Page 11 - Complementarity and Variational Inequalities in Electronics

Chapter 1

The Complementarity Problem

In this chapter, we show how complementarity problems can be used to develop
a suitable approach for the formulation and mathematical analysis of electrical
n
networks involving devices like ideal diodes. For U,V ∈ R , we use the notation
n

U,V = U i V i
i=1
√
n
for the euclidean scalar product on R and U = U,U to denote the cor-
responding norm. The identity matrix of order n is denoted by I n×n , whereas
n
id R stands for the identity mapping on R .Weset
n
n n
R =[0,+∞[
+
n
and denote by “≤” the partial order induced by R , that is,
+
n
U ≤ V ⇔ V − U ∈ R .
+
We will also use the notations
⎛ ⎞ ⎛ ⎞
min{U 1 ,V 1 } max{U 1 ,V 1 }
min{U 2 ,V 2 } max{U 2 ,V 2 }
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
min{U,V }= ⎜ . ⎟ and max{U,V }= ⎜ . ⎟ .
. .
⎜ ⎟ ⎜ ⎟
⎝ . ⎠ ⎝ . ⎠
min{U n ,V n } max{U n ,V n }
1.1 THE COMPLEMENTARITY RELATION
n
We say that two vectors U,V ∈ R satisfy the complementarity relation if
U ≥ 0,V ≥ 0 and U,V = 0.

The equation U,V = 0 being an orthogonality condition, we also present the
complementarity relation as

0 ≤ U ⊥ V ≥ 0
Complementarity and Variational Inequalities in Electronics. http://dx.doi.org/10.1016/B978-0-12-813389-7.00001-5
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