análisis_convexo_y_dualidad.pdf

8/19/2019 análisis_convexo_y_dualidad.pdf

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5/103

R = R ∪{−∞, +∞}

Γ


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7/103

R = R ∪{−∞,+∞}

R = R∪{−∞, +∞} = [−∞, +∞]

R =

] − ∞, +∞[ X

inf {f 0(x) | x ∈ C }

f 0 : X → R C ⊆ X

R A ⊆ R

inf A

sup A

A = {f 0(x) | x ∈ C } inf {f 0(x) | x ∈ C } R

inf C

f 0 := inf {f 0(x) | x ∈ C }.

A inf A ∈ A min A inf A

C δ C : X → R,

δ C (x) =

0 x ∈ C,+

∞ x /

∈ C.

α + (+∞) = (+∞) + α = +∞, ∀α ∈ R, f 0 + δ C (f 0 + δ C )(x) := f 0(x) + δ C (x)

inf C

f 0 = inf X

(f 0 + δ C )

C = ∅

inf C

f 0 ∈ {f 0(x) | x ∈ C } ⇔ inf X

(f 0 + δ C ) ∈ {f 0(x) + δ C (x) | x ∈ X }.


8/103

inf

{f (x)

| x

∈ X

},

f : X → R∪{+∞} f = f 0+ δ C f

R∪{−∞} R

λ ∈ R f, g : X → R

f + λg R R

(+∞) + α = α + (+∞) = +∞ ∀α ∈ R ∪ {+∞}. (−∞) + α = α + (−∞) = −∞ ∀α ∈ R ∪{−∞}.

α · (+∞) = +∞ α > 0 α · (+∞) = −∞ α


9/103

R = R ∪{−∞, +∞}

arg min f := {x ∈ X | f (x) = inf X f } inf X f inf X f

Γγ (f )

∅

= ∅ f ≡ +∞

Γγ (f ) × {γ } = epi f ∩ (X × {γ }).

R

(f i)i∈I (I = ∅) f i : X → R

supi∈I f i inf i∈I f i

(supi∈I

f i)(x) := sup{f i(x) | i ∈ I }

(inf i∈I

f i)(x) := inf {f i(x) | i ∈ I }

R

epi(supi∈I

f i) =i∈I

epi f i

epi(inf i∈I

f i) ⊇i∈I

epi f i,

|I | < +∞


10/103

epi(sup f i) =

{(x, α)

∈ X

×R

| sup f i(x)

≤ α

}= {(x, α) ∈ X ×R | f i(x) ≤ α, ∀i ∈ I }=i∈I

epi(f ).

i∈I

epi(f i) = {(x, α) ∈ X ×R | ∃i ∈ I , f i(x) ≤ α}

⊆ {(x, α) ∈ X ×R | inf f i(x) ≤ α}= epi(inf f i),

|I | < +∞ i ∈ I

f : X → R

∀x ∈ X, f (x) > −∞

dom f

= φ

∃x0

∈ X

f (x0) < +

∞

−∞ f −∞ inf X f > −∞ f

X

(X, τ )

x ∈ X N x(τ ) x τ f : X → R ∪ {+∞} τ τ x

∀λ < f (x), ∃N λ ∈ N x(τ ) : ∀y ∈ N λ, f (y) > λ. x ∈ X f τ X

f : X → R ∪ {+∞}

f

τ

X


11/103

epi(f ) X ×R τ × τ R τ R R

∀γ ∈R Γγ (f ) (X, τ ) ∀γ ∈R {x ∈ X | f (x) > γ } ∈ τ

∀x ∈ X f (x) ≤ lim inf y→x

f (y) := supN ∈N X(τ )

inf y∈N

f (y)

(i) ⇒ (ii) (x, λ) /∈ epi(f ) λ < f (x) γ ∈ R

λ < γ < f (x)

(i)

N γ ∈ N x(τ ) ∀y ∈ N γ , f (y) > γ (y, γ ) /∈ epi(f ) (N γ ×]−∞, γ [)∩epi(f ) = ∅ N γ ×]−∞, γ [∈ N (x,λ)(τ ×τ R) epi(f )C

(ii) ⇒

(iii) Γγ (f )× {

γ }

= epi(f )∩

(X × {

γ }

) Γγ (f )× {

γ }

X ×R Γγ (f ) X

(iii) ⇒ (iv) (iv) ⇒ (v) x ∈ X γ < f (x) N = {y ∈ X | f (y) > γ } ∈ N x(τ ) x γ ≤ inf

y∈N f (y) γ ≤ sup

N ∈N x(τ )inf

y∈N f (y)

γ < f (x) (v)

(v) ⇒ (i) λ < f (x) (v) λ < supN ∈N X(τ )

inf y∈N

f (y) N ∈ N x(τ ) :λ < inf

y∈N f (y)

(X, τ ) f τ ⇔ ∀x ∈ X xn → x, f (x) ≤

lim inf n→+∞

f (xn) := supn∈N

inf k≥n

f (xk)

{f i}i∈I τ f i : X →R ∪ {+∞} sup

i∈I f i τ I min

i∈I f i

i∈I

f i τ

f : X → R ∪ {+∞} τ γ ∈ R

Γγ (f ) = {x ∈ X | f (x) ≤ λ} X τ f τ Γγ (f )


