Đối ngẫu liên hợp cho bài toán tối ưu đa mục tiêu và ứng dụng

R
R
+
N
R
n
n−
R
n
+
n−
R
n
++
n−
a
T
x R
n
x x ∈ R
n
{x
n

}
N(x, X) X x
∇f(x) f x
∂f(x)
∂

f(x)
intX X
cl(X) X
conv(X) X
co(X) X

X
E
{f(x) : x ∈ X} f(x) X
f

f
F : X ⇒ Y X Y
1951
X
max(min){f(x)| A ⊂ X}
X
∗
max(min){g(p)| A
∗
⊂ X
∗
},
g f A

∗
A
f : R
n
→ R ∪ {±∞}
f
H
(p) =



− inf{f(x) : p
T
x ≥ 1} p ∈ R
n
\ {0}
− sup{f(x) : x ∈ R
n
} p = 0.
f

(p) =
1
sup{f(x) : p
T
x ≤ 1, x ≥ 0}
R
n
+
R

n
+
R
n
R
n
+
R
n
+
R
n
+
k k > 1
R
k
+
x = (x
1
, x
2
, , x
n
), x

=
(x

1

, x

2
, , x

n
) ∈ R
n
x ≤ x

x < x

x
i
≤ x

i
(x
i
<
x

i
) i = 1, 2, , n.
X R
n
X X
X
0
= {p ∈ R

n
+
| p
T
x ≤ 1 ∀x ∈ X}.
X
X
∗
= {p ∈ R
n
+
| p
T
x ≥ 1 ∀x ∈ X}.
y = 0
X {x + λy| λ ≥ 0} ⊂ X ∀x ∈ X
X 0
X.
F X
x, y ∈ X, (1 − λ)x + λy ∈ F, 0 < λ < 1 ⇒ [x, y] ⊂ F,
[x, y] = {z| z = λx + (1 − λ)y, λ ∈ [0, 1]}
0
1
X.
X X
X R
n
X
X.
f : X → R

f X
f(λx
1
+ (1 − λ)x
2
) ≤ λf(x
1
) + (1 − λ)f(x
2
) ∀x
1
, x
2
∈ X, ∀λ ∈ [0; 1].
f X −f X
f X
f(λx
1
+ (1 − λ)x
2
) < λf(x
1
) + (1 − λ)f(x
2
) ∀x
1
= x
2
∈ X, ∀λ ∈ (0; 1).
f X −f X

f X
f(λx
1
+ (1 − λ)x
2
) ≤ max{f(x
1
); f(x
2
)} ∀x
1
, x
2
∈ X, ∀λ ∈ [0; 1].
f X −f X
f X
α ∈ R {x ∈ X : f(x) ≤ α}
f X α ∈ R
{x ∈ X : f(x) ≥ α}
f
f f : R
n
→ R
−f
f R
n
f
f(x) = min{(q
i
)

T
x − α
i
: i = 1, 2, , s}
q
1
∈ R
n
, q
2
∈ R
n
, , q
s
∈ R
n
0
α
i
∈ R, i = 1, 2, , s
f X
f(x
1
) ≤ f(x
2
) ∀x
1
, x
2
∈ X, x

1
≤ x
2
.
f X
f(x
1
) < f(x
2
) ∀x
1
, x
2
∈ X, x
1
< x
2
.
f X −f
X
f
R
n
f
f(x) = min{(q
i
)
T
x : i = 1, 2, , s}
q

1
∈ R
n
+
, q
2
∈ R
n
+
, , q
s
∈ R
n
+
0
f(x) x
0
∈ X
lim inf
x→x
0
f(x) ≥ f(x
0
). f(x)
x
0
∈ X lim sup
x→x
0
f(x) ≤ f(x

0
). f
X f
X.
f(x) x
0
f(x)
x
0
f
X f X.
f(x)
X α ∈ R {x ∈ X : f(x) ≤ α}
f(x) X
α ∈ R {x ∈ X : f(x) ≥ α}
f(x)
X f(x) X
f(x) X f(x)
X
{f
α
: α ∈ I}
X
{f
α
: α ∈ I}
X f(x) := sup{f
α
(x) : α ∈ I} ∀x ∈ X
{f

α
: α ∈ I} X
g(x) := inf{f
α
(x) : α ∈ I} ∀x ∈ X
f
α
X α ∈ I
I f(x) = max{f
α
(x) : α ∈ I} g(x) =
min{f
α
(x) : α ∈ I} X.
f
α
X
α ∈ I f(x) = max{f
α
(x) : α ∈ I} X f
α
X α ∈ I g(x) = min{f
α
(x) : α ∈ I}
X.
X R
n
+
X 0 ≤ x


