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FICHA CATALOGRÁFICA
Serviço de Processamento Técnico da UFPI
Biblioteca Comunitária Jornalista Carlos Castello Branco
C331a
Carvalho, Francisco Gilberto de Sousa.
Algoritmo do ponto proximal generalizado em espaços
de Hilbert para o problema de desigualdade variacional
/ Francisco Gilberto de Sousa Carvalho. Teresina:
2010.
63 f.
Orientador: Prof. Dr. Jurandir de Oliveira Lopes
Dissertação (Mestrado em Matemática)–Universidade
Federal do Piauí.
1. Algoritmo. 2. Ponto proximal. I. Título.
CDD: 005.1
T
C
T C
T
x ∈ C u ∈ T(x)
u x − x ∀ x ∈ C
C H T : H ⇒ H
S
∗
T C
S
∗
{x
k
} ⊂ H
x ∈ H x
k
x
k+
∈ T
k
(x
k+
)
T
k
(x) = T(x) + λ
k
(x − x
k
) λ
k
∈ ( λ) λ >
{x
k
} T
C D
g
C
S
∗
= ∅
x ∈ S
∇g(x) − ∇g(y) x ∈ S S H
g
X Y
X Y X Y T
X
∗
X
∗∗
x ∈ X
x ∈ X
∗∗
x(x
∗
) = x
∗
(x) ∀ x
∗
∈ X
∗
x = kx
k X X
∗∗
X = k(X)
k(X) = X
∗∗
H
|| · ||
{x
k
} ⊂ H x ∈ H x
k
→ x
=
k→∞
||x
k
− x|| =
k→∞
x
k
− x x
k
− x
{x
k
} ⊂ H x ∈ H x
k
x
=
k→∞
x
k
− x y ∀ y ∈ H
x
k
→ x x
k
x
ε > n ∈ N ||x
k
− x|| <
ε
||y||
|x
k
− x y| ||x
k
− x||||y|| ε ∀ k n
H = l = {x = {x
k
}
∞
k=
(x
k
) < ∞} com x y =
∞
k=
x
k
y
k
{e
k
} ⊂ H e
k
n
= δ
kn
e
k
k→∞
e
k
−
y =
k→∞
e
k
y =
k→∞
y
k
=
∞
k=
(y
k
) < ∞
||e
k
|| = ∀ k k = j e
k
e
j
= ||e
k
− e
j
|| =
√
{e
k
}
C ⊂ H
{x
k
} ⊂ C
C C ⊂ H
{x
k
} ⊂ C C
H
H
[ ]
{x
k
} ⊂ H {x
k
}
[ ]
C ⊂ H H f : C → R
f C
minf(x) sujeito a x ∈ Ck
C f
x ∈ C
f(x) f(x) ∀ x ∈ C
V x
f(x) f(x) ∀ x ∈ C ∩ V
x = x x
f : C → R C
{x
k
} ⊂ C ||x
k
|| → ∞ x
k
→ x ∈ C k → ∞
k→∞
f(x
k
) = ∞
f : C → R
x ∈ C ⊂ H {x
k
} ⊂ H x
k
x
k → ∞
k→∞
f(x
k
) f(x)
f
k→∞
f(x
k
) f(x)
f C
C
H C ⊂ H
f : C → R
f C f
x ∈ C
f(x ) =
x∈C
f(x)
C = ∪
∞
n=
f
−
(−n ∞) f
−
(−n ∞)
C
C = ∪
n
n=
f
−
(−n ∞)
n ∈ N f(x) > −n x ∈ C f
c =
x∈C
f(x) > −∞ c f(x) ∀ x ∈ C
{x
k
} ⊂ C c =
k→∞
f(x
k
) C
{x
k
} x
k
j
x ∈ C c =
k
j
→∞
f(x
k
j
) f(x ) f(x ) =
c
H f : H → R
H f H
f R > f(x) f( )
x ∈ H ||x|| R
B
R
( )
f f : B
R
( ) → R
x ∈ B
R
( ) f(x ) =
x∈H
f(x) R
C
H f : H → R
x ∈ C f(x) =
x∈C
f(x)
f : C → R c ∈ R
(c) = {x ∈ C f(x) c}
C ⊂ H f : C → R C
c ∈ R (c)
C ⊂ H
C C C ⊂ H
x y ∈ C α ∈ R α
αx + ( − α)y ∈ C
α x + + α
k
x
k
α
i
i = k
k
i=
=
x x
k
αx + ( − α)y C
H
H {x ∈ H a x c} a ∈ H c ∈ R
H {x ∈ H a x = c} a ∈ H c ∈ R
C ⊂ H p ∈ N
x
i
∈ C α ∈ [ ] i = p
p
i=
α
i
=
p
i=
α
i
x
i
∈
C
(⇒) C ⊂ H
p ∈ N
p ∈ N x
i
∈ C e α
i
∈ [ ]
p
i=
α
i
= x =
p
i=
α
i
x
i
p = α = e x = x ∈ C
p = n p = n +
n+
i=
α
i
= −
n
i=
α
i
= α
n+
α
n+
= α
i
= ∀ i = n
x =
p
i=
x
i
+ x
n+
= x
n+
∈ C
α
n+
∈ [ ) ( − α
n+
) >
x =
n+
i=
α
i
x
i
=
n
i=
α
i
x
i
+ α
n+
x
n+
= ( − α
n+
)
n
i=
α
i
( − α
n+
)
x
i
+ α
n+
x
n+
y =
n
i=
β
i
x
i
β
i
=
α
i
− α
n+
i = n
n+
i=
α
i
= − α
n+
=
n
i=
α
i
n
i=
β
i
=
− α
n+
n
i=
α
i
=
− α
n+
( − α
n+
) =
y ∈ C
y
x = ( − α
n+
