tóm tắt luận án tiến sĩ hàm giá trị tối ưu và ánh xạ nghiệm hữu hiệu trong các bài toán tối ưu véc tơ có tham số

VIỆN KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM
VIỆN TOÁN HỌC
———————–
THÁI DOÃN CHƯƠNG
HÀM GIÁ TRỊ TỐI ƯU VÀ ÁNH XẠ NGHIỆM
HỮU HIỆU TRONG CÁC BÀI TOÁN TỐI ƯU
VÉCTƠ CÓ THAM SỐ
Chuyên ngành: Lý thuyết tối ưu
Mã số: 62 46 20 01
TÓM TẮT LUẬN ÁN TIẾN SĨ TOÁN HỌC
Hà Nội - 2011
Công trình này được hoàn thành tại Viện Toán học, Viện Khoa học và
Công nghệ Việt Nam.
Người hướng dẫn khoa học:
1. GS.TSKH. Nguyễn Đông Yên
2. TS. Nguyễn Quang Huy
Phản biện 1: GS.TSKH. Nguyễn Xuân Tấn
Phản biện 2: GS.TSKH. Nguyễn Hữu Công
Phản biện 3: PGS.TS. Nguyễn Thị Bạch Kim
Luận án sẽ được bảo vệ tại Hội đồng chấm luận án cấp viện họp tại Viện
Toán học - Viện Khoa học và Công nghệ Việt Nam
vào hồi 9 giờ 00 ngày 28 tháng 07 năm 2011
Có thể tìm hiểu luận án tại: - Thư viện của Viện Toán học
. - Thư viện quốc gia
f : X → Y
X Y
Y
K ⊂ Y y
1

K

y
2
⇔ y
2
−y
1
∈ K.
Y = R K = R
+
¨o
¨o
Θ
C[Θ, R
n
] Θ R
n
||f|| := max
x∈Θ
f(x)
n
∀f ∈ C[Θ, R
n
],
|| · ||
n
n
R
n
X × Y X

Y
||(x, y)|| := ||x|| + ||y|| ∀(x, y) ∈ X × Y.
Ω (X, d)
T
P := C[Ω, R
s
] × C[Ω × T, R
m
] × C[T, R
m
].
p := (f, g, b) ∈ P
(SVO)
p
: min
R
s
+
f(x) x ∈ C(p),
C(p) := {x ∈ Ω | g(x, t) − b(t) ∈ −R
m
+
∀t ∈ T }
R
k
+
:= {x = (x
1
, , x
k

) ∈ R
k
| x
i
≥ 0 ∀i = 1, , k}, k = 1, 2,
R
k
.
C : P ⇒ Ω p ∈ P C(p)
A Y intA clA N (y)
y ∈ Y
¯x ∈ S(p) ¯x
(SVO)
p
¯x ∈ C(p) x ∈ C(p)
f(x) − f(¯x) ∈ −R
s
+
\{0
s
}. 0
s
0 R
s
.
S : P ⇒ Ω p ∈ P S(p)
¯x ∈ S
w
(p) ¯x
(SVO)

p
¯x ∈ C(p) x ∈ C(p)
f(x) − f(¯x) ∈ −intR
s
+
.
F : Y ⇒ Z
domF := {y ∈ Y | F (y) = ∅} F.
F : Y ⇒ Z
y
0
∈ Y V ⊂ Z F (y
0
) ⊂ V
U ∈ N (y
0
) F (y) ⊂ V y ∈ U.
y
0
∈ F V ⊂ Z
V ∩ F (y
0
) = ∅ U ∈ N (y
0
) V ∩ F (y) = ∅ y ∈ U.
Θ
f : Θ → R
s
K ⊂ R
s

f K K Θ x
1
, x
2
∈ Θ,
t ∈ [0, 1],
f(tx
1
+ (1 − t)x
2
) ∈ tf(x
1
) + (1 − t)f(x
2
) − K,
f K K Θ intK = ∅,
y ∈ R
s
, x
1
, x
2
∈ Θ, x
1
= x
2
, t ∈ (0, 1),
f(x
1
), f(x

2
) ∈ y − K f(tx
1
+ (1 − t)x
2
) ∈ y − intK.
C : P ⇒ Ω
p
0
:= (f
0
, g
0
, b
0
) ∈ dom C.
C p
0
.
Ω
p
0
:= (f
0
, g
0
, b
0
) ∈ P
t ∈ T g(·, t) R

m
+
Ω
C(p
0
) ˆx ∈ Ω
g
0
(ˆx, t) − b
0
(t) ∈ −intR
m
+
∀t ∈ T.
C p
0
S
p
0
:= (f
0
, g
0
, b
0
) ∈ P. S
p
0
x
0

