Page 99 - Complementarity and Variational Inequalities in Electronics

90 Complementarity and Variational Inequalities in Electronics

The matrix M is positive semidefinite, and thus (M, ) ∈ PD0 n .Wehave
∗
D( ) ∞ = D( ∞ ) = R × R × R + , (D( ∞ )) ={0}×{0}× R + , N 0 (M) =
3
3
{x ∈ R : x 1 = x 2 ,x 3 = 0}, and K(M, ) ={x ∈ R : x 1 = 0,x 2 = 0,x 3 ≥ 0}.
3
Therefore D( ) ∞ ∩ N 0 (M) ∩ K(M, ) ={0} and R(M, ) = R .
4.8 EXISTENCE AND UNIQUENESS RESULTS
We start this section with a basic result for positive definite matrices.

n
Theorem 6. Suppose that ∈ (R ;R ∪{+∞}) and let M ∈ R n×n be a posi-
n
tive definite matrix. Then, for each q ∈ R , problem V I(M,q, ) has a unique
solution.
Proof. We know that (M, ) ∈ PD n , and thus from Corollary 5 we may con-
n
clude that for any q ∈ R , problem V I(M,q, ) has at least one solution. If u 1
and u 2 are two solutions of problem VI(M, ,q), then from Proposition 15 we
obtain M(u 1 − u 2 ),u 1 − u 2 ≤ 0. We also have M(u 1 − u 2 ),u 1 − u 2 ≥ 0,
and thus M(u 1 − u 2 ),u 1 − u 2 = 0, from which we deduce that u 1 − u 2 = 0,
that is, the uniqueness holds.

Requiring some additional structural properties on as specified in (4.7),
(4.8), and (4.9), the uniqueness of the solution of problem VI(M,q, ) can also
be proved for matrices that are not assumed to be positive definite. The following
result is a generalization of the well-known existence and uniqueness theorem
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in complementarity theory. Recall that ∈ D (R ;R ∪{+∞}) means that

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(∀x ∈ R ) : (x) = 1 (x 1 ) + 2 (x 2 ) + ··· + n (x n ), (4.48)
where, for all 1 ≤ i ≤ n,wehave

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i ∈ (R ;R ∪{+∞}) (4.49)
and
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(∀λ ≥ 0, ∀x ∈ R ) : i (λx) = λ i (x). (4.50)

Theorem 7. Suppose that
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∈ D (R ;R ∪{+∞})

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and let M ∈ R n×n be a P-matrix. Then, for each q ∈ R , problem V I(M,q, )
has a unique solution.