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(V, ·) f : V → R ∪ {+∞}

lim

x→∞f (x) = +∞

(V, ·) f : V → R ∪ {+∞}

(i) f (ii) ∀γ ∈ R, Γγ (f ) dim V < +∞ f f

(V, ·) dim V −∞ x∗ ∈ X f X f (x∗) = min

x∈X f (x)

τ (xn)n∈N f lim

n→+∞f (xn) = inf X f

inf X f > −∞ n ≥ 1, xn ∈ X inf X f ≤f (xn) ≤ inf X f + 1n inf X f = −∞ xn ∈ X f (xn) ≤ −n inf X f = +∞ f ≡ +∞ x∗ ∈ X f (xn) ≤ max{inf X f + 1n , −n} ≤ max{inf X f +1, −1} =: γ 0 ∈R γ 0 > inf X f xn ∈ Γγ 0(f ) n ≥ 1 Γγ 0(f ) (xnk) τ x∗

∈ X lim

k→∞

f (xnk) = inf X f

τ

f

f (x∗) ≤ lim

k→∞f (xk) = inf X f inf X f > −∞ f (x∗) = inf X f

arg min f = ∅

arg min f =

γ>inf X f

Γγ (f ) =

γ 0>γ>inf X f

Γγ (f )

γ 0 ∈R γ 0 > inf X f inf X f


13/103

τ

∃γ 0 > inf X f : Γγ 0(f )

f : R → R f (x) = x21+x2

Γγ (f ) γ < 1 Γ1(f ) = R

τ

f : X → R ∪ {+∞} τ K ⊆ X τ x∗ ∈ K f (x∗) = minK f g := f + δ K .

X

C ⊂ X C ∞

C ∞ =

d ∈ X

∃tk → ∞, ∃xk ∈ C lim xktk = d

.

C +∞ C ∞

C ∞

C ∞

= C ∞

C C ∞

C ∞ = {d ∈ X | d + C ⊂ C }= {d ∈ X | x + td ∈ C t > 0}

x ∈ C

C C ∞ = C

C C ∞ = {0} X

C C ∞ C

C C = { x ∈ Rn | Ax ≤ b } A m × n b ∈ Rm C ∞ = { d ∈ Rn | Ad ≤ 0 }


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(C i)i∈I X

i∈I C i∞ ⊆ i∈I (C i)∞

i∈I

C i

∞

⊇ i∈I

(C i)∞ I

f : X → R ∪ {+∞} f ∞

f ∞(d) = inf

lim inf

k→∞

f (tkdk)

tk

tk → +∞, dk → d

C

(δ C )∞ = δ C ∞

f, g : X → R ∪ {+∞} dom f ∩ dom g = ∅

h = f + g

f

g

h

d ∈ X f ∞(d) g∞(d) +∞ −∞ h∞(d) ≥ f ∞(d) + g∞(d).

f epi(f ∞) = (epi f )∞

(epi f )∞ ⊆ epi(f ∞) (d, µ) ∈ (epif )∞ tk → ∞ (dk, µk) ∈ epi f t−1k (dk, µk) → (d, µ) f (dk) ≤ µk t−1k f (t−1k dk · tk) ≤ t−1k µk

f ∞(d) ≤ µ (d, µ) ∈ epi(f ∞)

(d, µ) ∈ epi(f ∞) tk → ∞ dk → d

f ∞(d) = limk→∞

t−1k f (tkdk) (d, µ) ∈ epi(f ∞) ε > 0 k f (tkdk) ≤ (µ−ε)tk zk = tk(dk, µ+ε) ∈ epi(f ∞) t−1k zk = (dk, µ+ε) → (d, µ+ε)

(d, µ + ε) ∈ (epi f )∞ (epi f )∞ ε (d, µ)

∈ (epi f )∞

(f i)i∈I X R ∪ {+∞} supi∈I

f i

∞

≥ supi∈I

{(f i)∞}

inf i∈I

f i

∞

≤ inf i∈I

{(f i)∞} .

I

f : X → R ∪ {+∞} λ > inf f

[Γλ(f )]∞ ⊆ {d ∈ X | f ∞(d) ≤ 0 }.


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d ∈ [ Γλ(f ) ]∞ xk ∈ Γλ(f ) tk → ∞ lim

k→∞t−1k xk = d dk = t

−1k xk → d xk ∈ Γλ(f ) t−1k f (tkdk) =

t−1k f (xk) ≤ t−1k λ → 0 f ∞(d) ≤ 0

(f i)i∈I X R ∪ {+∞}

S X

C = { s ∈ S | f i(x) ≤ 0 ∀i ∈ I }

C ∞ ⊆ { d ∈ S ∞ | (f i)∞(d) ≤ 0 ∀i ∈ I }

C i = {x f i(x) ≤ 0} C = S ∩

i∈I

C i

C ∞ ⊆ S ∞ ∩

i∈I (C i)∞

f :R

n

→ R ∪ {+∞} f ∞(d) > 0 d = 0 f

f ∞(d) > 0 d = 0 λ > inf f

0 ∈ [ Γλ(f ) ]∞ ⊆ { d ∈ Rn | f ∞(d) ≤ 0 } = {0}. {0} Γλ(f )

i = 0, 1, . . . , m f i : Rn → R ∪ {+∞}

C = { x ∈ Rn | f i(x) ≤ 0 ∀i } dom f 0 ∩ C = ∅

(P ) inf { f 0(x) | x ∈ C }.

f = f 0 + δ C

(P ) inf { f (x) | x ∈ Rn }.