∈ S(p
0
) V (x
0
) ∈ N (x
0
) Ω
¯x ∈ V (x
0
) ∩ S(p
0
)
f
−1
0
(f
0
(¯x)) ∩ C(p
0
) ⊂ V (x
0
).
C p
0
p
0
∈ P C(p) = Ω p ∈ P S
p
0
x

0
∈ S(p
0
) V (x
0
) ∈ N (x
0
) Ω
¯x ∈ V (x
0
) ∩ S(p
0
)
f
−1
0
(f
0
(¯x)) ∩ [Ω \ V (x
0
)] = ∅.
p
0
:= (f
0
, g
0
, b
0
) ∈ P.

C p
0
.
S p
0
.
Ω f
0
R
s
+
Ω.
f
0
f
0
(x
1
) = f
0
(x
2
) x
1
= x
2
.
Ω
p
0

:= (f
0
, g
0
, b
0
) ∈ P.
t ∈ T g(·, t) R
m
+
Ω
C(p
0
);
x
0
∈ S(p
0
) σ ∈ intR
s
+
argmin{σ, f
0
(x) | x ∈ C(p
0
)} = {x
0
},
·, · R
s

.
S p
0
.
S
p
0
:= (f
0
, g
0
, b
0
) ∈ P. S p
0
S(p
0
) = S
w
(p
0
). C
p
0
p
0
∈ P. C(p) = Ω
p ∈ P, S p
0
S(p

0
) = S
w
(p
0
).
Ω
p
0
:= (f
0
, g
0
, b
0
) ∈ P. C
p
0
. f
0
R
s
+
Ω S
p
0
.
Ω
p
0

:= (f
0
, g
0
, b
0
) ∈ P. g(·, t) R
m
+
Ω
t ∈ T C(p
0
). S(p
0
) = S
w
(p
0
) S
p
0
.
K ⊂ R
m
intK = ∅
T
K R
n
R
m

CO
K
[R
n
, R
m
].
P := CO
K
[R
n
, R
m
] × C[T, R].
p := (f, b) ∈ P
(CSVO)
p
: min
K
f(x) x ∈ C(p),
C(p) := {x ∈ R
n
| g
t
(x) ≤ b(t) ∀t ∈ T} g
t
: R
n
→ R
t ∈ T (t, x) → g

t
(x) T × R
n
.
C : P ⇒ R
n
S : P ⇒ R
n
(X, d) x ∈ X Ω ⊂ X
d(x, Ω) := inf {d(x, y) | y ∈ Ω}, d(x, ∅) := +∞
X = R
k
k = 1, 2, , R
k
|| · ||
k
. cone(Ω)
Ω ⊂ R
k
Ω {0
k
}.
cone(∅) = {0
k
}. h : R
k
→ R ∪ {+∞} x ∈ R
k
h(x) = +∞. h x ∂h(x)
∂h(x) := {v ∈ R

k
| v, y − x ≤ h(y) − h(x) ∀y ∈ R
k
}.
K ⊂ R
m
K
∗
:= { ∈ R
m
| , y ≥ 0 ∀y ∈ K}.
F : X ⇒ Y F
gphF := {(x, y) ∈ X × Y | y ∈ F (x)}.
F
(x
0
, y
0
) ∈ gphF U ∈ N (x
0
) V ∈ N (y
0
) κ > 0
d(y
2
, F (x
1
)) ≤ κd(x
1
, x

2
) ∀x
1
, x
2
∈ U, ∀y
2
∈ V ∩ F (x
2
).
p := (f, b) ∈ P.
C(p) ˆx ∈ R
n
g
t
(ˆx) < b(t) ∀t ∈ T.
x ∈ C(p). T
p
(x) := {t ∈ T | g
t
(x) = b(t)}
x
p := (f, b) ∈ P x ∈ S(p).  ∈ K
∗
||||
m
= 1 x ∈ argmin{ ◦ f(z) | z ∈ C(p)}, x
.  ◦ f(z) := , f(z)
z ∈ R
n

.
p
0
:= (f
0
, b
0
) ∈ P (p
0
, x
0
) ∈ gphS.
C(p
0
)
T
0
⊂ T
p
0
(x
0
) |T
0
| < n
∂(
0
◦ f
0
)(x

