Page 85 - Complementarity and Variational Inequalities in Electronics

76 Complementarity and Variational Inequalities in Electronics

Then ∞ ≡ {0} , D( ∞ ) ={0}, D( ) ∞ = R, and the inclusion in (4.29) is
strict.
Let us now denote by B(M, ) the solutions set of problem SCP ∞ (M, ):

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B(M, ) ={z ∈ R : z solution of SCP ∞ (M, )}.
Note that problem SCP ∞ (M, ) has at least one (trivial) solution since 0 ∈
B(M, ).
Let us also set:

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∗
K(M, ) ={x ∈ R : Mx ∈ (D( ∞ )) }, (4.31)
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N 0 (M) ={x ∈ R : Mx,x = 0}, (4.32)
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N − (M) ={x ∈ R : Mx,x ≤ 0}, (4.33)
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N + (M) ={x ∈ R : Mx,x ≥ 0}. (4.34)
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Proposition 15. Let : R → R be a proper convex lower semicontinu-
ous function, and let M ∈ R n×n .If u 1 and u 2 are two solutions of problem
VI(M, ,q), then

u 1 − u 2 ∈ N − (M).
Proof. If u 1 and u 2 are two solutions of VI(M,q, ), then

Mu 1 + q,u 2 − u 1 + (u 2 ) − (u 1 ) ≥ 0

and
Mu 2 + q,u 1 − u 2 + (u 1 ) − (u 2 ) ≥ 0,

from which we deduce that
M(u 1 − u 2 ),u 1 − u 2 ≤ 0.

The structure of the set B(M, ) can be specified in several situations that
are described in the following proposition.
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Proposition 16. Let : R → R be a proper convex lower semicontinuous
function with closed domain, and let M ∈ R n×n .
a) We have
B(M, ) = D( ) ∞ ∩ N − (M) ∩ K(M, ).

b) If D( ∞ ) = D( ) ∞ , then
B(M, ) = D( ) ∞ ∩ N 0 (M) ∩ K(M, ).