Page 23 - Complementarity and Variational Inequalities in Electronics

The Convex Subdifferential Relation Chapter | 2 13

with ϕ 1 ,ϕ 2 : R → R real-valued convex (and thus continuous) functions. Let
A ∈ R 2×2 be given. Then

w ∈ ∂( (A))(x)
⇔
T
w ∈ A ∂ (Ax)
⇔

w 1 a 11 a 21 ∂ϕ 1 (a 11 x 1 + a 12 x 2 )
∈ ,
w 2 a 12 a 22 ∂ϕ 2 (a 21 x 1 + a 22 x 2 )
that is, also

w 1 ∈ a 11 ∂ϕ 1 (a 11 x 1 + a 12 x 2 ) + a 21 ∂ϕ 2 (a 21 x 1 + a 22 x 2 )

and
w 2 ∈ a 12 ∂ϕ 1 (a 11 x 1 + a 12 x 2 ) + a 22 ∂ϕ 2 (a 21 x 1 + a 22 x 2 ).

n
∗
Let ∈ 0 (R ;R ∪{+∞}) be given. The Fenchel transform of is the
function defined by
n
∗
(∀z ∈ R ) : (z) = sup {
x,z − (x)}.
x∈D( )
n
∗
The function : R → R∪{+∞} is proper convex and lower semicontinuous.
It is also called the conjugate function of . A well-known result in convex
analysis (see e.g. [53], [79]) ensures that

∗
∗
z ∈ ∂ (x) ⇐⇒ x ∈ ∂ (z) ⇐⇒ (x) + (z) =
x,z .
Example 8. Let : R → R be defined by
2
(∀x ∈ R) : (x) = x .
Simple calculations (see Fig. 2.7)give

z z 2 1 2
2
∗
(z) = sup{xz − x }= z( ) − ( ) = z .
x∈R 2 2 4

Example 9. Let : R → R be defined by
(∀x ∈ R) : (x) =|x|.