)y + α
n+
x
n+
C x =
n+
i=
α
i
x
i
∈ C x =
p
i=
α
i
x
i
∈ C ∀ p ∈
N com x
i
∈ C e α
i
∈ [ ]
p
i=
α
i
=
(⇐)
p
i=
α
i
x
i
∈ C ∀ p ∈ N com x
i
∈ C e α
i
∈ [ ]
p
i=
α
i
=
p =
i=
α
i
x
i
= α x + α x ∈ C comx x ∈ C e α α ∈ [ ] tal que α + α =
α = − α ( − α )x + α x = α x + α x ∈ C
C
C
j
⊂ H j ∈ I I
C = ∩
j∈I
C
j
x y ∈ C C C
j
x y ∈ C
x y ∈ C
j
j ∈ I C
j
j ∈ I
α ∈ [ ] αx + ( − α)y ∈ C
j
j ∈ I αx + ( − α)y ∈ C
C
C C C
x y ∈ C α ∈ [ ] {x
k
}
k∈N
{y
k
}
k∈N
k→∞
x
k
= x
k→∞
y
k
= y C αx
k
+
( − α)y
k
∈ C k ∈ R αx + ( − α)y ∈ C αx + ( − α)y =
k→∞
[αx
k
+ ( − α)y
k
] αx + ( − α)y ∈ C C
x y ∈ C ε > ε > B(x ε ) ⊂ C B(y ε ) ⊂ C
ε = {ε ε } B(x ε) ⊂ C B(y ε) ⊂ C
αx + ( − αy) ∈ C
B(αx + ( − α)y ε) ⊂ C
z ∈ B(αx + ( − α)y ε) ||z − αx − ( − α)y|| < ε q = z − αx − ( − α)y
z = αx + ( − α)y + q ||q|| < ε z = αx + αq + ( − α)y + q − αq =
α(x + q) + ( − α)(y + q) ||x + q − x|| = ||q|| < ε x + q ∈ B(x ε) ⊂ C
||y + q − y|| = ||q|| < ε y + q ∈ B(y ε) ⊂ C
C z = α(x + q) + ( − α)(y + q) ∈ C z ∈ C B(αx + ( − α)y ε) ⊂ C
αx + ( − α)y ∈ C C
C ⊂ H C
C H C
H C
C C ∈ H C
C C
K ⊂ H
d ∈ K ⇒ td ∈ K ∀ t >
K K
C ⊂ H x ∈ C
x C
N
C
=
{d ∈ H d x − x ∀ x ∈ C} x ∈ C
∅ x ∈ C
K ⊂ H
K
∗
= {y ∈ H y d ∀ d ∈ K}
d ∈ H
C x ∈ C
dist(x + td C) = o(t) t >
T
C
(x) C x
∅ = K ⊂ H
(K
∗
)
∗
= conv (K)
K K = (K
∗
)
∗
∅ = C ⊂ H
(N
C
(x))
∗
= ((T
C
(x))
∗
)
∗
= T
C
(x)
C ⊂ H f :
C → R ∪ {∞} C x y ∈ C α ∈ [ ]
f(αx + ( − α)y) αf(x) + ( − α)f(y)
f
x = y α ∈ ( )
f γ > x y ∈ C
α ∈ [ ]
f(αx + ( − α)f(y)) αf(x) + ( − α)f(y) − γα( − α)||x − y||
f : R → R f(x) = x
f : R → R f(x) = e
x
f : R → R f(x) = x
D(f) = {x ∈ H f(x) < ∞}
f
E
f
= {(x c) ∈ C × R f(x) c}
f
f D(f) = ∅
f : H → R ∪{∞}
H × R
C ⊂ H f : C → R ∪{∞}
C f H × R
f E
f
x y ∈ C (x f(x)) ∈ E
f
(y f(y)) ∈ E
f
f E
f
α ∈ [ ]
(αx + ( − α)y αf(x) + ( − α)f(y) = α(x f(x)) + ( − α)(y f(y)) ∈ E
F
f(αx + ( − α)y) αf(x) + ( − α)f(y)
f
⇐ f (x c ) ∈ E
f
(y c ) ∈ E
f
f(x) c f(y) c f α ∈ [ ]
f(αx + ( − α)y) αf(x) + ( − α)f(y)
αc + ( − α)c
α(x c ) + ( − α)(y c ) = (αx + ( − α)y αc + ( − α)c ) ∈ E
f
f E
f
f(x) sujeito a x ∈ C
C ∈ H f : C → R ∪ {∞}
C
C ⊂ H f : C → R
C
f
x
∗
∈ C
y ∈ C f(y) < f(x
∗
)
x(α) = αy + ( − α)x
∗
f α ∈ ( ]
f(x(α)) αf(y) + ( − α)f(x
∗
)
= f(x
∗
) + α(f(y) − f(x
∗
))
< f(x
∗
)
α > x(α)
x
∗
f(x(α)) < f(x
∗
) x ∈ C
x
∗
S ⊂ C v
∗
∈ R
f(x
∗
) = v
∗
∀ x
∗
∈ S x x
∗
∈ S α ∈ [ ] f
f(αx + ( − α)x
∗
) αf(x) + ( − α)f(x
∗
)
= αv
∗
+ ( − α)v
∗
= v
∗
f(αx + ( − α)x
∗
) = v
∗
αx + ( − α)x
∗
∈ S
S
f x x
∗
∈ S x = x
∗
α ∈ ( ) x x
∗
αx+ ( − α)x
∗
∈ C
C
f(αx + ( − α)x
∗
) f(x) = f(x
∗
) = v
∗
f(αx + ( − α)x
∗
) < αf(x) + ( − α)f(x
∗
)
= αv
∗
+ ( − α)v
∗
= v
∗
f x
H ∇f(x) d ∈ H
α→
f(x + αd) − f(x)
α
= d ∇f(x)
∇f(x) f
x f x
d f
(x d)
∇f(x) f
(x d) d ∈ H
f
(x d) = d ∇f(x)
f x
f x f
(x ·) : H → R
C ⊂ H
f : C → R C
