Page 19 - Complementarity and Variational Inequalities in Electronics

The Convex Subdifferential Relation Chapter | 2 9

FIGURE 2.4 (x) =|x| and ∂ (x).

Example 2. Let : R → R be defined by

(∀x ∈ R) : (x) =|x|.

Then
⎧
−1 if x< 0
⎪
⎪
⎨
∂ (x) = [−1,+1] if x = 0
⎪
⎪
+1 if x> 0.
⎩
The function is indeed differentiable at x = 0. If x> 0, then (x) = x and
∂ (x) = 1. If x< 0, then (x) =−x and ∂ (x) =−1. If x = 0, then
|v|≥ wv, ∀v ∈ R,

if and only if w ∈[−1,+1] (see Fig. 2.4).

Example 3. Let : R → R be defined by

2
(∀x ∈ R) : (x) = max{0,x − 1}.
Then
⎧
2x if x< −1
⎪
⎪
⎪
⎪
⎪ [−2,0] if x =−1
⎪
⎪
⎨
∂ (x) = 0 if x ∈]−1,+1[
⎪
⎪
⎪ [0,+2] if x = 1
⎪
⎪
⎪
⎪
2x if x> 1.
⎩
The function is indeed differentiable at x ∈ R\{−1,1}.If x< −1or x> 1,
2
then (x) = x − 1 and ∂ (x) = 2x.If x ∈]−1,+1[, then (x) = 0 and