0
) ∩ cone


t∈T
0
(−∂g
t
(x
0
))

= ∅

0
∈ K
∗
x
0

0
. x
0

¯
 z ∈ R
n
||z||
n
= 1

u, z · ¯u, z ≥ 0 u ∈ ∂( ◦ f
0
)(x
0
), ¯u ∈ ∂(
¯
 ◦ f
0
)(x
0
).
S (p
0
, x
0
).
W ∈ N (p
0
) C(p) p ∈ W.
{(p
k
, x
k
) := (f
k
, b
k
, x
k
)}

∞
k=1
⊂ gphS (p
k
, x
k
) →
(p
0
, x
0
) := (f
0
, b
0
, x
0
) ∈ gphS, k 
k
∈ K
∗
||
k
||
m
=
1 u
k
∈ ∂(
k

◦ f
k
)(x
k
), u
k
t
k
i
∈ −∂g
t
k
i
(x
k
), t
k
i
∈ T
p
k
(x
k
), λ
k
i
> 0
i = 1, 2, , n,
u
k

=
n

i=1
λ
k
i
u
k
t
k
i
,
{u
k
t
k
1
, , u
k
t
k
n
} R
n
.

0
∈ K
∗

x
0
argmin{
0
◦ f
0
(z) | z ∈ C(p
0
)} = {x
0
}.
{p
k
}
∞
k=1
⊂ P p
0
x
k
∈ S(p
k
) x
k
→ x
0
k → +∞
f : P × X → Y
C : P ⇒ X P, X Y
|| · ||. K ⊂ Y

K 
K
Y
y 
K
y

⇔ y

− y ∈ K ∀y, y

∈ Y.
min
K

f(p, x) | x ∈ C(p)

p ∈ P.
y ∈ A A ⊂ Y
K (y − K) ∩ A = {y}. A K
E(A|K). E(∅|K) := ∅.
F : P ⇒ Y
F (p) := {f(p, x) | x ∈ C(p)}.
F(p) := E(F (p)|K), p ∈ P
F : P ⇒ Y
p ∈ P, S(p)
S(p) = {x ∈ C(p) | f(p, x) ∈ F(p)}.
G : P ⇒ Y
G
αG(p) + (1 − α)G(p


) ⊂ G(αp + (1 − α)p

) ∀p, p

∈ P, ∀α ∈ [0, 1].
G K K
αG(p) + (1 − α)G(p

) ⊂ G(αp + (1 − α)p

) + K, ∀p, p

∈ P, ∀α ∈ [0, 1].
(A) C f
K
(B) F K
(A) (B).
¨a Ω ⊂ Y ¯y ∈ clΩ.
Ω ¯y
T
C
(Ω; ¯y) := {v ∈ Y | ∀{¯y
n
} ⊂ Ω, ¯y
n
→ ¯y, ∀{t
n
} ⊂ (0, +∞), t
n

→ 0,
∃{v
n
} ⊂ Y, v
n
→ v ¯y
n
+ t
n
v
n
∈ Ω ∀n ∈ N}.

Ω ⊂ P × Y, u ∈ P
Π
u

Ω := {y ∈ Y | (u, y) ∈

Ω}.
G : P ⇒ Y
(¯p, ¯y) ∈ gphG. D
C
G(¯p, ¯y) : P ⇒ Y
G (¯p, ¯y)
D
C
G(¯p, ¯y)(u) = E

Π

u
T
C
(epiG; (¯p, ¯y))|K

∀u ∈ P,
epiG := {(p, y) ∈ P × Y | p ∈ domG, y ∈ G(p) + K}
G.
G : P ⇒
Y (¯p, ¯y) ∈ gphG
{t
n
} ⊂ (0, +∞), t
n
→ 0 {h
n
} ⊂ P, h
n
→ h ∈ P
{y
n
} ⊂ Y ¯y + t
n
y
n
∈ G(¯p + t
n
h
n
) n {y

n
}
G : P ⇒ Y
¯p ∈ P U ∈ N (¯p)
G(p) ⊂ E(G(p)|K) + K ∀p ∈ U.
F
F.
(B) (¯p, ¯y) ∈ gphF.
D
C
F (¯p, ¯y)(u) ⊂ E

Π
u
T
C
(gphF ; (¯p, ¯y))|K

∀u ∈ P,
epiD
C
F (¯p, ¯y) = T
C
(epiF ; (¯p, ¯y)).
(¯p, ¯y) ∈ gphF. D
C
F (¯p, ¯y)(0) = ∅,
(B) (¯p, ¯y) ∈ gphF.
F ¯p.
D

