Báo cáo khoa học: "D­ưới vi phân giới hạn của hàm giá trị tối ưu trong một số bài toán "bệnh tật" quy hoạch trơn" pps

(a)
ϕ : X ×Y → R R := [−∞, ∞]
G : X ⇒ Y
ϕ(x, y) y ∈ G(x).
µ(x) := inf{ϕ(x, y) |y ∈ G(x)}
M(·)
M(x) := {y ∈ G(x) |µ(x) = ϕ(x, y)}.
ϕ G
µ X = R
n
Y = R
m
ϕ C
1
G(x) x
g
i
(x, y)  0, i = 1, . . . , p; h
j
(x, y) = 0, j = 1, . . . , q; (1.4)
g
i
: X × Y → R (i = 1, . . . , p h
j
: X × Y → R (j = 1, . . . , q
ϕ G
µ
µ
X ϕ
ϕ x ∈ X
ϕ

ϕ : X → R
x ε ≥ 0 x
∗
X
∗
X
ε − subgradient ϕ x
lim inf
x→x
ϕ(x) − ϕ(x) − x
∗
, x − x
||x − x||
≥ −ε.
ˆ
∂
ε
ϕ(x) ε ϕ x
ˆ
∂
0
ϕ(x) ⊂
ˆ
∂
ε
ϕ(x)
ε ≥ 0
ˆ
∂ϕ(x) :=
ˆ

∂
0
ϕ(x) ϕ x.
F : X ⇒ X
∗
X
X
∗
F (x) x → x
x→x
F (x) := {x
∗
∈ X
∗
| ∃ x
k
→ x x
∗
k
w
∗
−→ x
∗
x
∗
k
∈ F (x
k
) k = 1, 2, },
w

∗ ∗
X
∗
ϕ x
∂ϕ(x) :=
x
ϕ
−→x
ε↓0
ˆ
∂
ε
ϕ(x).
ϕ x
∂
∞
ϕ(x) :=
x
ϕ
−→x
ε,λ↓0
λ
ˆ
∂
ε
ϕ(x).
X
ϕ x
ε = 0 ∂ϕ(x) = ∅
X f : X → R x

f x
∂
CL
f(x) :=

x
∗
∈ X
∗
|x
∗
, v  lim sup
x

→x,t→0
+
f(x

+ tv) −f(x

)
t
, ∀v ∈ X

.
∂
CL
f(x) =
∗
[∂f(x) + ∂

∞
f(x)],
∗
∗
X
∗
ϕ : R
n
→ R x
∂
CL
ϕ(x) = ∂ϕ(x) = {∇ϕ(x)}.
F : X ⇒ Y
F := {x ∈ X |F (x) = ∅}, F := {(x, y) ∈ X × Y |y ∈ F (x)}.
ϕ
G(x) :=

y ∈ Y |ϕ
i
(x, y)  0, i = 1, , m,
ϕ
i
(x, y) = 0, i = m + 1, , m + r

,
ϕ
i
: X × Y → R i = 1, , m + r
L(x, y, λ) = ϕ(x, y) + λ
1

ϕ
1
(x, y) + ··· + λ
m+r
ϕ
m+r
(x, y),
λ
1
, , λ
m+r
λ := (λ
1
, , λ
m+r
) ∈ R
m+r
(x, y) ∈ M M(·)
Λ(x, y) :=

λ ∈ R
m+r
|L
y
(x, y, λ) := ∇
y
ϕ(x, y) +
m+r

i=1

λ
i
∇
y
ϕ
i
(x, y) = 0,
λ
i
≥ 0, λ
i
ϕ
i
(x, y) = 0 i = 1, , m

.
(x, y)
∇ϕ
m+1
(x, y), , ∇ϕ
m+r
(x, y)
w ∈ X × Y ∇ϕ
i
(x, y), w = 0 i = m + 1, , m + r
∇ϕ
i
(x, y), w < 0 i = 1, , m ϕ
i
(x, y) = 0.