f C
x ∈ C y ∈ C
f(y) f(x) + ∇f(x) y − x
x ∈ C y ∈ C
∇f(y) − ∇f(x) y − x
(a) ⇒ (b) ⇒ (c)
f x y ∈ C α ∈ ( ] d = y − x
f(x + αd) = f(αy + ( − α)x) αf(y) + ( − α)f(x)
α(f(y) − f(x)) f(x + αd) − f(x)
α >
α →
+
f(y) − f(x)
α→
+
f(x + αd) − f(x)
α
= ∇f(x) d = ∇f(x) y − x
f(y) − f(x) ∇f(x) y − x f(y) f(x) + ∇f(x) y − x
x y
f(x) f(y) + ∇f(y) x − y
(c) ⇒ (b) ⇒ (a)
x y ∈ C α ∈ ( )
f(y) − f(x) = ∇f(x + α(y − x)) y − x
(x + α(y − x))
∇f(x + α(y − x) y − x = α
−
∇f(x + α(y − x)) α(y − x)
α
−
f(x) α(y − x)
= ∇f(x) y − x
d = y − x
f(x) f(x + αd) − α∇f(x + αd) d
f(y) f(x + αd) + ( − α)∇f(x + αd) d
(b) x x + αd x + αd
− α α
( − α)f(x) + αf(y) ( − α)(f(x + αd) − α∇f(x + αd) d)
+α(f(x + αd) + ( − α)f(x + αd) d)
= f(x + αd)
= f(( − α)x + αy)
f
f : H → R x ∈ H
d ∈ D(f(x)) ∇f(x) d
d ∈ H ∇f(x) d < d ∈ D(f(x))
D(f(x)) = {x ∈ H f(x) < ∞} f x
d ∈ D(f(x)) t >
> f(x + td) − f(x) = t(∇f(x) d + o(t)/t)
t >
t →
+
∇f(x) d
∇f(x) d <
f(x + td) − f(x) = t(∇f(x) d + o(t)/t)
t >
∇f(x) d + o(t)/t ∇f(x) d/ <
f(x + td) − f(x) < d ∈ D(f(x))
f : H → R
x
k
x
f(x
k
) f(x) ∀ x ∈ H
f E
f
E
f
H ×R H ×R
H × R H R f
H f : H → R ∪{∞}
C ⊂ H D(f)∩C = ∅
f C
H f : H → R ∪{∞}
||x||→∞
f(x) = ∞
f H
{f(x) x ∈ H} > −∞
R >
{f(x) x ∈ H} = {f(x) ||x|| R}
f {x ||x|| R}
H
f : H → R ∪{∞}
x ∈ D(f)
y ∈ H f x
f(x) f(x ) + y x − x ∀ x ∈ H
f x f
x ∂f(x ) x ∈ D(f) ∂f(x) = ∅
∂f
D(∂f) = {x ∈ H ∂f(x) = ∅}
f
f f x
f : H → R ∪ {∞}
x ∈ H ∂f(x)
d ∈ H
f
(x d) = {y d y ∈ ∂f(x)}
[ ]
f : H → R x ∈ H
∂f(x) ∂f(x) =
{∇f(x)}
f x ∈ H
f(z) f(x) + ∇f(x) z − x ∀ z ∈ H
∇f(x) ∈ ∂f(x) f
x
y ∈ ∂f(x) d ∈ H
f(x) + y d f(x + d) = f(x) + ∇f(x) d + o(||d||)
y − ∇f(x) d o(||d||)
y − ∇f(x) d
o(||d||)
||d||
||d||
d →
y − ∇f(x) d ∀ d ∈ H
d = y − ∇f(x)
y − ∇f(x) y − ∇f(x)
y − ∇f(x) = ∂f(x) =
{∇f(x)}
∂f(x) = {y}
∂f(x)
∂d
= y d ∀ d ∈ H
d H
y
i
= ∂f(x)/∂x
i
∀ i ∈ N
∂f(x)
∂d
d f
x
f e g
H (D(f)) ∩ D(g) = ∅
∂(f + g) = ∂f + ∂g
∂f + ∂g ⊂ ∂(f + g)
∂(f+g) ⊂ ∂f+∂g x ∈ D(∂f)∩D(∂g) w ∈ ∂(f+g)(x )
w = w + w w ∈ ∂f(x ) e w ∈ ∂g(x )
x = w = e f( ) = g( ) =
f e g x → f(x + x ) − f(x ) −z x e x → g(x + x ) − g(x ) − z x
∈ ∂f( ) + ∂g( )
E = {(x λ) ∈ H × R f(x) λ}
E = {(x λ) ∈ H × R g(x) −λ}
∈ ∂(f + g)( )
= (f + g)( ) = {f(x) + g(x) x ∈ H}
E ∩E = ∅ [ ]
H × R E e E w α ∈ H × R
x w + αλ ∀ (x λ) ∈ E
x w + αλ ∀ (x λ) ∈ E
α =
{x ∈ H x w = } D(f) e D(g)
α =
x w −λ −f(x) ∀ x ∈ D(f)
x w −λ g(x) ∀ x ∈ D(g)
−w ∈ ∂f( ) w ∈ ∂g( )
f : H → R C ⊂ H
x ∈ H f C
∃ y ∈ ∂f(x) tal que y x − x ∀ x ∈ C
∈ ∂f(x) + N
C
(x)
x f H
∈ ∂f(x)
y x − x x ∈ C −y ∈ N
C
(x)
∀ x ∈ H f(x) f(x) + y x − x f(x)
x f C
x f C d ∈ T
C
(x)
d = {t
k
} {d
k
} ⊂ H
{d
k
} → d {t
k
} →
+
k → ∞ x + t
k
d
k
∈ C k k
f(x + t
k
d
k
) − f(x) = t
k
∂f(x)
∂d
k
+ o(t
k
)
t
k
> k → ∞
∂f
∂d
(x) ∀ d ∈ T
C
(x)
y ∈ H −y ∈ ∂f(x)
y ∈ N
C
(x)
(−∂f(x)) ∩ N
C
(x) = ∅
