ < d
∗
(0, ∂f(x)). x

z → f(z) +σ||x− z|| z ∈ X f(z)+σ||x − z|| < f(x)