(f 0)∞(d) > −∞ d = 0 f i, i ≥ 1

(f i)∞(d) ≤ 0 ∀i ⇒ d = 0,

(

P )

f ∞(d) ≥ (f 0)∞(d) + δ C ∞(d) f ∞(d) ≥ (f 0)∞(d) d ∈ C ∞

f ∞(d) > 0 d = 0 f


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(E, d)

f : E →R∪{+∞}

x0 ∈ dom(f ) ε > 0 x̄ ∈ E

f (x̄) + εd(x0, x̄) ≤ f (x0), ∀x = x̄ f (x̄) < f (x) + εd(x, x̄).

ε = 1

f : E → R+ ∪ {+∞}

F : E

→ 2E

F (x) = {y ∈ E | f (y) + d(x, y) ≤ f (x)},

y ∈ F (y), y ∈ F (x) F (y) ⊆ F (x).

v : dom(f ) → R

v(y) := inf z∈F (y)

f (z).

y ∈ F (x)

d(x, y) ≤ f (x) − v(x),

diam(F (x)) ≤ 2(f (x) − v(x)).

(xn)n∈N x0

xn+1 ∈ F (xn), f (xn+1) ≤ v(xn) + 2−n.

F v(xn) ≤ v(xn+1) v(y) ≤ f (y)

v(xn+1) ≤ f (xn+1) ≤ v(xn) + 2−n ≤ v(xn+1) + 2−n

0

≤ f (xn+1)

−v(xn+1)

≤ 2−n.

diam(F (xn)) → 0 F (xn) ̄x ∈ E

n∈N

F (xn) = {x̄}.

̄x ∈ F (x0) (i) x̄ ∈ F (xn) ∀n F (x̄) ⊂ F (xn)

F (x̄) = {x̄}.

x = x̄ x /∈ F (x̄) (ii)


17/103

f : X → R ε > 0 ε

ε − arg min f ={x ∈ X | f (x) ≤ inf f + ε} inf f > −∞,

{x ∈ X | f (x) ≤ −1ε}

ε, λ > 0

x0 ∈ ελ − arg min f x̄ ∈ E

f (x̄) ≤ f (x0)

d(x0, x̄) ≤ λ

∀x

∈ E, f (x̄)

≤ f (x) + εd(x, x̄)

V

τ

V × V → V (v, w) → v + w

R× V → V (λ, v) → λv

τ ×τ V ×V τ R×V R×V

(V, · )

(V, τ )

(V, τ )∗

V ∗

V

R

τ

V V ∗ (V, τ ) v∗ ∈ V ∗ v ∈ V

v, v∗ := v∗(v),

·, · : V × V ∗ → R V V ∗ V V ∗ v ∈ V v, · : V ∗ → R V ∗

V ∗

V


18/103

H

⊆ V

{v

∈V | f (v) = α} f : V → R α ∈ R

[ = α] = {v ∈ V | f (v) = α} [ ≤ α] = {v ∈ V | f (v) ≤ α}

[ = α] τ ∈ (V, τ )∗

[ ≤ α] [ ≥ α] [ < α] [ > α] (V, τ )

C ⊆ V x, y ∈ C λ ∈ [0, 1]

λx + (1 − λ)y ∈ C.

f : V → R ∪ {+∞} x, y ∈ V λ ∈ [0, 1]

f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y).

f : V → R ∪ {+∞} epi f V × R

(V, τ )

A, B ⊆ V

A ∩ B = ∅

A A B ∈ (V, τ )∗ α ∈ R A ⊆ [ ≤ α] B ⊆ [ ≥ α]

(V, τ ) A B

A B

∈ (V, τ )∗ α ∈ R ε > 0 A ⊆ [ ≤

α − ε] B ⊆ [ ≥ α + ε].

(V, τ )

v1

= v2 V

v∗ ∈ (V, τ )∗ v∗(v1) = v∗(v2)

(V, τ )

C ⊆ V τ

C =

{S | C ⊆ S S }.

u /∈ C A := C B := {u} S

C ⊆ S u /∈ S


19/103

V ∗ V ∗ V v, · v ∈ V σ(V ∗, V ) V ∗ V

V ∗

(V ∗, σ(V ∗, V ))

(V, τ ) V ∗ : V ∗ → R

σ(V ∗, V )

v ∈ V

∀v∗ ∈ V ∗, (v∗) =

v, v∗ V (V ∗, σ(V ∗, V ))∗ v → v, · v

∈ V V ∗

v∗

→ v, v∗

= v∗(v)

σ(V ∗, V ) V (V ∗, σ(V ∗, V ))∗

: V ∗ → R σ(V ∗, V ) v ∈ V = v, · v U = {v∗ ∈ V ∗ |(v∗) < 1} ∃ε > 0 v1,...,vn ∈ V {v∗ ∈ V ∗ | vi, v∗ ≤ ε, ∀i = 1,...,n} ⊆ U

ni=1

Kervi, · ⊆ Ker .

