Page 27 - Complementarity and Variational Inequalities in Electronics

The Convex Subdifferential Relation Chapter | 2 17

n
Indeed, let U,V ∈ R satisfy the complementarity relation 0 ≤ U ⊥ V ≥ 0.
Then (∀h ≥ 0) :
V,h ≥ 0, and since
V,U = 0, we see that

(∀h ≥ 0) :
V,h − U ≥ 0,

meaning that −V ∈ ∂
R (U). Reciprocally, if −V ∈ ∂
R (U), then U ≥ 0
n
n
+ +
and (∀h ≥ 0) :
V,h − U ≥ 0. Setting h = 2U, we obtain
V,U ≥ 0. Then,
setting h = 0, we get
V,U 0. Thus V ⊥ U. Moreover, for any H ≥ 0, we
may set h = H + U to see that
V,H ≥ 0. It results in V ≥ 0. Thus U,V ∈ R n
satisfy the complementarity relation.
n
Aset K ⊂ R is a cone if
(∀α> 0) : αK ⊂ K.

If K is a nonempty closed convex cone, then

0 ∈ K,
K + K ⊂ K,

and
(∀α> 0) : αK ⊂ K.

Remark 5. The relation K + K ⊂ K is a consequence of the convexity and
the cone property. Indeed, if z ∈ K + K, then z = z (1) + z (2) with z (1) ∈ K
and z (2) ∈ K.The set K is a cone and thus 2z (1) ,2z (2) ∈ K. This results in
1
1
z = 2z (1) + 2z (2) ∈ K since K is convex.
2 2
∗
The dual cone of K is denoted by K and defined by
n
∗
K ={w ∈ R :
w,v ≥ 0, ∀v ∈ K}.
∗
We remark that K is the set of vectors w forming an acute angle with all the
vectors v of K, that is,

w,v
cos(≺ w,v ) = ≥ 0,∀v ∈ K.
||w||||v||
o
∗
The set K =−K is called the polar cone of K (see Fig. 2.10).
Remark 6. If K is a nonempty closed convex cone, then (K ) = K.
∗ ∗
n
Example 13. Let K = V where V is a real-vector subspace of R . Then
∗
⊥
K = V .