M : domG ⇒ Y
(x, y) h : domG → Y
h(x) = y  > 0, δ > 0 h(x) ∈
G(x) h(x) − h(x)  x − x x ∈ domG ∩ B
δ
(x)
B
δ
(x) := {x ∈ X |x − x < δ}.
µ(.) x ∈ M y ∈
M(x)
∇ϕ
1
(x, y), , ∇ϕ
m+r
(x, y)
ˆ
∂µ(x) ⊂

λ∈Λ(x,y )

∇
x
ϕ(x, y) +
m+r

i=1
λ
i
∇

x
ϕ
i
(x, y)

.
ˆ
∂µ(x) =

λ∈Λ(x,y)

∇
x
ϕ(x, y) +
m+r

i=1
λ
i
∇
x
ϕ
i
(x, y)

M : domG ⇒ Y (x, y)
M : domG ⇒ Y
(x, y)
X = R, Y = R
2

x = 0, y = (0, 0)
µ(.) ϕ(x, y) = −y
2
, y = (y
1
, y
2
) ∈ G(x)
G(x) :=

y = (y
1
, y
2
) ∈ R
2
|ϕ
1
(x, y) = y
2
− y
2
1
 0,
ϕ
2
(x, y) = y
2
+ y
2

1
− x  0

.
µ(x) =



−x x  0
−
x
2
;
M(x) =

y = (y
1
, y
2
) ∈ G(x) |y
2
=



x x  0
x
2

,

Λ(x, y) = {(t, 1 − t) |0  t  1}.
∇ϕ
1
(x, y) = (0, 0, 1), ∇ϕ
2
(x, y) = (−1, 0, 1)
[−1, 0]
[−1, −
1
2
],
M(.) (x, y)
X = Y = R x = y = 0 µ(.)
ϕ(x, y) = (x − y
2
)
2
, G(x) = {y ∈ R | ϕ
1
(x, y) = −(1 + y)
2
 0}.
µ(x) =

x
2
x  0
0 ;
M(x) =


{0} x  0
{−
√
x,
√
x}
Λ(x, y) = {0}.
∇ϕ
1
(
x, y) = (0, −2) = (0, 0) {0}
ˆ
∂µ(0) = {0}
M(.)
(x, y)
X = R
2
, Y = R
2
x = (0, 0), y = (0, 0)
µ(.) ϕ(x, y) = −y
2
, y = (y
1
, y
2
) ∈ G(x)
G(x) :=

y = (y

1
, y
2
) ∈ R
2
|ϕ
1
(x, y) = y
2
+ y
4
1
x
1
+ g(y
1
)  0
ϕ
2
(x, y) = y
2
− y
4
1
− x
2
 0
ϕ
3
(x, y) = y

2
1
− 5  0
ϕ
4
(x, y) = −y
2
− 1  0

;
g(y
1
) =

0 y
1
= 0
y
4
1
sin
4
2π
y
1
.
∇ϕ
1
(x, y) = (0, 0, 0, 1), ∇ϕ
2

(x, y) = (0, −1, 0, 1), ∇ϕ
3
(x, y) = (0, 0, 0, 0),
∇ϕ
4
(x, y) = (0, 0, 0, −1)
(x, y)
Λ(x, y) = {(t, 1 − t, 0, 0) |0  t  1}.
ˆ
∂µ(x) ⊂ {0}× [−1, 0].
M(.)
G(.)
∂µ(x) ⊂

λ∈Λ(x,y )

∇
x
ϕ(x, y) +
m+r

i=1
λ
i
∇
x
ϕ
i
(x, y)


,
∂
∞
µ(x) ⊂

λ∈Λ(x,y )

m+r

i=1
λ
i
∇
x
ϕ
i
(x, y)

,
Λ(x, y)
Λ
∞
(x, y) :=

λ ∈ R
m+r
|
m+r

i=1

λ
i
∇
y
ϕ
i
(x, y) = 0, λ
i
≥ 0, λ
i
ϕ
i
(x, y) = 0 i = 1, , m

.
M(·)
(x, y)
∂
∞
µ(x) = {0} ϕ
i
y ϕ
i
∇
y
ϕ
i
(x, y)
X = Y = R.
ϕ(x, y) = −y y ∈ G(x),

G(x) = {y ∈ R |ϕ
1
(x, y) = g(y) − x  0},
g(y) =



−(y +
1
2
)
2
+
5
4
y  0
e
−y
µ(x) = {ϕ(x, y) = −y |y ∈ G(x)}
G(x) =














R x ≥
5
4
(−∞, −
1
2
−

5
4
− x] ∪ [−
1
2
+

5
4
− x, +∞) 1  x <
5
4
(−∞, −
1
2
−

5
4

− x] ∪ [−ln x + ∞) 0 < x < 1
(−∞, −
1
2
−

5
4
− x] x  0;
µ(x) =

−∞ x > 0
1
2
+

5
4
− x ;
M(x) =

{y ∈ R |y = −
1
2
−

5
4
− x} x  0
∅

x = 0 y = −
1
2
−
√
5
2
. ∇ϕ
1
(x, y) = (−1,
√
5) = (0, 0)
M(·) (0, −
1
2
−
√
5
2
)
Λ(x, y) = {
1
√
5
}
ˆ
∂µ(x) ⊂ {
−1
√
5

