class="bi x0 y0 w1 h1"
class="bi x1 y1 w2 h2"
class="bi x1 y2 w3 h3"
class="bi x1 y3 w4 h4"
class="bi x1 y4 w4 h5"
class="bi x1 y5 w3 h6"
f X
f
X
X
class="bi x1 yb w3 h9"
X Y
X
∗
Y
∗
. (x
∗
, x) ∈ X
∗
× X x
∗
, x := x
∗
(x).
τ
·
∗
X
∗
σ

x
∗
, x − ¯x
x − ¯x
≤ ε

,
x
Ω
−→ ¯x x → ¯x x ∈ Ω ¯x ∈ Ω

N
ε
(¯x; Ω) := ∅
ε = 0

N(¯x; Ω) :=

N
0
(¯x; Ω)
Ω ¯x
(ii) Ω ¯x ∈ Ω N(¯x; Ω)
X
∗
N(¯x; Ω) := Lim sup
x → ¯x
ε↓0

N

w
∗
−→ x
∗
x
∗
k
→ x
∗
σ

X
∗
, X

.
N(¯x; Ω) := ∅ ¯x ∈ Ω.
U X, f : U → Y ¯x ∈ U.
(i) f ¯x
∇f(¯x) : X → Y
lim
x,u→¯x
x=u
f(x) −f(u) −

∇f(¯x), x −u

x − u
= 0.
∇f(¯x) f ¯x.

, ϕ
x
∈ X
∗∗
:=

X
∗

∗
.
Φ : X → X
∗∗
, x → Φ(x) := ϕ
x
Φ(x) = ϕ
x
 x ∈ X. Φ X X
∗∗
.
x ϕ
x
X
X
∗∗
X ⊂ X
∗∗
. Φ(X) = X
∗∗
, X

∈ X : v
k
→ v, ¯x + t
k
v
k
∈ Ω ∀k

.
X

N(¯x; Ω) =

T (¯x; Ω)

−
,
K
−
:=

x
∗
∈ X
∗
|x
∗
, v ≤ 0 ∀v ∈ K

K ⊂ X.

∗
, −y
∗
) ∈ N

(¯x, ¯y); gph F

.
(ii) F (¯x, ¯y)

D
∗
F (¯x, ¯y) : Y
∗
⇒ X
∗

D
∗
F (¯x, ¯y)(y
∗
) :=

x
∗
∈ X
∗
|(x
∗
, −y

, y
k
) → (¯x, ¯y), x
∗
k
w
∗
−→ x
∗
,
y
∗
k
·
−→ y
∗
: (x
∗
k
, −y
∗
k
) ∈

N
ε
k

(x
k


x ∈ X | ϕ(x) < ∞

epi ϕ :=

(x, α) ∈ X × R | α ≥ ϕ(x)

.
(ii) ϕ ϕ = ∅ ϕ(x) > −∞ x ∈ X.
(iii) ϕ x lim inf
u→x
ϕ(u) ≥ ϕ(x).
(iv) δ > 0 ϕ u ∈ B
δ
(x)
ϕ x.
(v) ϕ x ϕ
¯x ∈ X ϕ(¯x) ∈ R.
(i) ϕ ¯x

∂ϕ(¯x) ⊂ X
∗

∂ϕ(¯x) :=

x
∗
∈ X
∗
| (x

¯x; g
−1
(Θ)

= ∇g(¯x)
∗
N(¯y; Θ)

N

¯x; g
−1
(Θ)

= ∇g(¯x)
∗

N(¯y; Θ).
f : X → Y ¯x ∈ X
F : X ⇒ Y ¯y − f(¯x) ∈ F (¯x), ¯y ∈ Y.
y
∗
∈ Y
∗
,
D
∗
N
(f + F )(¯x, ¯y)(y
∗


(y
∗
).
ψ : X → R ¯x ∈ X ϕ : X → R
¯x.
∂(ϕ + ψ)(¯x) = ∇ϕ(¯x) + ∂ψ(¯x)

∂(ϕ + ψ)(¯x) = ∇ϕ(¯x) +

∂ψ(¯x).
g : X → Y ¯x ∈ Ω, Ω := g
−1
(K)
K ⊂ Y. (P )

f(x) → inf,
x ∈ Ω,
f : X → R ¯x; f Ω
(P ).
¯x (P ) δ > 0
f(x) ≥ f(¯x) x ∈ Ω ∩B
δ
(¯x).
Y := R
m
, K := {0
R
p
} × R