∂f(x) N
C
(x)
a ∈ H − { } c ∈ R
−a y > c > a d ∀ y ∈ ∂f(x) ∀ d ∈ N
C
(x)
∈ N
C
(x) c > a y < −c < y ∈ ∂f(x)
∂f
∂a
(x) =
y∈∂f(x)
a y <
a d > d ∈ N
C
(x) td ∈ N
C
(x) t >
t → ∞ c ∈ R
a d d ∈ N
C
(x)
a ∈ (N
C
(x))
∗
= ((T
C
(x))
∗
)
∗
= T
C
(x)
C = H
∈ ∂f(x)
S ⊂ H g : H → R
D
g
: S × S → R
+
D
g
(x y) = g(x) − g(y) − ∇g(y) x − y
∇g(y) g y
g D
g
g
B ) g S
B ) g S
B ) Γ (y δ) = {x ∈ S D
g
(x y) δ} Γ (x δ) = {y ∈ S D
g
(x y) δ}
δ > y ∈ S x ∈ S
B ) {x
k
} {y
k
} ⊂ S x
k
x y
k
x
k→∞
D
g
(x
k
y
k
) =
k→∞
[D
g
(x x
k
)−
D
g
(x y
k
)] =
B ) {x
k
} ⊂ S {y
k
} ⊂ S y
k
y
k→∞
D
g
(x
k
y
k
) =
x
k
y
g B − B
B ) {x
k
} ⊂ S {y
k
} ⊂ S x
k
x y
k
y x = y
k→∞
|∇g(x
k
) − ∇g(y
k
) x − y| >
g B − B
B ) y ∈ H x ∈ S ∇g(x) = y
g B − B
B ) {y
k
} ⊂ S S x
S
k→∞
D
g
(x y
k
) = ∞ ∀ x ∈ S S g
D
g
(x y) x ∈ S y ∈ S D
g
(x y) =
x = y
g : H → R g(x) = ||x|| D
g
(x y) = ||x − y||
S = R
n
++
g(x) =
n
i=
x
i
x
i
R
n
++
=
D
g
(x y) =
n
i
(x
i
x
i
y
i
+ y
i
− x
i
)
g S
D
g
(x y) − D
g
(x z) − D
g
(z y) = ∇g(y) − ∇g(z) x − y ∀x ∈ S ∀y z ∈ S
∇
x
D
g
(x y) = ∇g(x) − ∇g(y) ∀ x y ∈ S
D
g
(x y) y ∈ S
x ∈ S y z ∈ S
D
g
(x y) − D
g
(x z) − D
g
(z y) = g(x) − g(y) − ∇g(y) x − y
−g(x) + g(z) + ∇g(z) x − z
−g(z) + g(y) + ∇g(y) z − y
= ∇g(y) z − y + y − x + ∇g(z) x − z
= ∇g(y) z − x − ∇g(z) z − x
= ∇g(y) − ∇g(z) z − x
x z ∈ S D
g
(x y) = g(x) − g(y) − ∇g(y) x − y
∇
x
D
g
(x y) = ∇g(x) − −
∂
∂x
∇g(y) x − y − ∇g(y)
∂
∂x
(x − y)
= ∇g(x) − x − y − ∇g(y)
= ∇g(x) − ∇g(y)
y ∈ S
G : S −→ R
x −→ G(x) = D
g
(x y)
x x ∈ S x = x ∇G(x ) = ∇g(x ) − ∇g(y) ∇G(x ) =
∇g(x ) − ∇g(y)
∇G(x ) − ∇G(x ) x − x = ∇g(x ) − ∇g(y) − ∇g(x ) + ∇g(y) x − x
= ∇g(x ) − ∇g(x ) x − x >
g S G = D
g
(· y) S
y ∈ S
D
g
( y)
g D
g
( y)
g(x) = D
g
(x y) = ∞ ∀ x ∈ S
g S
y ∈ S
∂
x
D
g
(x y) =
{∇g(x) − ∇g(y)} x ∈ S
∅ x ∈ S
∂
x
D
g
(x y) x ∈ S
x ∈ S ∂
x
D
g
(x y) = ∅
x S
f
y
(·) = D
g
(· y) x ∈ ∂S ξ ∈ ∂f
y
(·)
f
y
g f
y
B
S
g B z ∈ S ∇f
y
(z) = ∇g(z) −
∇g(y) = ξ
= ξ − ξ z − x = ∇f
y
(z) − ξ z − x com ξ ∈ ∂f
y
(x)
f
y
x = z z ∈ S x ∈ S S
∂f
y
(x) = ∅
g
{ε
k
} ⊂ ( )
k→∞
ε
k
= x
k
= ( − ε
k
)x + ε
k
y S
x
k
∈ S ∀ k
ε
k
ξ y − x = ξ x
k
− x
f
y
(x
k
) − f
y
(x)
= g(x
k
) − g(y) − ∇g(y) x
k
− y − [g(x) − g(y) − ∇g(y) x − y]
= g(x
k
) − g(x) + ∇g(y) x − x
k
−∇g(x
k
) x − x
k
+ ∇g(y) x − x
k
= ∇g(y) − ∇g(x
k
) x − x
k
= ∇
g
(x
k
) − ∇
g
(y) x
k
− x
x
k
= ( − ε
k
)x + ε
k
y
x
k
− ε
k
x
k
= ( − ε
k
)x + ε
k
y − ε
k
x
k
x
k
− x =
ε
k
− ε
k
(y − x
k
)
∇g(u) − ∇g(v) u − v = Dg(u v) + Dg(v u)
∀ u v ∈ S
ε
k
ξ y − x ∇g(x
k
) − ∇g(y)
ε
k
− ε
k
(y − x
k
)
=
ε
k
− ε
k
∇g(x
k
) − ∇g(y) y − x
k
= −
ε
k
− ε
k
∇g(x
k
) − ∇g(y) x
k
− y
= −
ε
k
− ε
k
(D
g
(x
k
y) + D
g
(y x
k
))
ε
k
ξ y − x −
ε
k
− ε
k
(D
g
(x
k
y) + (D
g
(y x
k
))
ε
k
−ε
k
> ∀ k
( − ε
k
)ξ y − x −(D
g
(x
k
y) + D
g
(y x
K
))
( − ε
