Page 76 - Complementarity and Variational Inequalities in Electronics

A Variational Inequality Theory Chapter | 4 67

For example, the matrix

⎛ ⎞
−1 −10
⎜
M = ⎝ 1 ⎟
−10 ⎠
0 0 0
is weakly positive semidefinite since σ(M) ={−1 − i,−1 + i,0}.

4.2.14 Proper Convex Functions

n
We denote by (R ;R ∪{+∞}) the set of proper convex lower semicontinuous
n
functions : R → R ∪{+∞} with closed domain, that is,
n
n
(R ;R ∪{+∞}) ={ ∈ 0 (R ;R ∪{+∞}) : D( ) = D( )}.
n
Example 24. Let K be a nonempty closed convex subset of R . Then
n
K ∈ (R ;R ∪{+∞}).
n n
We denote by D (R ;R ∪{+∞}) the set of functions : R → R ∪{+∞}
of “diagonal” structure:
n
(∀x ∈ R ) : (x) = 1 (x 1 ) + 2 (x 2 ) + ··· + n (x n ), (4.7)
where, for all 1 ≤ i ≤ n,wehave

n
i ∈ (R ;R ∪{+∞}) (4.8)
and
n
(∀λ ≥ 0, ∀x ∈ R ) : i (λx) = λ i (x). (4.9)
It is clear that

n n n
D (R ;R ∪{+∞}) ⊂ (R ;R ∪{+∞}) ⊂ 0 (R ;R ∪{+∞}).
Example 25. Let α 1 ≥ 0,α 2 ≥ 0,...,α n ≥ 0. We set
n
(∀x ∈ R ) : (x) = α 1 |x 1 |+ α 2 |x 2 | + ··· + α n |x n |.
n
It is clear that ∈ D (R ;R ∪{+∞}).
Example 26. Let K 1 ,K 2 ,...,K N be nonempty closed convex subsets of R.
Then
n
(∀x ∈ R ) : K 1 ×K 2 ×···×K n (x) = K 1 (x 1 ) + K 2 (x 2 ) + ··· + K n (x n ).
n
∈ D (R ;R ∪{+∞}).
It is thus clear that K 1 ×K 2 ×···×K n