Page 105 - Complementarity and Variational Inequalities in Electronics

96 Complementarity and Variational Inequalities in Electronics

matrix M is positive semidefinite, and thus (see e.g. Proposition 3.4.3 in [46])
there exists c> 0 such that

n 2
(∀x ∈ R ) : Mx,x ≥ c||P X ⊥x|| . (4.65)
1
Thus N − (M) = X 1 . Part a) is then a direct consequence of Corollary 6, part b)
follows from Theorem 8, and part c) is a consequence of Proposition 15.
Example 48. Let

⎛ ⎞
2 −20
⎟
⎜
M = ⎝ 1 1 0 ⎠
0 0 1
and set
3
(∀x ∈ R ) : (x) = 3 .
R
+
Let us first note that a necessary condition of solvability of Section 4.3 is
T
(∀v ∈ ker(M )) : q,v + ∞ (v) ≥ 0.

It reduces here to
3 T
(∀v ∈ R ∩ ker(M )) : q,v ≥ 0.
+
T
We have ker(M ) ={0}, and the necessary condition of solvability is always
satisfied.
The matrix M is positive semidefinite, and
T
ker(M + M ) ={0}.

Thus
T
D( ) ∞ ∩ ker(M + M ) ∩ K(M, ) ={0}.
3
Therefore, for all q ∈ R , problem VI(M,q, ) has at least one solution.
The solution is unique. Indeed, let x be a solution of problem VI(M,q, )
and suppose that if y denotes another solution of problem VI(M,q, ), then
T
x − y ∈ ker(M + M ), and thus x = y.
Example 49. Let
⎛ ⎞
2 −10
M = ⎝ 0 1 0 ⎠
⎜
⎟
0 0 0