k
)ξ y − x + D
g
(x
k
y) −D
g
(y x
k
)
( − ε
k
)ξ y − x + f
y
(x
k
) −D
g
(y x
k
)
x
k
x ∈ ∂S ⊂ S f
y
S
k → ∞
ξ y − x + D
g
(x y) B −∞
∂f
y
(x) = ∅
T : H → H
x − y T( x) − T (y) ∀ x y ∈ H
D(T) = {x ∈ H T (x) = ∅} T {(x u) x ∈
H u ∈ T(x)} T
T = ∇f f
T = ∇f
T : H → H
H H
x ∈ H H
T : H ⇒ H
T : H ⇒ H
x − y u − v ∀ x y ∈ H u ∈ T (x) v ∈ T(y)
T = ∂f f
T T : H → H D(T )∩D(T ) = ∅
T + T
T
T
T T(x) ⊂ T(x) x T = T
T
f : H → R ∪{∞}
T = ∂f
C
Proj
C
: H → C x ∈ H
C x H
C
T T : H ⇒ H
D(T ) ∩ (D(T )) = ∅ T + T
G(T) = {(y v) y ∈ H v ∈ T(y)} R(T) = ∪
x∈H
T(x)
T
T : H ⇒ H
∀ u ∈ R(T) e ∀ x ∈ D(T)
(y v)∈G(T )
v − u x − y < ∞
T = ∂f f T
[ ]
T T : H ⇒ H
R(T + T ) = ∪
u∈D(T )∩D(T )
(T u + T u)
R(T ) + R(T ) = ∪
u∈D(T ) v∈D(T )
(T u + T v)
X conv(X) X
T : H ⇒ H F ⊂ H
u ∈ F x ∈ H
(v y)∈G(T )
v − u x − y < ∞
conv(F) ⊆ R(T )
(conv(F)) ⊆ R(T )
u ∈ conv(F)
u =
λ
i
u
i
u
i
∈ F λ
i
x
i
∈ H e c
i
∈ R
v − u
i
x
i
− y c
i
∀ (y v) ∈ G(T )
v x
i
− v y + u
i
y c
i
+ u
i
x
i
x =
λ
i
x
i
u ∈ conv(F)
F ⊂ R(T ) e F ⊂ R(T )
u ∈ F y
ε
εy
ε
+ T y
ε
= u
y = y
ε
e v = u − εy
ε
u − εy
ε
− u x − y
ε
d
√
ε||y
ε
|| k d e k u ∈ R(T )
F ⊂ R(T )
u ∈ F
o
e B(u r) ⊂ F ||w|| < r x(w) ∈ H e c(w) ∈ R x
c w
v − u − w x(w) − y < c(w) ∀ (v y) ∈ G(T )
(u − εy
ε
y
ε
) ∈ G(T )
−εy
ε
− w x(w) − y
ε
< c(w)
w y
ε
c (w)
|w y
ε
| T ||y
ε
|| T
y
ε
y u − εy
ε
→ u u − εy
ε
∈ T (y
ε
)
u ∈ T (y) (conv(F)) ⊂ R(T )
T T
T
D(T ) ∩ D(T ) = ∅ e R(T ) = H
T + T
R(T + T ) = H
F = R(T ) + R(T ) T = T + T
F
F = R(T ) + R(T ) u ∈ R(T ) + R(T )
x ∈ D(T ) ∩ D(T ) w ∈ T (x)
u = w + (u − w)
w ∈ T (x)
R(T ) = H y ∈ H u − w ∈ T (y) T
u − w ∈ R(T ) e x ∈ D(T ) γ ∈ R
(z s)∈G(T )
s − (u − w) x − z γ
(z s) ∈ G(T )
s − (u − w) x − z γ
v ∈ T (z) z ∈ D(T ) ∩ D(T ) T
v − w z − x
w ∈ T (x) ) e ( )
(s + v) − u x − z γ
s ∈ T (z) v ∈ T (z) s + v ∈ (T + T )(z)
(z t)∈G(T +T )
t − u x − z +∞
F = R(T ) + R(T ) T = T + T
R(T + T ) = H
y ∈ C
∂
x
D
g
(· y) D
g
(· y) T : H ⇒ H
C g
C y ∈ C e λ > T(·) = T(·) + λ∂
x
D
g
(· y)
D(T) ∩ C = ∅
T
R(T) = H
T T
T = λ∂
x
D
g
(· y) T = T
T = λ∂
x
D
g
(· y) D(T ) ∩D(T ) = ∅ T = T + T
T = T T = λ∂
x
D
g
(· y)
λ∂
x
D
g
(· y)
D(T) ∩ C = ∅ R(∂
x
D
g
(· y)) = H B
T
R(T) = H
T : H ⇒ H D(T )
T
[ ]
T : H ⇒ H T
u − v x − y = u ∈ T(x) v ∈ T(y) u ∈ T(y) v ∈ T (x)
T = ∂f(x) f : H → R T
∂f x y ∈ H
u − v x − y = u ∈ ∂f(x) v ∈ ∂f(y) f : H → R
f(z) := f(z) + u x − z
f z w ∈ H α ∈ [ ]
f(αz + ( − α)w) = f(αz + ( − α)w) + u x − αz − ( − α)w
αf(z) + ( − α)f(w) + u x + αx − αx − αz − ( − α)w
= αf(z) + ( − α)f(w) + αu x − z + ( − α)u x − w
= α[f(z) + u x − z] + ( − α)[f(w) + u x − w]
= αf(z) + ( − α)f(w)
f
∂f(z) = ∂f(z) − {u} = {w − u w ∈ ∂f(z)} z = x f(x) = f(x)
w = u ∈ ∂f(x) ∈ ∂f(x) x f H
u − v x − y =
u x − y = v x − y
u ∈ ∂f(x)
u y − x f(y) − f(x)
f(x) − f(y) u x − y
