Page 44 - Complementarity and Variational Inequalities in Electronics

34 Complementarity and Variational Inequalities in Electronics

n n
The mapping P : R → R ;y ø P (y), called the proximal operator (see
e.g. [75]) and defined by
n
(∀y ∈ R ) : P (y) = (id R + ∂ ) −1 (y), (3.5)
n
is thus a well-defined singled-valued operator. Moreover, it is easy to check that
y ∈ x + ∂ϕ(x) ⇐⇒ x = (id R + ∂ϕ) −1 (y)
n
1 2
⇐⇒ x = argmin n{ ||v − y|| + (v)}.
v∈R
2
If K is a nonempty closed convex set, then

≡ P K ,
P K
n
where P K denotes the projector from R onto K, that is,
1 2
P K (x) = argmin v∈K { ||v − x|| }.
2
n
Remark 8. For all x ∈ R ,wehave
P R (x) = max{0,x}
n
+
and
P R (x) = min{0,x}.
n
−
Using the proximal operator, we see that (3.1) can be formulated as the
equivalent fixed point problem

u = (id R + ∂ ) −1 (u − F(u)).
n
Remark 9. The fixed point formulation can be used to propose a numerical
n
method to solve a variational inequality. For a given x 0 ∈ R , we may consider
the recurrence
u k+1 = (id R + ∂ ) −1 (u k − F(u k )).
n
This simple iteration is a prototype that has been used to develop more ad-
vanced numerical methods and algorithms. We refer the reader to the book of
F. Facchinei and J.-S. Pang [37], Chapter 12, for more details.
n
1
Finally, we recall that if F =∇G for some G ∈ C (R ;R), then any solution
∗
x of the optimization problem
min G(x) + (x)
x∈R n