∇g
i
(¯x) = 0, λ
j
≥ 0, ∀j ∈ I(¯x),
λ
i
g
i
(¯x) = 0, ∀i = 1, 2, , m.
dimX < ∞

T (¯x; Ω)

−
= ∇g(¯x)
∗
N

g(¯x); K


N

g(¯x); K

=

y
∗


g(¯x); K

∇f(¯x) + ∇g(¯x)
∗
y
∗
= 0.
dimX < ∞

N

¯x; Ω

= ∇g(¯x)
∗

N

g(¯x); K

.
g : X → Y ¯x ∈ Ω := g
−1
(K)
K ⊂ Y

N

¯x; Ω

∗

N

g(¯x); K

.
X Y g : X → Y
K ⊂ Y ¯x ∈ Ω := g
−1
(K).
¯x 0 ∈ int

g(¯x) + ∇g(¯x)(X) −K

.
X Y g : X → Y
K ⊂ Y ¯x ∈ Ω := g
−1
(K).
¯x.

N

¯x; Ω

= ∇g(¯x)
∗

N

 ≥
ϕ(b) − ϕ(a)
b − a
b − c, (2.1)
lim inf
k→∞
x
∗
k
, b − a ≥ ϕ(b) − ϕ(a), (2.2)
c = a
lim
k→∞
x
∗
k
, b − a = ϕ(b) − ϕ(a). (2.3)
X
X
ϕ : X → R x
k
ϕ
−→ c
x
∗
k
∈

∂ϕ(x
k

U ⊂ X T
e
(x) := T (x)
x ∈ U T
e
(x) := ∅ x ∈ X\U.
X f : X → X
∗
u
∗
, u ≥ 0 x ∈ X u ∈ X ⊂ X
∗∗
u
∗
∈ D
∗
N
f(x)(u);
u
∗
, u ≥ 0 x ∈ X u ∈ X ⊂ X
∗∗
u
∗
∈ D
∗
M
f(x)(u);
u
∗

), x
1
− x
2

= x
2
− x
1
< 0, (x
1
, x
2
) ∈ Q ×

R\Q

,
x
1
> x
2
. f (c) ⇒ (d).
f
f : R
n
→ R
n
f
J

D
∗
f(x)(u);
f
C
X f : X → X
∗
f C
f x ∈ C
z, u ≥ 0 ∀u ∈ C − C ⊂ X
∗∗
, ∀z ∈ D
∗
M
f(x)(u), ∀x ∈ C;
int C = ∅ f C x ∈ intC
z, u ≥ 0 ∀u ∈ intC −intC ⊂ X
∗∗
, ∀z ∈

D
∗
f(x)(u).
ϕ : X → R
¯x ∈ X ¯x
∗
∈ ∂ϕ(¯x) ϕ
(¯x, ¯x
∗
) ∂

ϕ(¯x, ¯x
∗
) : X
∗∗
⇒ X
∗
∂
2
M
ϕ(¯x, ¯x
∗
)(u) := D
∗
M

∂ϕ

(¯x, ¯x
∗
)(u) ∀u ∈ X
∗∗
.
¯x
∗
∈

∂ϕ(¯x)

∂
2

N
ϕ(¯x, ¯x
∗
)(u) := ∅ ∂
2
M
ϕ(¯x, ¯x
∗
)(u) := ∅,
¯x
∗
∈

∂ϕ(¯x)

∂
2
ϕ(¯x, ¯x
∗
)(u) := ∅ u ∈ X
∗∗
.
∂ϕ(¯x) = {¯x
∗
}, ∂
2
N
ϕ(¯x) ∂
2
N

∗∗
u
∗
∈ ∂
2
N
ϕ(x)(u).
x ∈ X u
∗
, u ≥ 0 u ∈ X ⊂ X
∗∗
u
∗
∈ ∂
2
M
ϕ(x)(u).
x ∈ X u
∗
, u ≥ 0 u ∈ X ⊂ X
∗∗
u
∗
∈

∂
2
ϕ(x)(u).
ϕ
⇒ ⇒ ⇒ X ⇔ ∇ϕ

2
, , b
m
) ∈ R
m
Θ(b) :=

x ∈ X | a
∗
i
, x ≤ b
i
, ∀i ∈ T

.
F : K → X
∗
K
X
∗
. x ∈ K

F (x), u −x

≥ 0 ∀u ∈ K,
(K, F ). F
K x ∈ K

F (x), u −x



x; Θ(b)