v ∈ ∂f(y)
v x − y f(x) − f(y)
f(x) − f(y) u x − y = v x − y f(x) − f(y )
(
f(x) − f(y) = u x − y
f(x) = f(y) + u x − y = f(y)
f(x) = f(x) f(x) = f(y) y f
∈ ∂f(y)
= w − u w ∈ ∂f(y) u ∈ ∂f(y)
g : H → R
g(z) := f(z) + v y − z
f g ∂g(z) = ∂f(z) − {v} = {w − v w ∈ ∂f(z)}
z = y g(y) = f(y) w = v ∈ ∂f(y) ∈ ∂g(y)
y g H
( f(y) = f(x) + v y − x = g(x) g(y) = f(y) g(y) = g(x)
x g H
∈ ∂g(x)
= w − v w ∈ ∂f(x) v ∈ ∂f(x)
∂f u − v x − y = u ∈ ∂f(x) v ∈ ∂f(y)
u ∈ ∂f(y) v ∈ ∂f(x) T = ∂f
T : H ⇒ H T : H ⇒ H
D(T ) ∩ D(T ) = ∅ T + T
x y ∈ H u ∈ (T + T )(x) v ∈ (T + T )(y)
u ∈ (T + T )(x) = T (x) + T (x) u ∈ T (x) u ∈ T (x) u = u + u
v ∈ T (y) v ∈ T (y) v = v + v
T + T u − v x − y = u − u − v −
v x − y = u − v x − y + u − v x − y = T T
u − v x − y = u − v x − y = T T
u ∈ T (y) u ∈ T (y) v ∈ T (x) v ∈ T (x)
u = u + u ∈ T (y) + T (y) = (T + T )( y) v = v + v ∈ T (x) + T (x) =
(T + T )(x)
T + T
H G
D(T) ⊂ H T : H ⇒ H G
{x
k
} ⊂ G x ∈ G
{w
k
} ⊂ H w
k
∈ T(x
k
) ∀ k
k→∞
w
k
x
k
− x
y ∈ G w ∈ T (x )
w x − y
k→∞
w
k
x
k
− y
S e S T : S ⇒ S
T S s
S V T(s ) S U s S
U V T(U) ⊂ V
f : H → R ∪ ∞
T : H ⇒ H T = ∂f T
D(T) H
{x
k
} D(∂f)
k→∞
x
k
= x e
k→∞
u
k
x
k
− x
u
k
∈ ∂f(x
k
)
u
k
y − x
k
f(y) − f(x
k
) y ∈ D(∂f)
k ∈ H f
k→∞
u
k
x
k
− y
k→∞
f(x
k
) − f(y) = f(x) − f(y)
f u ∈ ∂f(x) f(y) f(x) + u y − x
u x − y f(x) − f(y)
u ∈ ∂f(x)
u x − y
k→∞
u
k
x
k
− y
T = ∂f
T : H → H
T S : H ⇒ H
D(T) ∩ D(S) = ∅ T + S
{x
k
} D(T + S)
k→∞
x
k
= x ∈ D(T + S)
k→∞
u
k
x
k
− x
u
k
∈ (T + S)(x
k
)
y ∈ D(T + S)(x
k
) u
k
T
∈ T(x
k
) u
k
S
∈ S(x
k
)
u
k
= u
k
T
+ u
k
S
( )
k→∞
u
k
T
+ u
k
S
x
k
− x
k→∞
u
k
T
x
k
− x
k→∞
u
k
T
+ u
k
S
x
k
− x
k→∞
u
k
S
x
k
− x
k→∞
u
k
T
+ u
k
S
x
k
− x
T S u
T
∈ T(x) u
S
∈ S(x)
u
T
x − y
k→∞
u
k
T
x
k
− x
u
S
x − y
k→∞
u
k
S
x
k
− x
t ∈ D(T + S)
u
T
+ u
S
x − y
k→∞
u
k
T
+ u
k
S
x
k
− x
u = u
T
+ u
S
∈ T(x) + S(x) = (T + S)(x)
u x − y
k→∞
u
k
T
+ u
k
S
x
k
− x
y ∈ D(T + S) T + S
C H g
C = ∅ {λ
k
}
λ
k
> T
x ∈ C
x
k
T
k
: H ⇒ H T
k
(·) = T (·) + λ
k
D
g
(· x
k
)
x
k+
∈ H
∈ T
k
(x
k+
)
x
k+
T g
{x
k
} D(T)∩C = ∅
g C {x
k
}
S
k
x ∈ C
x
k
∈ C x
k+
∈ C
B
k
(·) = λ
k
∂D
g
(· x
k
) T
k
= T + B
k
T
k
= TB
k
T B
k
g R(T
k
) = H T
k
D(T
k
)
x
k+
C D(T
k
) = D(T) ∩ C
x
k+
∈ D(T
k
) x
k+
∈ C {x
k
} ⊂ C
{x
k
} D(T) ∩C =
∅ h
T C
(x) < ∞ A x ∈ C ∩ D(T)
g {x
k
} C
R(T
k
) = H
g
g h
T C
(x) C ∩ D(T )
k = x ∈ C
x
k
∈ C x
k
∈ D(T) ∩ C ∈ T
k−
(x
k
)
x
k
∈ D(T) ∩ C h
T C
h
T C
(x
k
) < ∞ h := h
T C
h(x
k
) =
v x
k
− y ∀ y ∈ C e ∀ v ∈ T (y)
u T (x
k
) y C T
u − v x
k
− y
u y − x
k
v y − x
k
x
k
∈ S
∗
x
k+
:= x
k
∈ T
k
(x
k+
)
T
k
(x
k+
) = T(x
k
) + λ
k
∂D
g
(x
k
x
k
)
= T (x
k
) + λ
k
(∇g(x
k
) − ∇g(x
k
))
= T (x
k
)
x ∈ D(T) ∩ C ∩ S
∗
x T
x
k+
= x
k
C
h(x
k
) >
S
k
:= {x ∈ C D
g
(x x
k
)
h(x
k
)
λ
k
}
S
k
x
S
k
x ∈ C e D
g
