Page 114 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 105

The condition

(∀e ∈ ker(M),e
= 0) : q,e
= 0
reads here as
(∀e 1 ∈ R,e 1
= 0) : (q 1 + q 2 )e 1
= 0.

3
This condition is satisfied if that q 1 + q 2
= 0. Thus, for all q ∈ R such that
q 1 + q 2 > 0, problem VI(M,q, ) has a unique solution.

4.10 COPOSITIVITY AND SOLVABILITY CONDITIONS
Our aim in this section is to show that our results established in Corollary 6 and
Theorem 8 recover some results established in the framework of complementar-
n
ity systems on K = R involving copositive plus matrices.
+
n
Let K ⊂ R be a nonempty closed convex cone. We set
∗
B(M,K) ={x ∈ K : Mx ∈ K and Mx,x = 0}. (4.72)
Here B(M, K ) ≡ B(M,K), and problem VI(M,q, K ) is equivalent to
the following complementarity problem CP(M,q,K):

⎧
⎪ u ∈ K
⎪
⎨
Mu + q ∈ K ∗
⎪
⎪
u,Mu + q = 0
⎩
⇔
K u ⊥ Mu + q ∈ K .
∗
Our results in Corollary 6 and Theorem 8 read here as follows.
n
Corollary 9. Let K ⊂ R be a closed convex cone. Let M be a matrix satisfying
(∀x ∈ K) : Mx,x ≥ 0. (4.73)

n
a) If B(M,K) ={0}, then for each q ∈ R , problem CP(M,q,K) has at least
one solution.
b) Suppose that B(M,K)
= {0}. If there exists x 0 ∈ K such that

T
(∀v ∈ B(M,K), v
= 0) : q − M x 0 ,v > 0, (4.74)
then problem CP(M,q,K) has at least one solution.
c) Moreover, if u 1 and u 2 are two solutions of problem CP(M,q,K), then
u 1 − u 2 ∈ N − (M). (4.75)