F : V ∗ → Rn F (v∗) = (

vi, v

∗

)ni=1

L : F (V ∗) → R L(y1,...,yn) = (v

∗)

v∗ ∈ V ∗

F (v∗) = (y1,...,yn) L

F (V ∗) Rn R

n

L(y) =n

i=1

αiyi.

∀v∗ ∈ V ∗, (v∗) =n

i=1

αivi, v∗ = ni=1

αivi, v∗ , v =

ni=1 αivi

V σ(V, V ∗) ·, v∗ v∗ ∈ V ∗ σ(V, V ∗) τ (V, σ(V, V ∗))∗ V ∗ (V, σ(V, V ∗)) (V, τ )


20/103

(V, τ ) V ∗

C

⊆ V

C τ − C σ(V, V ∗) − .

f : V → R ∪ {+∞} f

τ −

f

σ(V, V ∗) −

σ(V, V ∗) ⊆ τ

τ

σ(V, V ∗)

C σ(V, V ∗)

f

epi(f )

V ×R

f

τ

epi(f ) τ × τ R V ×R

(V, ·) V ∗

V

(V, ·)

σ(V, V ∗)

(V, ·) f : V → R ∪ {+∞} σ(V, V ∗) u ∈ V ∀v ∈ V, f (u) ≤ f (v).

(V, ·) f : X → R ∪ {+∞}

u

∈ V

∀v

∈ V, f (u)

≤ f (v).

f V f σ(V, V ∗)

σ(V, V ∗)

τ = σ(V, V ∗)


21/103

(V, ·) =(C ([0, 1];R), ·∞)

v∞ = sup {|v(t)| | t ∈ [0, 1]} .

C :=

v ∈ V

1/20

v(t)dt − 11/2

v(t)dt = 1

.

C C

δ C

Γγ (δ C ) = ∅ γ 1 vn : [0, 1] → R

vn(x) =

β n x ∈ [0, αn]

β nαn−

1

2

x + 12β n

1

2−αn

x ∈]αn, 1 − αn[−β n x ∈ [1 − αn, 1]

αn = 12 − 1n , β n = 1 + 1n vn ∈ C vn∞ = n+1n

d(0, C ) = 1

u ∈ C u∞ = 1

1/20 u(t)dt

≤ 12

11/2 u(t)dt

≤ 12

1/20 u(t)dt = 11/2 u(t)dt = 12 1/20 (1 − u(t))dt = 0 1 − u(t) ≥ 0 u ≡ 1 0, 12 u ≡ −1 12 , 1 u d(0, C ) (C ([0, 1];R), ·∞)

inf V f


22/103


23/103

clτ (f ) τ clτ (f )(x) ≤ lim inf y→x

clτ (f )(y) ≤ lim inf y→x

f (y)

h(x) := liminf y→x

f (y) = sup

V ∈N x

inf y∈V

f (y) f τ

h ≤ clτ (f ).

clτ (f ) ≤ f f τ f (x) ≤ lim inf y→x

f (y) = clτ f (x)

f : X → R ∪ {+∞}

clτ (f )(x) = min{lim inf d

f (xd) | D (xd)d∈D , xd → x}.

(X, τ )

clτ (f )(x) = min

{lim inf

n→∞

f (xn)

| (xn)n∈N xn

→ x

}.

liminf y→x

f (y) = supε>0

inf y∈Bτ (x,ε)

f (y).

xn → x ∀ε > 0, ∃n0(ε) ∈ N, ∀n ≥ n0, xn ∈ Bτ (x, ε) f (xn) ≥ inf

y∈Bτ (x,ε)f (y) inf

n≥n0f (xn) ≥ inf

y∈Bτ (x,ε)f (y) liminf

n→∞f (xn) = sup

k∈Ninf n≥k

f (xn) ≥inf

y∈Bτ (x,ε)f (y).

ε

xn → x

lim inf y→x

f (y) ≤ inf {lim inf n→∞

f (xn) | xn → x}.

n ≥ 1 xn ∈ Bτ (x, 1n ) inf

y∈Bτ (x,1

n)

f (y) ≥ f (xn) − 1n inf Bτ (x,

1

n)

f > −∞−n ≥ f (xn) inf

Bτ (x,1n)

f = −∞.

lim inf y→x

f (y) ≥ limn→∞

inf y∈Bτ (x, 1n )

f (y) ≥ lim supn→∞

f (xn) ≥ lim inf n→∞

f (xn)

f : X → R ∪ {+∞} inf X f = inf X clτ (f )

inf U f = inf U clτ (f ) U ∈ τ arg min f ⊆ arg min clτ (f ).

f ≥ clτ (f ) inf U f ≤ inf U clτ (f ) x ∈ U U ∈ τ U ∈ N x(τ ) clτ (f )(x) ≥ inf U f . x inf U clτ (f )(x) ≥ inf U f x ∈ arg min f

clτ (f )(x) ≤ f (x) = inf X

f = inf X

clτ (f )

x ∈ arg min clτ (f ).


24/103

f : X → R∪{+∞} (xn)n∈N f ̄

x ∈ X

(xnk)k∈N xnk → x̄ x̄ clτ (f )

X

limn→∞

f (xn) = inf X f

clτ (f )(x̄) ≤ limk→∞

f (xnk) = inf X

f = inf X

clτ (f ).

min{clτ (f )(x) | x ∈ X }

inf {f (x) | x ∈ X }.

