Page 30 - Complementarity and Variational Inequalities in Electronics

20 Complementarity and Variational Inequalities in Electronics

FIGURE 2.12 A maximal monotone set-valued function F with D(F) =]−∞,1] and the mini-
0
mal section β of F.

FIGURE 2.13 The function ϕ as defined in (2.3).

A classical result (see e.g. Proposition 1.3.15 in [46]) ensures that there exists
a proper convex lower semicontinuous function ϕ : R → R ∪{+∞} such that

(∀i ∈ R) : F(i) = ∂ϕ(i).

Remark 7. Note that there exists −⇔ ≤ a ≤ b ≤+∞ such that ]a,b[⊂
D(F) ⊂[a,b] and ϕ can be determined by the formula (see Fig. 2.13)

⎧
i
β (s)ds if i ∈[a,b]
⎨ 0
ϕ(i) = i 0 (2.3)
+∞ if i ∈ R\[a,b],
⎩
0
where i 0 ∈]a,b[, and β : D(F) → R denotes the minimal section of F, that is,
0
0
β (x) ∈ F(x) and |β (x)|= inf{|w|: w ∈ F(x)} (see Fig. 2.12). Remark that
the function ϕ in (2.3) is determined by F up to an additive constant.
Note also that

0
0
+
(∀i ∈]a,b[) : ∂ϕ(i) = β (i ),β (i ) ,
−