C
F(¯p, ¯y)(u) ⊂ E

Π
u
T
C
(gphF ; (¯p, ¯y))|K

∀u ∈ P.
(B) (¯p, ¯y) ∈ gphF.
F ¯p. D
C
F (¯p, ¯y)(0) = ∅,
D
C
F(¯p, ¯y)(u) = E

Π
u
T
C
(gphF ; (¯p, ¯y))|K

∀u ∈ P.
F
C : P ⇒ X

C : P × Y ⇒ X


C(p, y) = {x ∈ C(p) | y − f(p, x) ∈ K}.
¯p ∈ P ¯x ∈ C(¯p). f
(¯p, ¯x) ∇f(¯p, ¯x) (B)
{∇f(¯p, ¯x)(p, x) | (p, x) ∈ T
C
(gphC; (¯p, ¯x))} + K ⊂ Π
p
T
C
(epiF ; (¯p, ¯y))
p ∈ P, ¯y := f(¯p, ¯x). (B)
(A)

C ((¯p, ¯y), ¯x),
{∇f(¯p, ¯x)(p, x) | (p, x) ∈ T
C
(gphC; (¯p, ¯x))} + K = Π
p
T
C
(epiF ; (¯p, ¯y))
p ∈ P.
¯p ∈ P ¯x ∈ C(¯p) ¯y := f(¯p, ¯x) ∈ F(¯p).
F ¯p f
(¯p, ¯x) ∇f(¯p, ¯x)
D
C
F(¯p, ¯y)(p) = E

{∇f(¯p, ¯x)(p, x) | (p, x) ∈ T

C
(gphC; (¯p, ¯x))}|K

∀p ∈ P.
C : P ⇒ X
C(p) := {x ∈ X | g
t
(p, x) ≤ 0 ∀t ∈ T },
T t ∈ T, g
t
: P × X → R := R ∪ {±∞}
R
(T )
+
λ : T → R λ
t
T, 0
A(¯p, ¯x) := {λ ∈ R
(T )
+
| λ
t
g
t
(¯p, ¯x) = 0 ∀t ∈ T}.
ϕ : X → R ϕ
epiϕ := {(x, µ) ∈ X × R | µ ≥ ϕ(x)}.
ϕ
∗
: X → R ϕ

ϕ
∗
(v) := sup {v, x − ϕ(x) | x ∈ X} ∀v ∈ X.
cone


t∈T
epig
∗
t

P × X × R.
F
¯p ∈ P ¯x ∈ C(¯p) ¯y := f(¯p, ¯x) ∈ F(¯p).
F ¯p, f
(¯p, ¯x),

∇f(¯p, ¯x)(p, x)


(p, x) ∈P × X,

t∈T
λ
t
∂g
t
(¯p, ¯x)(p, x) ≤ 0, ∀λ ∈ A(¯p, ¯x)

+ K = Π

p
T
C
(epiF ; (¯p, ¯y)) ∀p ∈ P
p ∈ P
D
C
F(¯p, ¯y)(p) = E


∇f(¯p, ¯x)(p, x) | (p, x) ∈ P × X,

t∈T
λ
t
∂g
t
(¯p, ¯x)(p, x) ≤ 0 ∀λ ∈ A(¯p, ¯x)



K

.
F F
P, X Y
|| · ||.
X X
∗
· , ·.

A
∗
A x ∈ X ρ X B
ρ
(x)
K
K
∗
:= {y
∗
∈ Y
∗
| y
∗
, k ≥ 0 ∀k ∈ K}.
f : P → Y ¯p ∈ P
∇f(¯p): P → Y
lim
p,u→¯p
f(p) − f(u) − ∇f(¯p), p − u
p − u
= 0.
l : Ω ⊂ X → Y
¯x ∈ Ω η > 0  ≥ 0
l(x) − l(u) ≤ x − u ∀ x, u ∈ B
η
(¯x) ∩ Ω
( l(x) − l(¯x) ≤ x − ¯x ∀ x ∈ B
η
(¯x) ∩ Ω).