Θ(b) x
N

x; Θ(b)

:=

x
∗
∈ X
∗
|x
∗
, u−x ≤ 0, ∀u ∈ Θ(b)

x ∈ Θ(b)
N

x; Θ(b)

:= ∅ x ∈ X\Θ(b) (p, b) ∈ Z ×R
m
,
S(p, b) :=

x ∈ X | 0 ∈ f(p, x) + N


v
i
= 0 λ
i
≥ 0,
i = 1, 2, , m, λ
i
= 0 i = 1, 2, , m.
(x, b) ∈ gphΘ
I(x, b) := {i ∈ T |a
∗
i
, x = b
i
},
b
i
i b ∈ R
m
∅ = I ⊂ T b
I
b
i
i ∈ I
¯
I := T \I
¯x
∗
∈ N


I
1
(¯x,
¯
b, ¯x
∗
) :=

i ∈ I | ∃λ ∈ Ξ(¯x,
¯
b, ¯x
∗
) : λ
i
= 0

.
P Q P ⊂ Q ⊂ T
A
Q,P
:= span

a
∗
i
| i ∈ P } + pos{a
∗
j
| j ∈ Q\P


b
j
, ∀j ∈ T \Q

,
span

a
∗
i
| i ∈ P } :=


i∈P
λ
i
a
i
|λ
i
∈ R ∀i ∈ P

pos

a
∗
i
| i ∈ Q\P } :=



1
(¯x,
¯
b, ¯x
∗
)
{a
∗
j
|j ∈ I} x
∗
∈ X
∗
,
D
∗
M
S(¯p,
¯
b, ¯x)(x
∗
) ⊂ D
∗
N
S(¯p,
¯
b, ¯x)(x
∗
)
⊂

∗
v =

i∈Q
b
∗
i
a
∗
i
, J ⊂ P ⊂ Q ⊂ I

.
X x
∗
∈ X
∗
,
D
∗
N
S(¯p,
¯
b, ¯x)(x
∗
) = D
∗
M
S(¯p,
¯

b, ¯x
∗
),
x
∗
+ ∇
x
f(¯p, ¯x)
∗
v =

i∈Q
b
∗
i
a
∗
i
, J
1
(λ) ⊂ P ⊂ Q ⊂ I, F
Q
= ∅

.

f(p, ·); Θ(b)

S : Z × R
m

¯
b, ¯x);
(b) b
∗
∈ R
m
(P, Q) J ⊂ P ⊂ Q ⊂ I
−v ∈ B
Q,P
, b
∗
¯
Q
= 0, b
∗
Q\P
≤ 0 ∇
x
f(¯p, ¯x)
∗
v =

i∈Q
b
∗
i
a
∗
i
, (v, b

Q,P
, F
Q
= ∅, b
∗
¯
Q
= 0, b
∗
Q\J
≤ 0 (v, b
∗
) = (0, 0) S
(¯p,
¯
b, ¯x) {a
∗
j
}
j∈I
J
1
(λ) := {i ∈ I |λ
i
> 0}
¯
Q := T \Q.
X = Z = R, a
∗
1


− b

.
¯p = 1 {a
∗
j
}
j∈I(¯x,
¯
b)
S
(¯p,
¯
b, ¯x) ∈ gphS.
¯p = 0 {a
∗
j
}
j∈I(¯x,
¯
b)
S
(¯p,
¯
b, ¯x) ∈ gphS.
X = Z = R, a
∗
1
= −1, a

√
−p < max(−b
1
, −
b
2
2
) <
√
−p,

max(−b
1
, −
b
2
2
), −
√
−p,
√
−p

p ≤ 0,
max(−b
1
, −
b
2
2

m
p ∈ Z
Θ(b) :=

x ∈ X | a
∗
i
, x ≤ b
i
, ∀i ∈ T

,
T :=

1, 2, , m

. S : Z × R
m
⇒ X, (p, b) → S(p, b),
(¯p,
¯
b, ¯x) ∈ gphS. {a
∗
j
|j ∈ I(¯x,
¯
b)}
class="bi x1 y1ae w6 h1b"
class="bi x1 y1fd w5 h1c"