Page 17 - Complementarity and Variational Inequalities in Electronics

Chapter 2

The Convex Subdifferential

Relation

In this chapter, we first provide some notions and fundamental results of convex
analysis. Then we show how some tools from convex analysis can be used to
formulate ampere–volt characteristics that may present some vertical branches.

2.1 THE CONVEX SUBDIFFERENTIAL RELATION
n
We denote by 0 (R ;R∪{+∞}) the set of proper convex lower semicontinuous
n
n
functions from R to R ∪{+∞}. Recall that : R → R ∪{+∞} is said to be
convex if the epigraph of (see Fig. 2.1),
n
epi( ) ={(x,λ) ∈ R × R : (x) ≤ λ},
n
is a convex subset of R . The domain D( ) of is defined as
n
D( ) ={x ∈ R : (x) < +∞},
and the function is said to be proper if D( ) = ∅.
Let us also recall that is said to be lower semicontinuous (see Fig. 2.2)if

x n → x ⇒ (x) ≤ liminf (x n ).
n→+∞
n
A function : R → R∪{+∞} is lower semicontinuous if and only if epi( ) is
n n
a closed subset of R × R. It is also equivalent to say that the level set {x ∈ R :
(x) ≤ c} is closed for any c ∈ R ∪{+∞}.

FIGURE 2.1 Epigraph of a convex function.
Complementarity and Variational Inequalities in Electronics. http://dx.doi.org/10.1016/B978-0-12-813389-7.00002-7
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