≤ x, x ∈ X x

∈ X
X 0 ≤ x ≤ x

, x ∈ X
x

∈ X
f(x)
R
n
+
{x ∈ R
n
+
: f(x) ≥ 1}
X
f(x) R
n
+
X = {x ∈
R
n
+
: f(x) ≥ 1}.
f(x)
R
n
+

{x ∈ R
n
+
: f(x) ≤ 1}
X f(x)
R
n
+
X = {x ∈ R
n
+
: f(x) ≤ 1}.
F : X ⇒ R
m
.
F
x
0
{x
k
} ⊂ R
n
x
0
{y
k
} ⊂ R
m
, y
k

∈ F (x
k
) y
k
→ y
0
∈ F (x
0
). F
X F
X.
F
x
0
{x
k
} ⊂ R
n
, x
k
→ x
0
, y
k
∈ F(x
k
) y
k
→ y
0

y
0
∈ F (x
0
). F X F
X.
F
x
0
F x
0
F X F X.
F
x
0
∈ X V x
0
cl(∪
x∈V
F (x))
Q : U ⇒ R
n
Q(u) = {x ∈ V : g(x, u) ≤ 0},
V R
n
U R
m
g
V × U R
s

g V ×
u V Q
u.
g x u ∈ U Q(u) × u
V x ∈ V g(x, u) < 0 Q
u.
u
inf{f(x, u) : x ∈ Q(u)},
f : R
n
× U → R Q : U ⇒ R
n
w(u) = inf{f(x, u) : x ∈ Q(u)}.
w.
Q
u f Q(u) × u w
u
Q u u f
Q(u) × u w u
f f(x)
R
n
+
f(x) > 0 ∀x > 0
f
f

f
f


(p) =
1
sup{f(x) : p
T
x ≤ 1, x ≥ 0}
∀p ∈ R
n
+
1
+∞
= 0
f(x) > 0 ∀x > 0 sup{f(x) : p
T
x ≤ 1, x ≥ 0} > 0 ∀p ∈ R
n
+
f

R
n
+
x ∈ R
n
+
p ∈ R
n
+
p
T
x ≤ 1

f(x)f

(p) ≤ 1.
f(x) = 0 f(x)f

(p) = 0 ≤ 1. f(x) > 0
p
T
x ≤ 1
f(x)f

(p) = f(x)
1
sup{f(x

) : p
T
x

≤ 1, x

≥ 0}
≤ f(x)
1
f(x)
= 1.
f R
n
+
f


R
n
+
p = 0 x
0
> 0 p
T
(kx
0
) = 0 ≤ 1 ∀k ∈ N
f(kx
0
) = kf(x
0
) → +∞ k → +∞
f

(0) =
1
sup{f(x) : x ≥ 0}
=
1
+∞
= 0.
p = 0 θ > 0
sup{f(x) : θp
T
x ≤ 1, x ≥ 0} = sup{f(
1

θ
x

) : p
T
x

≤ 1, x

≥ 0}
=
1
θ
sup{f(x

) : p
T
x

≤ 1, x

≥ 0},
f

(θp) = θf

(p). f

f
(f


)

= f
f(x) =
1
sup{f

(p) : p
T
x ≤ 1, p ≥ 0}
∀x ∈ R
n
+
.
f
f
f f
f
f
R
n
+
f(x) = c
T
x, c = (c
1
, c
2
, , c

n
) > 0.
f f

R
n
+
f

(p) = min{
p
i
c
i
: i = 1, 2, n} ∀p ∈ R
n
+
.
f

R
n
R
n
f
R
n
f
R
n

(f

)

= f
F = {x ∈ R
n
+
: f(x) ≥ 1} f
F {y
1
, y
2
, , y
r
}
F y
i
∈ R
n
+
y
i
= 0 ∀i = 1, 2, , r f
R
n
+
F R
n
+

R
n
+
F
F = conv{y
1
, y
2
, , y
r
} + R
n
+
.
g(p) = min{(y
i
)
T
p : i = 1, 2, , r} ∀p ≥ 0.
f

= g. f f

F
∗
= {p ∈ R
n
+
: f


(p) ≥ 1}
f

(p) = max{γ ≥ 0 : p ∈ γF
∗
}.