Page 83 - Complementarity and Variational Inequalities in Electronics

74 Complementarity and Variational Inequalities in Electronics

which in turn is equivalent to

q 1 + q 2 ≥ 0,q 1 + q 3 ≥ 0.

4.4 A SPECTRAL CONDITION OF SOLVABILITY

n
Let ∈ (R ;R ∪{+∞}). Let us recall that D( ) is nonempty and convex
and here is assumed to be closed. Therefore the set D( ) ∞ is a well-defined
nonempty closed convex cone. Let us now consider the semicomplementarity
problem SCP ∞ (M, ):
⎧
⎪ z ∈ D( ) ∞
⎪
⎨
Mz ∈ (D( ∞ )) ∗ (4.27)
⎪
⎪
Mz,z ≤ 0.
⎩
Recall that (D( ∞ )) denotes the dual cone of D( ∞ ) and the second relation
∗
in (4.27) also reads:
Mz,h ≥ 0, ∀h ∈ D( ∞ ).
Note that the first relation in (4.27) involves the recession cone of the domain
of , whereas the second relation in (4.27) invokes the dual cone of the domain
of ∞ (and not the dual cone of the recession cone of the domain of ).
Let us now first check in the following Proposition that if D( ) ∞ =
D( ∞ ), then problem SCP ∞ (M, ) reduces to a complementarity problem
of the form
∗
D( ) ∞ z ⊥ Mz ∈ (D( ) ∞ ) .
n
Proposition 14. Let : R → R be a proper convex lower semicontinuous
function with closed domain, and let M ∈ R n×n .If D( ) ∞ = D( ∞ ), then
n
z ∈ R is a solution of problem SCP ∞ (M, ) if and only if z is a solution of the
complementarity problem CP(M,D( ) ∞ ):
⎧
⎪ z ∈ D( ) ∞
⎪
⎨
Mz ∈ (D( ) ∞ ) ∗ (4.28)
⎪
⎪
Mz,z = 0.
⎩
Proof. Let z be a solution of problem SCP ∞ (M, ). Then the second relation
in (4.27) reads
Mz,h ≥ 0,∀h ∈ D( ) ∞ ,
from which we deduce in particular that Mz,z ≥ 0. This, together with the
third relation in (4.27), ensures that Mz,z = 0.