L: X ⇒ Y
(¯x, ¯y) ∈ L l : L → Y
¯x l(¯x) = ¯y l(x) ∈ L(x) x ∈ L
¯x G: X ⇒ X
∗
Lim sup
x→¯x
G(x) :=

x
∗
∈ X
∗



∃ x
n
→ ¯x, ∃x
∗
n
w
∗
−→ x
∗
, x
∗
n
∈ F (x
n

) ∀n ∈ N

X
∗
w
∗
X
∗
x
Ω
−→ ¯x Ω ⊂ X x → ¯x x ∈ Ω. ε ↓ ε
0
ε → ε
0
ε ≥ ε
0
.
Ω ⊂ X
ε ≥ 0

N
ε
(¯x; Ω) :=

x
∗
∈ X
∗




lim sup
x
Ω
−→¯x
x
∗
, x − ¯x
x − ¯x
≤ ε

.
ε
Ω ¯x ∈ Ω ε = 0

N(¯x; Ω) :=

N
0
(¯x; Ω)
Ω ¯x ¯x /∈ Ω,

N
ε
(¯x; Ω) := ∅ ε ≥ 0.
Ω ¯x ∈ Ω
N(¯x; Ω) := Lim sup
x
Ω
−→¯x

ε↓0

N
ε
(x; Ω).
N(¯x; Ω) := ∅ ¯x /∈ Ω.

N(¯x; Ω) ⊂ N(¯x; Ω)
ϕ :
X → R ¯x ∈ X.
ϕ ¯x |ϕ(¯x)| < ∞
∂ϕ(¯x) := {x
∗
∈ X
∗
| (x
∗
, −1) ∈ N((¯x, ϕ(¯x)); ϕ)},

∂ϕ(¯x) := {x
∗
∈ X
∗
| (x
∗
, −1) ∈

N((¯x, ϕ(¯x)); ϕ)}.
|ϕ(¯x)| = ∞, ∂ϕ(¯x) =


∂ϕ(¯x) = ∅
ϕ ¯x

∂
+
ϕ(¯x) := −

∂(−ϕ)(¯x).
ϕ
ϕ ¯x |ϕ(¯x)| < ∞
∂ϕ(¯x) = {x
∗
∈ X
∗
| x
∗
, x − ¯x ≤ ϕ(x) − ϕ(¯x) ∀x ∈ X}.

∂ϕ(¯x) ⊂ ∂ϕ(¯x)
ϕ
¯x

∂ϕ(¯x) = ∂ϕ(¯x).
G: P ⇒ Y
(¯p, ¯y) ∈ gph G.
G (¯p, ¯y)

D
∗
G(¯p, ¯y)(y

∗
) := {p
∗
∈ P
∗
| (p
∗
, −y
∗
) ∈

N((¯p, ¯y); gph G))} ∀y
∗
∈ Y
∗
.
F
H : P × Y ⇒ Y

C : P × Y ⇒ X
H(p, y) := F(p) ∩ (y − K),

C(p, y) := {x ∈ C(p) | y = f(p, x)}.
¯p ∈ P, ¯x ∈ C(¯p) ¯y := f(¯p, ¯x) ∈ F(¯p).
y
∗
∈ K
∗

∂

+
y
∗
, f(¯p, ¯x) = ∅ f
(¯p, ¯x).
F ¯p H (¯p, ¯y, ¯y).

D
∗
F(¯p, ¯y)(y
∗
) ⊂

(p
∗
,x
∗
)∈

∂
+
y
∗
,f(¯p,¯x)

p
∗
+

D

∗
C(¯p, ¯x)(x
∗
)

.
f (¯p, ¯x) ∇f(¯p, ¯x) :=
(∇
p
f(¯p, ¯x), ∇
x
f(¯p, ¯x))

C (¯p, ¯y, ¯x),

D
∗
F(¯p, ¯y)(y
∗
) = ∇
p
f(¯p, ¯x)
∗
y
∗
+

D
∗
C(¯p, ¯x)


∇
x
f(¯p, ¯x)
∗
y
∗

.
F
C : P ⇒ X
C(p) := {x ∈ X | h(p, x) ∈ Θ},
h : P ×X → W ∅ = Θ ⊂ W.
F
C
¯p ∈ P, ¯x ∈ C(¯p) ¯y := f(¯p, ¯x) ∈ F(¯p). y
∗
∈ K
∗

∂
+
y
∗
, f(¯p, ¯x) = ∅ f
(¯p, ¯x). F
¯p H
(¯p, ¯y, ¯y).
h (¯p, ¯x)
∇h(¯p, ¯x)


D
∗
F(¯p, ¯y)(y
∗
) ⊂

(p
∗
,x
∗
)∈

∂
+
y
∗
,f(¯p,¯x)

p
∗
+ u
∗


(u
∗
, −x
∗
) ∈ ∇h(¯p, ¯x)

∗

N( ¯w; Θ)

,
¯w := h(¯p, ¯x).
f (¯p, ¯x)
∇f(¯p, ¯x) := (∇
p
f(¯p, ¯x), ∇
x
f(¯p, ¯x))

C
(¯p, ¯y, ¯x).

