Page 55 - Complementarity and Variational Inequalities in Electronics

46 Complementarity and Variational Inequalities in Electronics

FIGURE 4.1 K and K ∞ .

4.1 RECESSION TOOLS

A general linear variational inequality problem may invoke several kinds of
data, among which a closed convex set and a proper convex lower semicon-
tinuous function. In this section, we discuss the mathematical tools we can use
to describe the asymptotic behavior of these data, which are respectively the re-
cession cone and the recession function. We will see later that this material can
be used to derive both sufficient and necessary conditions for the existence of
inequality problems.
Let x 0 be some arbitrary element of K. The recession cone of K is defined
by
1
K ∞ = (K − x 0 ).
λ
λ>0
The set K ∞ is a nonempty closed convex cone described in terms of the
directions that recede from K (see Fig. 4.1). We note that this definition does
not depend on the choice of x 0 ∈ K. We have indeed the following result.

n
Proposition 4. Let K ⊂ R be a nonempty closed convex. Then
n
K ∞ ={d ∈ R :∃{λ n }⊂]0,+∞[,∃{d n }⊂ R n
with λ n →+∞,d n → d, and (∀n ∈ N) : λ n d n ∈ K}.

Proof. If d ∈ K ∞ , then for each λ> 0, there exists x(λ) ∈ K such that

x(λ) x 0
d = − .
λ λ
Let {λ n } be a sequence of positive real numbers such that λ n →+∞ and set
x n x 0
d n = with x n = x(λ n ). It is clear that λ n d n ∈ K and d n = d + → d as
λ n λ n
n
n →+∞. Conversely, if d ∈ R is such that there exist {λ n }⊂ R + \{0},{d n }⊂
n
R with λ n →+∞,d n → d, and λ n d n ∈ K, then for α> 0 and x 0 ∈ K,we