Page 84 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 75

If z is a solution of problem CP(M,D( ) ∞ ), then clearly z also is a solution
of problem SCP ∞ (M, ).
n
Remark 21. Let : R → R be a proper convex lower semicontinuous func-
tion with closed domain. Then we have the following inclusion:

D( ∞ ) ⊂ D( ) ∞ . (4.29)

Indeed, let e ∈ D( ∞ ). Then, for some x 0 ∈ D( ),wehave
1
∞ (e) = lim (x 0 + λe) = c< +∞.
λ→+∞ λ
Then, noting that

1 1
lim (x 0 + λe) = lim ( (x 0 + λe) − (x 0 ))
λ→+∞ λ λ→+∞ λ
1
= sup ( (x 0 + λe) − (x 0 )),
λ>0 λ
we see that

(∀λ> 0) : (x 0 + λe) ≤ cλ + (x 0 ),
so that (∀λ> 0) : x 0 + λe ∈ D( ), and thus

1
(∀λ> 0) : e ∈ (D( ) − x 0 ),
λ
so that e ∈ D( ) ∞ .
n
Example 40. If ≡ K where K ⊂ R is a nonempty closed convex set, then
) = K ∞ = D( K ) ∞ .
D(( K ) ∞ ) = D( K ∞
In this case, problem SCP ∞ (M, K ) reduces to the complementarity problem:
⎧
⎪ z ∈ K ∞
⎪
⎨
Mz ∈ (K ∞ ) ∗ (4.30)
⎪
⎪
Mz,z = 0
⎩
⇔
∗
K ∞ z ⊥ Mz ∈ (K ∞ ) .
Example 41. Let : R → R be the function defined by
2
(∀x ∈ R) : (x) = x .