F : (X, τ ) →

R

∪ {+

∞} G : (X, τ )

→R

(F + G)τ

= F τ

+ G.

p ∈]1, +∞[

(P ) inf v∈V

1

p

Ω|∇v(x)| pdx −

Ω

g(x)v(x)dx

v = 0 Ω ,

Ω ⊆RN g V V = C 1c (Ω) Ω g F : L p(Ω) → R ∪ {+∞}

F (v) =

1 p

Ω|∇v| pdx v ∈ C 1c (Ω),

+∞

F = F Lp

v ∈dom(F )

vn → v L p(Ω) lim inf

n→+∞F (vn) < +∞

limn→+∞

F (vn) supn F (vn) < +∞ supn

∇vn

p,Ω < +

∞

sup

vn

1,p,Ω < +

∞

vn

1,p,Ω =

vn

p,Ω +

∇vn

p,Ω

(vn)n∈N ⊂ C 1c (Ω) W 1,p(Ω) w ∈ W 1,p(Ω) (vnk)k∈N vnk w

W 1,p(Ω)

vnk w L p(Ω)

w = v

vnk v

W 1,p(Ω) (vnk)k∈N ⊆ C 1c (Ω) ⊆ W 1,p0 (Ω) W 1,p(Ω)

v ∈ W 1,p0 (Ω) dom(F ) ⊆ W 1,p0 (Ω) v ∈ W 1,p0 (Ω) = C 1c (Ω)

·1,p (vn)n∈N ⊂ C 1c (Ω)

vn → v W 1,p0 (Ω) ∇v p,Ω = limn→∞∇vn p,Ω dom(F ) = W 1,p0 (Ω)

v ∈ W 1,p0 (Ω) F (v) ≤ 1 p∇v p p v ∈ W 1,p0 (Ω) vn → v

L p(Ω)

vn v


25/103

W 1,p(Ω) ∇vn ∇v L p(Ω)N Φ : L p(Ω)N → R Φ(f ) = 1 p Ω

|f (x)| pdx

1 p

∇v p p ≤ lim inf n→∞1

p∇vn p p.

1 p∇v p p ≤ F

F (v) =

1 p∇v p p v ∈ W 1,p0 (Ω),+∞

g ∈ Lq(Ω) 1 p + 1q = 1 G : L p(Ω) → R

G(v) := − Ω

g(x)v(x)dx.

J (v) := F (v) + G(v)

J (v) =

1 p

Ω|∇v(x)| pdx −

Ω g(x)v(x)dx v ∈ W 1,p0 (Ω),

+∞

(P ) inf v∈Lp(Ω)

J (v) ≤ J (0) = 0 < +∞.

(vn)n ⊂ L p(Ω) (P ) limn→∞

J (vn) = inf Lp J

supn J (vn) < +∞ (vn)n∈N

⊆ C 1

c

(Ω) ⊆

W 1,p

0

(Ω) sup

n∇vn

p < +

∞

(vn)n∈N W 1,p0 (Ω)

v̄ ∈ W 1,p0 (Ω) (vnk)k∈N vnk → v̄ L p(Ω) ̄

v ∈ arg min J

v̄

(P ) inf v∈Lp(Ω)

J (v).

Γ

Γ

(X, d) {F n}n∈N X R u ∈ X

(Γ(d) − lim inf n→+∞

F n)(u) := inf {lim inf n→+∞

F n(un) : un → u},

(Γ(d) − lim supn→+∞

F n)(u) := inf {lim supn→+∞

F n(un) : un → u}.

Γ(d)− liminf F n ≤ Γ(d)− lim sup F n {F n}n∈N Γ F u ∈ X F (u) =(Γ(d) − limn→∞ F n)(u) u ∈ X


26/103

un → u

F (u) ≤

lim inf n→+∞

F n(un) ,

un → u

F (u) = lim F n(un).

Γ

Γ(d) − lim F = cld(F ) F = Γ(d) − lim F n F

d Γ

Γ

Γ X X Γ

F n X R Γ

Γ(d) − limn→+∞

F n = cld( inf n∈N

{F n}).

Γ

{F n

}n F G X R

F = Γ − limn→+∞

F n.

G

lim supn→+∞

(inf {F n + G}) ≤ inf {F + G}.

unk ∈ arg min{F nk + G} u ∈ X u ∈arg min{F + G} inf {F nk + G} → inf {F + G}

(F n + G)n∈N Γ F + G G ≡ 0 inf F > −∞ ε > 0 uε ε F

F (uε) ≤ inf F + ε. un → uε limn→+∞ F n(un) = F (uε)

lim supn→+∞

(inf F n) ≤ limn→+∞

F n(un) ≤ inf F + ε,

ε > 0 lim supn→+∞(inf F n) ≤ inf F inf F = −∞


27/103

uk = unk u uk ∈ arg min(F nk) Γ

F (u) ≤ lim inf k→+∞ F k(uk),

F (u) ≤ lim inf k→+∞

inf F k

u ∈ arg min F limk→+∞ inf F nk =

inf F

(X, τ ) x ∈ X f (x) = sup{g(N ) |N

∈ N x(τ )

} g R

∪ {+

∞} τ

N x(τ )

τ x f

V U : V → R

U Ω ⊂ V (xn)n∈N ⊆ Ω

(|U (xn)|)n∈N U (xn) = 0 n U (xn) → 0 V ∗

̄x

∈ X

U (x̄) = 0

lim inf n→+∞

U (xn) ≤ U (x̄) ≤ lim supn→+∞

U (xn).

