Page 82 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 73

4.3 A NECESSARY CONDITION OF SOLVABILITY

∗
If a solution of (4.1) exists, say, u , then
n
∗
∗
∗
(∀e ∈ R ) : Mu + q,e + (u + e) − (u ) ≥ 0,
and then, using Proposition 9, we get
n
∗
(∀e ∈ R ) : Mu + q,e + ∞ (e) ≥ 0.
Therefore, necessarily,
T
(∀e ∈ ker(M )) : q,e + ∞ (e) ≥ 0. (4.26)
Example 39. Let K = R + × R + × R + , = K , and

⎛ ⎞
1 −10
M = ⎝ −1 1 0 ⎠ .
⎟
⎜
−1 1 0
We have
⎛ ⎞
1 −1 −1
T
M = ⎝ −1 1 1 ⎠
⎜
⎟
0 0 0
and
⎛ ⎞ ⎛ ⎞
1 1
T
N(M ) = vect{⎝ 1 ⎠ , ⎝ 0 ⎠},
⎜
⎟
⎟ ⎜
0 1
and thus
U V

⎛ ⎞ ⎛ ⎞
1 1
T ⎜ ⎟ ⎜ ⎟
N(M ) ∩ K ∞ ={λ 1 ⎝ 1 ⎠ +λ 2 ⎝ 0 ⎠ : λ 1 ≥ 0,λ 2 ≥ 0}.
0 1
T
If e ∈ ker(M ) ∩ K ∞ , then there exist λ 1 ≥ 0,λ 2 ≥ 0 such that e = λ 1 U + λ 2 V
and

q,e = (λ 1 + λ 2 )q 1 + λ 1 q 2 + λ 2 q 3 = λ 1 (q 1 + q 2 ) + λ 2 (q 1 + q 3 ),

and the necessary condition of solvability is equivalent to
(∀λ 1 ≥ 0,λ 2 ≥ 0) : λ 1 (q 1 + q 2 ) + λ 2 (q 1 + q 3 ) ≥ 0,