(x x
k
) <
h(x
k
)
λ
k
x
k
∈ S
k
D
g
(x
k
x
k
) = <
h(x
k
)
λ
k
x
k
∈ C
N
k
:= N
S
k
S
k
D(N
k
) =
S
k
B B
k
(·) := N
k
(·) + λ
k
∂
x
D
g
(· x
k
)
B
k
D(N
k
)∩
[D(∇
x
D
g
(· x
k
))] = ∅
x
k
∈ D(N
k
) ∩ C = S
k
∩ D(∂D
g
(· x
k
)) D(N
k
) ∩
[D(∇D
g
(· x
k
))] = ∅ B
k
D(B
k
) S
k
D(N
k
) =
S
k
R(B
k
) = H
A
k
:= T + B
k
A
k
D(T) ∩ (D(B
k
)) = ∅ D(T) ∩ (D(B
k
)) = ∅
x
k
∈ D(T) ∩ C ∩S
K
= D(T) ∩ D(B
k
)
A
k
A
k
D(B
k
)
y ∈ D(A
k
) = D(T) ∩ D(B
k
) ⊂ D(T) ∩ C
∈ T(y) + N
k
(y) + λ
k
D
g
(y x
k
)
u
k
w
k
e v
k
H
u
k
∈ T(y) w
k
∈ N
k
(y) v
k
∈ λ
k
∂D
g
(y x
k
)
= u
k
+ w
k
+ v
k
y ∈ S
k
y ∈ D(A
k
) ⊂ D(B
k
) ⊂ (∂
x
D
g
(· x
k
)) = C
x ∈ C ⇔ D
g
(x x
k
) <
h(x
k
)
λ
k
D
g
(y x
k
) <
h(x
k
)
λ
k
D
g
(· x
k
)
= D
g
(x
k
x
k
) > D
g
(y x
k
) +
v
k
λ
k
x
k
− v
λ
k
(u
k
x
k
− y + w
k
x
k
− y) > D
g
(y x
k
)
w
k
∈ N
k
(y) e x
k
∈ S
k
w
k
x
k
− y
λ
k
D
g
(y x
k
) < u
k
x
k
− y
u
k
x
k
− y
z∈C∩D(T ) v∈T z
v x
k
− z = h(x
k
)
D
g
(y x
k
) <
h(x
k
)
λ
k
y ∈ S
k
N
k
(y) = w
k
= = u
k
+ v
k
∈ T
k
(y) T
k
y T
k
y = x
k+
y ∈ C x
k+
∈ C
) e ( )
T : H ⇒ H
C H g
C
D(T) ∩ C = ∅
S
∗
= ∅
T
λ
k
∈ ( λ) λ >
g g h
T C
< ∞ ∀ x ∈ C ∩ D(T )
) e ( )
{D
g
(z x
k
)} ∀ z ∈ S
∗
{x
k
}
k→∞
D
g
(x
k+
x
k
) =
x {x
k
} u ∈ T(x) u x
∗
−u =
D
g
(z x
k+
) D
g
(z x
k
) − D
g
(x
k+
x
k
) ∀ z ∈ S
∗
z ∈ S
∗
∃v
∗
∈ T(z)
v
∗
x − z para todo x ∈ C
e x
k+
∈ C e u
k
∈ T(x
k+
)
u
k
= λ
k
(∇g(x
k
) − ∇g(x
k+
))
y ∈ C
∇g(x
k
) − ∇g(x
k+
) x
k+
− y = D
g
(y x
k
) − D
g
(y x
k+
) − D
g
(x
k+
x
k
)
( ) e ( )
u
k
x
k+
− y = λ
k
D
g
(y x
k
) − D
g
(y x
k+
) − D
g
(x
k+
x
k
)
z ∈ S
∗
T
u
k
x
k+
− z v
∗
x
k+
− z
y = z
D
g
D
g
(z x
k+
) D
g
(z x
k
) − D
g
(x
k+
x
k
)
D
g
(z x
k
) D(z x ) ∀ k e ∀ z ∈ S
∗
z ∈ S
∗
{x
k
} {x ∈ C D
g
(z x) D
g
(z x )} B
{x
k
}
k→∞
D
g
(x
k+
x
k
) =
D
g
(x
k+
x
k
) D
g
(z x
k
) − D
g
(z x
k+
)
z ∈ S
∗
{D
g
(z x
k
)}
k→∞
D
g
(x
k+
x
k
) =
{x
k
}
{x
k
} C
A ⊂ H
A = A
w
A
w
A [ ]
C C C = C
w
C
C x
{x
k
j
} ⊂ {x
k
}
x
k
j
x
y = x k = k
j
u
k
j
x
k
j
+
− x = λ
k
j
D
g
(x x
k
j
) − D
g
(x x
k
j
+
) − D
g
(x
k
j
+
x
k
j
)
B e B
j → ∞
d ) {x
k
j
}
d ) x
k
j
x
d )
k→∞
D
g
(x
k
j
+
x
k
j
) = B
{x
k
j
+
} e {x
k
j
}
x
k
j
+
x
(d ) e (d ) B {x
k
j
+
} e {x
k
j
}
j→∞
D
g
(x x
k
j
+
) − D
g
(x x
k
j
)
=
{λ
k
} (d ) ( ) em ( )
j→∞
u
k
j
x
k
j
+
− x =
x
k
j
+
x e u
k
j
∈ T(x
k
j
+
) ∀ j
( ) e ( ) z ∈ S
∗
u ∈ T(x)
u x − z
j→∞
u
k
j
x
k
j
+
− z
y = z ∈ S
∗
u
k
x
k+
− z = λ
k
D
g
(z x
k
) − D
g
(z x
k+
) − D
g
(x
k+
x
k
)
D
g
(z x
k
)
{λ
k
}
j → ∞
u x − z
u ∈ T (x) z ∈ S
∗
v
∗
∈ T(z) v
∗
y − z y ∈ C
T
u x − z v
∗
x − z
u x − z =
D(T)
C T C
T : H ⇒ H C
H g C
B − B
D(T) ∩ C = ∅
S
∗
= ∅
T D(T) H
λ
k
∈ ( λ] λ >
g g h
T C
(x) < ∞ ∀ x ∈ C ∩ D(T )
T C
{x
k
} ( ) e ( )
x
g {x
k
}
(b) {x
k
}
x {x
k
}
u ∈ T(x)
u x
∗
− x = ∀ x
∗
∈ S
∗
x
∗
u
∗
∈ T(x