}.
∂µ(x) ⊂ {
−1
√
5
}.
ˆ
∂µ(x) = ∂µ(x) = ∅.
X = R, Y = R
2
ϕ(x, y) = −y
2
y = (y
1
, y
2
) ∈ G(x),
G(x) = {y = (y
1
, y
2
) ∈ R
2
| ϕ
1
(x, y) = (y
2
1
+ (y
2

− 1)
2
−
1
4
)(y
2
1
+ (y
2
+ 1)
2
− 1)  0
ϕ
2
(x, y) = y
1
− x = 0, x ≥ 0}.
µ(x) =







−1 −

1
4

− x
2
0  x 
1
2
1 −
√
1 − x
2
1
2
< x  1
+∞ x > 1;
M(x) =







{y = (y
1
, y
2
) ∈ G(x), y
2
=

1 +


1
4
− x
2
0  x 
1
2
−1 +
√
1 − x
2
1
2
< x  1
}
∅ x > 1;
x :=
1
2
, y := (
1
2
, 1) ∈ M(x)
∇ϕ
1
(x, y) = (0,
13
4
, 0), ∇ϕ

2
(x, y) = (−1, 1, 0)
Λ(
1
2
,
1
2
, 1) = ∅.
ˆ
∂µ(x) = ∅.
∂µ(x) = ∅.
∂
CL
µ(x) = co[∂µ(x) + ∂
∞
µ(x)],
∂
CL
µ(x) = ∅.
x := 1, y := (1, −1) ∈ M(x)
ˆ
∂µ(x) = ∂µ(x) = ∂
CL
µ(x) = ∅.
X = Y = R,
ϕ(x, y) = −y y ∈ G(x),
G(x) = {y ∈ R | ϕ
1
(x, y) = y

3
− x  0}.
G(x) = {y ∈ R | y ∈ (−∞,
3
√
x]};
µ(x) = −
3
√
x;
M(x) = {y ∈ R | y =
3
√
x}.
x := 0, y := 0 ∇ϕ
1
(x, y) = (−1, 0) Λ(0, 0) = ∅.
ˆ
∂µ(x) = ∅.
∂µ(x) = ∅.
∂
CL
µ(x) = co[∂µ(x) + ∂
∞
µ(x)]
∂
CL
µ(x) = ∅.
X = R, Y = R
2

ϕ(x, y) = −y
2
y = (y
1
, y
2
) ∈ G(x),
G(x) = {y = (y
1
, y
2
) ∈ R
2
| ϕ
1
(x, y) = y
2
1
− 100  0
ϕ
2
(x, y) = g(y
1
) − y
2
= 0
ϕ
3
(x, y) = (y
8

1
− x)y
2
= 0},
g(y
1
) =

0 y
1
= 0
y
4
1
cos(
2π
y
1
)
G(x) =

{(y
1
, 0) ∈ R
2
, y
1
= 0 y
1
=

4
2k+1
, k = 0, ±1, ±2, } x  0
G(0) ∪ {(±
8
√
x, g(
8
√
x)) |
4
√
x  100}
µ(x) =

0 x  0
{0, −g(
8
√
x)}
M(x) =

{(y
1
, 0) ∈ R
2
y
1
= 0 y
1

=
4
2k+1
, k = 0, ±1, ±2, } x  0
{(y
1
, y
2
) ∈ G(x), y
2
= {0, g(
8
√
x)}}
x := 0, y := (
4
2k+1
, 0), k = 0, ±1, ±2, k = 2n, n = 0, ±1, ±2,
∇ϕ
1
(x, y) = (0,
8
4n + 1
, 0), ∇ϕ
2
(x, y) = (0,
32π
(4n + 1)
2
, −1), ∇ϕ