V U U (x) → +∞ x → +∞ U V

U

U U V U V

(E, d) G : E → 2E ̄

x

G x̄ ∈ G(x̄)

f : E →R∪{+∞} x ∈ E y ∈ G(x) f (y) + d(y, x) ≤ f (x) G

G

Grafo(G) = {(x, y) ∈ E × E | y ∈ G(x)}. Grafo(G) f : E → R+ ∪ {+∞} x ∈ E y ∈ G(x) f (y) + d(y, x) ≤ f (x) G


28/103

{F n : Rn ×Rm → R, n ∈ N}

F : Rn ×Rm → R (x, y) ∈

Rn

×Rm

xn → x yn → y lim supn→+∞ F n(xn, yn) ≤ F (x, y)

yn → y xn → x lim inf n→+∞ F n(xn, yn) ≥ F (x, y) (F n) F (x̄n, ȳn) (x, y) ∈ Rn ×Rm

F n(x, ȳn) ≤ F n(x̄n, ȳn) ≤ F n(x̄n, y).

(x̄n, ȳn) → (x̄, ȳ)

F (x, ȳ) ≤ F (x̄, ȳ) ≤ F (x̄, y).


29/103

V f : V → R epi f V × R f ∀n ≥ 2, ∀v1,...,vn ∈ V, ∀λ1,...,λn ∈R+ ∪ {0} :

ni=1

λi = 1

f

ni=1

λivi

≤

ni=1

λif (vi).

f : V

→R

λ ≥ 0

λf

g : V → R f + g A : W → V f ◦ A

θ : R → R θ ◦ f (f i)i∈I V R f = sup

i∈I f i

W

g : V

×W

→R

h : V → R h(v) = inf w∈W

g(v, w)

g : Rm → R F : V → Rm F (x) = (f 1(x),...,f m(x)) f i : V → R g◦F g = g(y1,...,ym) yi i = 1,...,m

(i) (ii) V


30/103

V f : V → R

−f

(V, ·)

f : V → R ∪ {+∞}

f x0 ∈ dom(f )

f

x0 ∈ dom(f )

f x0 ∈ dom(f )

f

int(dom(f )) = ∅

f int(dom(f )) = ∅

int(epi(f )) = ∅

(i) ⇒ (ii) ⇒ (iii) ⇒ (i) (i) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇒ (i)

(i) ⇒ (ii) x0 = 0 ̄

f (x) = f (x + x0)

ε > 0

M ∈R

x

x ≤ ε

f (x) ≤ M x1, x2 xi ≤ ε2 i = 1, 2 x1 = x2 α := x1−x2 > 0

y = x1 + ε2α (x1 − x2) y − x1 = ε2

y ≤ ε f (y) ≤ M x1 = 2α2α+ε y + ε2α+ε x2 f

f (x1) ≤ 2α2α + ε

f (y) + ε

2α + εf (x2).

f (x1) − f (x2) ≤ 2α2α + ε

[f (y) − f (x2)] ≤ 2α2α + ε

[M − f (x2)]

0 = 12x2 +

12(−x2) f (0) ≤

12f (x2) +

12f (−x2) −f (x2) ≤

f (−x2) − 2f (0) ≤ M − 2f (0)

f (x1) − f (x2) ≤ 4α2α + ε

[M − f (0)] ≤ 4 M̄

ε x1 − x2,

̄M > 0

x1 x2 |f (x1) − f (x2)| ≤ 4 M̄ ε x1 − x2

(ii) ⇒ (iii) (iii) ⇒ (i)


31/103

(i) ⇒ (iv) x̄ ∈ int(dom(f )) ε > 0 yε = x̄ + ε(x̄ − x0) ̄x = 11+ε yε +

ε1+ε x0 ε > 0 yε ∈ dom(f ) U

x0

f

M

U ε := 1

1 + εyε +

ε

1 + εU

̄x z ∈ U ε ∃x ∈ U : z =1

1+ε y + ε1+ε x f

f (z) ≤ 11 + ε

f (y) + ε

1 + εf (x) ≤ 1

1 + εf (y) +

ε

1 + εM := M ε,

f M ε x̄

(iv) ⇒ (v) (v) ⇒ (vi) x ∈ int(dom(f )) λ > f (x) (x, λ) ∈ int(epi(f )) γ ∈ R f (x) < γ < λ U x ∀y ∈ U, f (y) < γ

(x, λ) ∈ U ×]γ, +∞[ ⊆ epi(f ) (vi) ⇒ (i) U a < b U ×]a, b[⊆ epi(f ) ∀x ∈U, (x, a+b2 ) ∈ epi(f ) ∀x ∈ U, f (x) ≤ a+b2 0 : B(x0, ε) ⊆ dom(f ) ê1, ..., ên

