Page 13 - Complementarity and Variational Inequalities in Electronics

The Complementarity Problem Chapter | 1 3

FIGURE 1.1 Ideal diode model.

We see that the ideal diode is described by the complementarity relation

V ≤ 0,i ≥ 0,V i = 0 ⇔ 0 ≤−V ⊥ i ≥ 0.

1.3 THE COMPLEMENTARITY PROBLEM
n
n
Let F : R → R be a given function. The complementarity problem consists in
n
finding x ∈ R such that x and F(x) satisfy the complementarity relation
⎧
⎪ x ≥ 0
⎪
⎨
F(x) ≥ 0
⎪
⎪
x,F(x) = 0
⎩
⇔
0 ≤ x ⊥ F(x) ≥ 0
⇔
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n
R x ⊥ F(x) ∈ R .
+ +
The complementarity problem is also equivalent to the equation
min{x,F(x)}= 0.

Letting α> 0, it is also possible to give an equivalent fixed point formulation of
the complementarity problem as follows:

0 ≤ x ⊥ F(x) ≥ 0
⇔