Elliptic Variational Inequalities - Computational Methods for Quantitative Finance - page 268

H

Theorem A.1.2 (Projection onto closed convex subsets) Let be a Hilbert space

and K ⊂ H be a closed and convex subset . Then , for every f ∈ H there exists a

unique u ∈ K such that

f − u = min

v ∈ K

f − v . (A.4)

Moreover , u is characterized by the variational inequality: find

u ∈ K : (f − u, v − u) ≤ 0 ∀ v ∈ K . (A.5)

We define u = P f as the projection of f onto K .

K

Proof The set { f − v : v ∈ K } is bounded from below by 0, hence d :=

inf v ∈ K f − v ≥ 0, exists. Let (v n ) n ⊂ K denote a minimizing sequence, i.e.

d n :=

−

v n →

→∞ −

f d as n . Then (v n ) n is Cauchy, since by ( A.2 ) with f v n

−

in place of u and with f v m in place of v it holds

1
2 (d n +

v n + v m v n − v m

2 2

d m ) = f − + .

2 2

Since K is convex, (v n + v m )/ 2 ∈ K and hence f − (v n + v m )/ 2 ≥ d . There-

fore,

v n − v m

2

1
2 (d n +

d m ) d 2 ≤ − → 0as n, m →∞ ,

2

H =

and hence (v n ) n is Cauchy. Since is complete, there exists a unique u

K ∈ K

lim n →∞ v n , and since is closed, u . Since

= { − : ∈ K }≤ − = −

v n + v n −

d inf f v v f u f u

≤ d n + u − v n −→

n →∞

d,

u satisfies ( A.4 ).
To

show ( A.5 ), let u ∈ K verify ( A.4 ) and let w ∈ K be arbitrary. Then

{ ( 1 − λ)u + λw : 0 <λ< 1 }⊂ K and hence

f − u ≤ f −[ ( 1 − λ)u + λw ] = (f − u) − λ(w − u) ,

so that for 0 ≤ λ ≤ 1 holds

2 2

2 λ(f − u, w − u) + λ 2

2 ,

f − u ≤ f − u − w − u

from where we find that

2 .

∀ w ∈ K , ∀ 0 ≤ λ ≤ 1 : 2 (f − u, w − u) ≤ λ w − u

Choosing here λ = 0 implies ( A.5 ).

Conversely, let u satisfy ( A.5 ). Then, for all v ∈ K ,

2 2 2

u − f − v − f = 2 (f − u, v − u) − u − v ≤ 0

which implies ( A.4 ).
To show that u in ( A.4 ) is unique, we assume that there is u = u , u ∈ K

such

u

that d = f − u = f − . Then