Rn xi := x0 + εêi i = 1,...,n

S =

ni=0

λixi

n

i=0

λi = 1, λi > 0, i ∈ {0, 1,...,n}

x0

f

ni=0

λixi

≤ max{f (xi) | i = 0, 1,...,n} < +∞.

f S

Rn

n = 1


32/103

I f : I → R x0 ≤ y ≤ x1 I

f (y) − f (x0)y − x0 ≤

f (x1) − f (x0)x1 − x0 ≤

f (x1) − f (y)x1 − y .

x ∈ I

∆x(y) := f (y) − f (x)

y − x y ∈ I \{x}

y = x1−yx1−x0x0 + y−x0x1−x0

x1

f : I

⊆R

→R I

f

I

f I

f (y) ≥ f (x) + f (x)(y − x), ∀x, y ∈ I f C 2 f (x) ≥ 0, ∀x ∈ I

f

I

f I

f (y) > f (x) + f (x)(y − x), ∀x, y ∈ I

x = y

f (x) > 0, ∀x ∈ I

f ∈ C 2

(convexidad) ⇒ (i) (i) ⇒ (ii) gx(y) := f (x) − f (y) + f (x)(y − x) gx(x) = 0 gx(y) = −f (y)+f (x) gx(y) ≥ 0 y ∈ I ∩]−∞, x] gx(y) ≤= y ∈ I ∩[x, +∞]

gx y = x 0

(ii) ⇒ (convexidad) lx(y) = f (x) + f

(x)(y − x) ∀x ∈ I , f (y) ≥ lx(y) f (x) = lx(x) f (y) = supy∈I lx(y) f (ii) ⇒ (convexidad estricta) x0 < x1 xλ = x0 + λ(x1 − x0) lλ(y) = f (xλ) + f ‘(xλ(y − xλ) f (x0) > lλ(x0) f (x1) > lλ(x1) f (xλ) =l − λ(xλ = λlλ(x1) + (1 − λ)lλ(x0) f (xλ) < (1 − λ)f (x0) + λf (x1)


33/103

f (x) = ax2 + bx + c R a ≥ 0 a > 0

f (x) = eax R a = 0

f (x) = xα ]0, +∞[ α ≥ 1 α > 1

f (x) = −xα ]0, +∞[ 0 ≤ α ≤ 1 0 < α


34/103


35/103

y ∈ Y α ∈R lα(x) = x, y − α f

∀x ∈ X, x, y − α ≤ f (x),

α ≥ supx∈X

{x, y − f (x)}.

f : X → R f ∗ : Y → R

f ∗(y) = supx∈X

{x, y − f (x)}.

f ∗(y) = supx∈dom(f )

{x, y − f (x)}

dom(f ) = ∅ ∃x0 ∈ X f (x0) = −∞ f ∗ ≡ +∞ f ∗ Y ·, ·

∀x ∈ X, ∀y ∈ Y, f (x) + f ∗(y) ≥ x, y.

y ∈ Y f ∗(y) α ·, y ·, y−α f

f ∗(0) = − inf X f −f ∗(y) = inf x∈X {f (x)−x, y} −·, y X Y f x, y − f (x) x

y

f ∗(y)

y

f

≤ g f ∗

≥ g∗

(f i)i∈I X R inf i∈I

f i

∗= sup

i∈I f ∗i ,

supi∈I

f i

∗≤ inf

i∈I f ∗i .

λ > 0 (λf )∗(y) = λf ∗( yλ )

α ∈ R

(f + α)∗ = f ∗ − α

x0 ∈ X f x0(x) = f (x − x0) f ∗x0(y) = f ∗(y) + x0, y


36/103

A : X → X (f ◦ A)∗ = f ∗ ◦ A∗−1

y0

∈ Y

f y0(x) = f (x) +

x, y0

f ∗y0(y) = f ∗(y

−y0)

f (x) = exp(x) R

f ∗(0) = 0

y > 0 f ∗(y) ≥ t(−y) − exp(t) → ∞ t → −∞ f ∗(y) = +∞

y < 0

f ∗(y)

f ∗(y) = y ln y − y

f ∗(y) =y ln y − y y ≥ 0,

+∞ y


37/103

X = Y f (x) = 12x2 = x, x f (y) = f ∗(y) 12x2

K X

δ K (x) =

0

x ∈ K,

+∞ x /∈ K

K

σK (y) := δ ∗K (y) = sup

x∈K x, y.

K = {x0} σ{x0}(y) = x0, y.

(X, ·) K = {x ∈ X | x ≤ 1} σK (y) = y∗

K

σK (y) = δ K 0(y)

K 0 = {y ∈ Y | ∀x ∈ K, x, y ≤ 0}

K K (K 0)0 = K K = {x ∈ X | ∀y ∈ Y, x, y ≤ σK (y)}

K

σ

K (y) = δ

K ⊥(y)

K ⊥ = {y ∈ Y | ∀x ∈ K, x, y = 0}

K K (K ⊥)⊥ = K

K ⊆ X σK

σ

K = {x ∈ X | ∀y ∈ Y, x, y ≤ σ(y)}

(X,Y, ·, ·)

Γ(X ) = {f : X → R | f }Γ0(X ) = Γ(X ) \ {ω, ω} ω ≡ +∞, ω ≡ −∞

f ∈ Γ0(X ) f


38/103

f (x, r) /∈ epi(f )

(y, s) ∈ Y ×R

\ {(0, 0)}

α ∈R

∀(z, λ) ∈ epi(f ), x, y + sr < α ≤ z, y + sλ. s ≥ 0 x0 ∈ dom(f ) λ > f (x0)

x ∈ dom(f ). λ = f (x)

x, y + sr < α ≤ x, y + sf (x)

s(f (x) − r) > 0 s > 0

∀z ∈ dom(f )

x, ys + r r

x /∈ dom(f ). s > 0 s = 0 s = 0

∀(z, λ) ∈ epi(f ), x, y < α ≤ z, y.

dom(f )

= φ

∈ (X, τ )∗

∀z ∈ X, f (z) ≥ (z).

