Page 104 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 95

and

T T
M (D( )) = {M x}.
x∈D( )
T
Condition (4.60) means that if q ∈ M (D( )) + R + (M, ∞ ), then q ∈
R(M, ).
ii) Note that if B(M, ) ={0} (see also Corollary 6), then condition (4.60)
is trivially satisfied on the empty set.
Remark 27. If 0 ∈ D( ) (which is the case for most practical problems), then
we may choose x 0 = 0to see (4.60) in the more legible form

(∀v ∈ B(M, ),v
= 0) : q,v + ∞ (v) > 0. (4.62)

Theorem 8 may be applied to the class of positive semidefinite matrices, that
is,
n
(∀x ∈ R ) : Mx,x ≥ 0.
This last class of (not necessarily symmetric) matrices is of particular interest
for various problems in engineering, and it is then worthwhile to specify our
results in this framework.

n
Corollary 7. Let : R → R ∪{+∞} be a proper convex lower semicontinu-
ous function with closed domain, and let M ∈ R n×n be a positive semidefinite
matrix.
n
T
a) If D( ) ∞ ∩ ker(M + M ) ∩ K(M, ) ={0}, then for each q ∈ R , problem
VI(M,q, ) has at least one solution.
T
b) Suppose that D( ) ∞ ∩ ker(M + M ) ∩ K(M, )
= {0}. If there exists x 0 ∈
D( ) such that
T
(∀v ∈ D( ) ∞ ∩ ker(M + M ) ∩ K(M, ), v
= 0)
and
T
q − M x 0 ,v + ∞ (v) > 0, (4.63)
then problem VI(M,q, ) has at least one solution.
c) If u 1 and u 2 are two solutions of problem VI(M,q, ), then
T
u 1 − u 2 ∈ ker(M + M ). (4.64)
n
T
Proof. Setting X 1 = ker(M + M ), we may write R = X 1 ⊕ X . We denote
⊥
1
n
(resp. P ⊥) the orthogonal projector from R onto X 1 (resp. X ). The
⊥
X 1
by P X 1
1