3
(x, y) = (0, 0, (
4
4n + 1
)
8
)
M(.)
(0,
4
4n+1
, 0).
Λ(0,
4
4n + 1
, 0) = {(0, 0, (n +
1
4
)
8
)}.
ˆ
∂µ(x) ⊂ {0}.
k = 2n + 1, n = 0, ±1, ±2,
ˆ
∂µ(x) ⊂ {0}
∂µ(x) ⊂ {0}, ∂
∞
µ(x) ⊂ {0}.
∂

CL
µ(x) = co[∂µ(x) + ∂
∞
µ(x)],
∂
CL
µ(x) ⊂ {0}.
ˆ
∂µ(x)
∂µ(x)
ˆ
∂µ(x) = ∂µ(x) = ∅. ∂
CL
µ(x) = ∅
∂
∞
µ(x) = {0}
ϕ
i
, i = 1, 2, 3
y
X = R
2
, Y = R
2
ϕ(x, y) = −y
2
y = (y
1
, y

2
) ∈ G(x),
G(x) = {y = (y
1
, y
2
) ∈ R
2
| ϕ
1
(x, y) = y
2
+ y
4
1
x
1
+ g(y
1
)  0
ϕ
2
(x, y) = y
2
− y
4
1
− x
2
 0

ϕ
3
(x, y) = y
2
1
− 5  0
ϕ
4
(x, y) = −y
1
− 1  0},
g(y
1
) =

0 y
1
= 0
y
4
1
sin
4
(
2π
y
1
)
x := (0, 0)
M(x) = {(0, 0)}∪ {(

2
k
, 0) |k = ±1, ±2, }.
y := (
2
k
, 0), k = ±1, ±2,
∇ϕ
1
(x, y) = (
16
k
4
, 0, 0, 1), ∇ϕ
2
(x, y) = (0, −1,
−32
k
3
, 1),
∇ϕ
3
(x, y) = (0, 0,
4
k
, 0), ∇ϕ
4
(x, y) = (0, 0, 0, −1)
Λ(0, 0,
2

k
, 0) = {(1, 0, 0, 0)}.
ˆ
∂µ(x) ⊂ {(
16
k
4
, 0)}.
∂µ(x) ⊂ {(
16
k
4
, 0)}, ∂
∞
µ(x) ⊂ {(0, 0)}.
∂
CL
µ(x) = co[∂µ(x) + ∂
∞
µ(x)],
∂
CL
µ(x) ⊂ {(
16
k
4
, 0)}.
y := (0, 0)
∇ϕ
1

(x, y) = (0, 0, 0, 1), ∇ϕ
2
(x, y) = (0, −1, 0, 1),
∇ϕ
3
(x, y) = (0, 0, 0, 0), ∇ϕ
4
(x, y) = (0, 0, 0, −1).
(x, y)
Λ(0, 0, 0, 0) = {(t, 1 −t, 0, 0) | 0  t  1}
ˆ
∂µ(x) ⊂ {0}× [−1, 0].
∂µ(x) ⊂ {0}× [−1, 0], ∂
∞
µ(x) ⊂ {(0, 0)}.
∂
CL
µ(x) = co[∂µ(x) + ∂
∞
µ(x)],
∂
CL
µ(x) ⊂ {0} × [−1, 0].
∂
∞
µ(x) = {(0, 0)} ϕ
i
, i =
1, 2, 3, 4
y

X = R, Y = R
2
ϕ(x, y) = −y
2
y = (y
1
, y
2
) ∈ G(x),
G(x) = {y = (y
1
, y
2
) ∈ R
2
|ϕ
1
(x, y) = y
2
− y
2
1
 0
ϕ
2
(x, y) = y
2
+ y
2
1

− x  0}.
µ(x) =

−x x  0
−
x
2
M(x) =

y = (y
1
, y
2
) ∈ G(x) | y
2
=

x x  0
x
2

.
x := 0, y := (0, 0)
∇ϕ
1
(x, y) = (0, 0, 1), ∇ϕ
2
(x, y) = (−1, 0, 1)
M(·)
(x, y). Λ(0, 0, 0) = {(t, 1 − t) | 0  t  1}.

ˆ
∂µ(x) ⊂ [−1, 0].
∂µ(x) ⊂ [−1, 0], ∂
∞
µ(x) ⊂ {0}.
∂
CL
µ(x) = co[∂µ(x) + ∂
∞
µ(x)],
∂
CL
µ(x) ⊂ [−1, 0].
ˆ
∂µ(x) = ∂µ(x) = [−1, −
1
2
].
∂
∞
µ(x) = {0} ϕ
1
ϕ
2
y