∗
)
u
∗
x − x
∗
∀ x ∈ C
x
u
∗
x − x
∗
T u ∈ T(x) e u
∗
∈ T(x
∗
)
u − u
∗
x − x
∗
= u x − x
∗
u
∗
x − x
∗
u x − x
∗
u
∗
x − x
∗
=
u − u
∗
x − x
∗
=
u ∈ T(x) e u
∗
∈ T(x
∗
)
T
u ∈ T(x
∗
) e u
∗
∈ T(x)
u
∗
x − x = u
∗
x − x
∗
+ u
∗
x
∗
− x
x x
{x
k
} {x
k
}
g
z ∈ S
∗
β
k
(z) :=
∇g(x
k
) x
k
− z − g(z) β
k
(z) −g(z) D
g
(z x
k
)
{β
k
(z)}
z z ∈ S
∗
{x
j
k
} e {x
i
k
}
{x
k
} x
j
k
z e x
i
k
z {β
k
(z )} e {β
k
(z )}
l l ∈ R
k→∞
β
k
(z ) = l e
k→∞
β
k
(z ) = l
k→∞
(β
k
(z ) − β
k
(z )) =
k→∞
∇g(x
k
) z − z = l − l
k = j
k
k = i
k
k→∞
g(x
j
k
) − ∇g(x
i
k
) z − z =
z = z
g {x
k
}
S
∗
{x
k
}
{x
k
}
V N
V
D(N
V
)
{x
k
} {w
k
} x y x
k
y ∈ D(N
V
) = V
N
V
w
k
x
k
− y N
V
(x )
∈ N
V
(x
) w = w x − y =
k→∞
w
k
x
k
− y
T C
{x
k
} S
∗
= ∅ {x
k
}
{x
k
}
B ⊂ H {x
k
} ⊂ B T = T + N
B
N
B
B N
B
(x) = { } ∀ x ∈ B {x
k
}
T x = x
x
k
= x
k
∀ k
k =
x
k
= x
k
∈ T(x
k+
) + λ
k
∂
x
D
g
(x
k+
x
k
) = T(x
k+
) + λ
k
∂
x
D
g
(x
k+
x
k
)
T = T B x
k+
y
∈ T(y) + λ
k
∂
x
D
g
(y x
k
)
T(y) + λ
k
∂
x
D
g
(· x
k
) y
x
k+
∈ T (y) + λ
k
D
g
(y x
k
)
x
k+
= x
k+
T
T
T = T + ∂(δ
B
)
N
B
T
T T
T + N
C
= T + N
B
+ N
C
D(T + N
C
) = D(T) ∩C ∩B ⊂ B
B T + N
C
{x
k
} {x
k
}
T x
∗
{x
k
} {x
k
} x
∗
∈ B T(x) = T (x) ∀ x ∈ B
T(x
∗
) = T (x
∗
) x
∗
{x
k
}
{x
k
}
{x
k
} S
∗
= ∅
S
∗
= ∅ x
∗
{x
k
} S
∗
S
∗
= ∅ {x
k
}
S
∗
= ∅
S
∗
= ∅
C ⊂ H δ
C
: C →
R ∪ {∞}
δ
C
(x) =
x ∈ C
∞ x ∈ C
C
x ∈ V δ
C
= ∞ d ∈ R
d < ∞= δ
C
(x) (x d) ∈ E
δ
C
x ∈ C δ
C
(x) = δ
C
(x) d ∀ d ∈ [ +∞) (x d) ∈ E
δ
C
E
δ
C
= C × [ +∞)
δ
C
E
δ
C
C ×[ +∞) C
C E
δ
C
δ
C
x ∈ C δ
C
(x) = δ
C
(x) d ∀ d ∈ [ +∞)
s ∈ ∂f(x)
x ∈ C δ
C
(x) = ∞
δ
C
(y) ∞ ∀ y ∈ H
∂δ
C
(x) = ∅
x ∈ C δ
C
(x) =
δ
C
(y) s y − x ∀ y ∈ H
y ∈ C
s y − x +∞
y ∈ C δ
C
(y) =
s y − x
∂δ
C
(x) = {s ∈ H s y − x ∀ y ∈ C}
∂δ
C
(x) =
{s ∈ H s y − x ∀ y ∈ C} x ∈ C
∅ x ∈ C
δ
C
N
C
= ∂δ
C
C N
C
: H → H
δ
C
x N
C
= ∅ x ∈ C N
C
= se x ∈ C N
C
(x)
D(T) = {x ∈ H Tx = ∅}
T : H ⇒ H
D(N
C
) = C
N
C
T : H ⇒ H C = ∅
H D(T) ∩ C = ∅
h
T C
: D(T) ∩ C −→ R ∪ {∞}
h
T C
:=
(x y)∈G
C
(T )
v x − y
G
C
(T) = {(v y) v ∈ T(y) y ∈ C} T
h
T C
T C e h
T C
h
T C
D(T) ∩ C
h
T C
D(T ) ∩ H
h
T C
x = y A
h
T c
A H
D(T) ∩C = ∅ h
T C
S
∗
S
∗
h
T C
h
T C
h = h
T C
A := T + N
C
D(A) = D(T) ∩ C
h(x) = A y ∈ D(T) ∩ C v ∈ T y
v x − y
w ∈ N
C
(x) y ∈ C ∩ D(T)
w x − y
N
C
x ∈ C A A
− (v + w) x − y
y ∈ D(T) ∩C = D(A) v + w ∈ A(y) = T (y) + N + C
A A
∈ A(x) = T(x) + N
C
(x)
A x ∈ S
∗
x ∈ C ∩S
∗
y ∈ C v ∈ T x
v y − x
w ∈ Ty
w y − x
A
h(x) =
(y w)∈G
C
(T )
w x − y
A A h(x) =
C
T = N
C
δ
C
x ∈ C = D(T)
h
N
C
C
(x) =
(y v)∈G
C
(N
C
)
v x − y
N
C
h
N
C
(x) =
x ∈ C x ∈ C C = ∅
α =
a y − y y ∈ C
a x − y > λa ∈ N
C
(y ) λ >
h
N
C
C
(x) =
(y v)∈G
C
(N
C
)
v x − y λa x − y
λ → ∞ h
N
C
C
(x) = ∞ x ∈ C
h = δ
C
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