D
∗
F(¯p, ¯y)(y
∗
) =

∇
p
f(¯p, ¯x)
∗
y
∗
+ u
∗



(u
∗
, −∇
x
f(¯p, ¯x)
∗
y
∗
) ∈ ∇h(¯p, ¯x)
∗

N( ¯w; Θ)

.
C(p) :=

x ∈ X | g
i
(p, x) ≤ 0, i = 1, , m,
g
i
(p, x) = 0, i = m + 1, , m + r

,
g
i
, i = 1, , m + r P × X.
h : P × X → R

m+r
h(p, x) := (g
1
(p, x), , g
m+r
(p, x))
Θ ⊂ R
m+r
Θ :=

(α
1
, , α
m+r
) ∈ R
m+r
| α
i
≤ 0, i = 1, , m,
α
i
= 0, i = m + 1, , m + r

.
F C
¯p ∈ P, ¯x ∈ C(¯p) ¯y := f(¯p, ¯x) ∈ F(¯p).
y
∗
∈ K
∗


∂
+
y
∗
, f(¯p, ¯x) = ∅ f
(¯p, ¯x). F
¯p H
(¯p, ¯y, ¯y) g
i
, i = 1, , m + r,
(¯p, ¯x)
∇g
1
(¯p, ¯x), , ∇g
m+r
(¯p, ¯x)

D
∗
F(¯p, ¯y)(y
∗
) ⊂

(p
∗
,x
∗
)∈


∂
+
y
∗
,f(¯p,¯x)

λ∈Λ(¯p,¯x,x
∗
)

p
∗
+
m+r

i=1
λ
i
∇
p
g
i
(¯p, ¯x)

,
Λ(¯p, ¯x, x
∗
) :=

λ := (λ

1
, ,λ
m+r
) ∈ R
m+r


x
∗
+
m+r

i=1
λ
i
∇
x
g
i
(¯p, ¯x) = 0,
λ
i
≥ 0, λ
i
g
i
(¯p, ¯x) = 0 i = 1, , m


D

∗
F(¯p, ¯y)(y
∗
) =

λ∈Λ(¯p,¯x,∇
x
f(¯p,¯x)
∗
y
∗
)

∇
p
f(¯p, ¯x)
∗
y
∗
+
m+r

i=1
λ
i
∇
p
g
i
(¯p, ¯x)


,

C (¯p, ¯y, ¯x) f
(¯p, ¯x) ∇f(¯p, ¯x) := (∇
p
f(¯p, ¯x), ∇
x
f(¯p, ¯x)).
F
C
t ∈ T, g
t
: P × X → R
P, X Y
(¯p, ¯x) ∈ gph C

N

(¯p, ¯x); C

=

λ∈A(¯p,¯x)


t∈T
λ
t


∂g
t
(¯p, ¯x)

.
¯p ∈ P, ¯x ∈ C(¯p) ¯y := f(¯p, ¯x) ∈ F(¯p).
y
∗
∈ K
∗

∂
+
y
∗
, f(¯p, ¯x) = ∅ f
(¯p, ¯x). F
¯p H (¯p, ¯y, ¯y) g
t
, t ∈ T,
(¯p, ¯x).
(¯p, ¯x).

D
∗
F(¯p, ¯y)(y
∗
) ⊂

(p

∗
,x
∗
)∈

∂
+
y
∗
,f(¯p,¯x)

p
∗
+ u
∗


(u
∗
, −x
∗
) ∈

λ∈A(¯p,¯x)


t∈T
λ
t


∂g
t
(¯p, ¯x)


.

D
∗
F(¯p, ¯y)(y
∗
) =

∇
p
f(¯p, ¯x)
∗
y
∗
+ u
∗


(u
∗
, −∇
x
f(¯p, ¯x)
∗
y

∗
) ∈

λ∈A(¯p,¯x)


t∈T
λ
t

∂g
t
(¯p, ¯x)


,

C (¯p, ¯y, ¯x) f
(¯p, ¯x) ∇f(¯p, ¯x) := (∇
p
f(¯p, ¯x), ∇
x
f(¯p, ¯x)).
¨o
•
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