∀k ≥ 1 ∀z ∈ X

f (z) ≥ (z) ≥ (z) + k(α − z, y)

(x) + k(α − x, y) → +∞ k → +∞

∀x ∈ X, ∀r < f (x), ∃ ∈ (X, τ )∗ : f ≥ , f (x) ≥ (x) > r f ∈ Γ(X ) f > −∞ f = +∞

f : X → R f ∗ ∈ Γ(X ) f : X → R f ∗∗ : X → R

f ∗∗(x) = supy∈Y

{x, y − f ∗(y)}.

f ∗∗ ∈ Γ(X ) f ∗∗ ≤ f


39/103

f : X → R

f ∗∗(x) = sup{

g(x) |

g ∈

Γ(X ), g ≤

f }

f ∗∗ Γ(X ) f Γ f

f ∈ Γ(X ) f = f ∗∗.

Γ(X )

h = sup{g(x) | g ∈ Γ(X ), g ≤ f } ∈ Γ(X )

f ∗∗ ≤ h g ∈ Γ(X ) g ≤ f g∗ ≥ f ∗ g∗∗

≤ f ∗∗

g ∈ Γ(X ) g∗∗

= g

g ∈ Γ(X ) {i}i∈I

g = sup i.

(yi, ri) ∈ Y × R i(x) = x, yi − ri

∗i (y) =

ri yi = y,

+∞

g ≥ i ∀i ∈ I, ∀x ∈ X, g∗∗(x) ≥ ∗∗i (x) = x, yi − ri = i(x) g∗∗ ≥

supi∈I i = g

g

∗∗

= g

f̄

f : X → R

f ∗∗ ≤ f̄ ≤ f.

f f̄ epi( f̄ ) = epi(f ) f ∗∗ = f̄ f : X → R −∞

f (x) =

−∞ x ∈ C,+

∞ x /

∈ C,

C

C = X f = f̄ ≥ f ∗∗ ≡ −∞ f̄ = f ∗∗

f f f̄ = f ∗∗ f ω ̄f f

ω∗X ≡ +∞ = ωY ω∗X ≡ −∞ = ωY ∗ : Γ0(X ) →Γ0(Y )

∗ : Γ0(Y ) → Γ0(X ) ∗


40/103

f : R → R (x0, f (x0)) x0

(X, τ )

(Y, σ)

·, · f ∈ Γ0(X )

f (x0) = f ∗∗(x0) = sup

y∈Y {x, y − f ∗(y)}.

y0 ∈ Y

f (x0) = x0, y0 − f ∗(y0) ∈ R.

∀x ∈ X

f (x) ≥ x, y0 − f ∗(y0) = x − x0, y0 − f ∗(x0),

(x) := x − x0, y0 − f ∗(x0)

f f x0 f (x0) ∈ R

f (t) =

−√ t t ≥ 0,+∞

Γ0(R)

f : X → R x0 ∈ X y ∈ Y

f x0

∀x ∈ X, f (x0) + x − x0, y ≤ f (x). ∂f (x0) f x0 ∂f (x0) = ∅ f x0

f (x) = |x| x ∈ R ∂f (0) = [−1, 1]


41/103

f (x0) = −∞ ∂f (x0) = Y f (x0) = +∞

∂f (x0) =

∅ dom(f ) = ∅,Y

f ≡ +∞.

f : X → R ∪ {+∞} x ∈ dom(f )

y ∈ ∂f (x)

f (x) + f ∗

(y) ≤ x, y

f (x) + f ∗(y) = x, y

f ∈ Γ0(X )

y ∈ ∂f (x) x ∈ ∂f ∗(y).

(i) ⇒ (ii) z ∈ X f (x) + z − x, y ≤ f (z) f (x) + z, y − f (z) ≤ x, y f (x) + f ∗(y) ≤ y, x (ii) ⇒ (iii)

(iii) ⇒ (i) f ∗∗ y

x ∈ ∂f ∗(y) f ∗(y) + f ∗∗(x) = x, y f ∈ Γ0(X )

f (x) = f ∗∗(x)

Ω ⊆Rn f : Ω → R ∇f : Ω →U ⊆ Rn (x, y) y ∈ U x ∈ ∂ f ∗(y) x z → z, y − f (z) y − ∇f (x) = 0 x = (∇f )−1(y) ∈ Ω

g(y) := (∇f )−1

(y), y − f ((∇f )−1

(y)),

f ∗(y) = g(y).

y = ∇f (x) x = ∇g(y)

f : X → R ∪ {+∞} ∂f (x) Y


42/103

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