Page 14 - Complementarity and Variational Inequalities in Electronics

4 Complementarity and Variational Inequalities in Electronics

0 ≤ x ⊥ αF(x) ≥ 0

⇔
min{x,αF(x)}= 0
⇔
max{−x,−αF(x)}= 0
⇔
x = max{0,x − αF(x)}.

Remark 1. The fixed point formulation can be used to propose a numerical
n
method to solve the complementarity problem. Let x 0 ∈ R be given. We may
+
consider the recurrence:
x k+1 = max{0,x k − αF(x k )}.

This simple iteration is a prototype that has been used to develop more ad-
vanced numerical methods and algorithms. We refer the reader to the book of
F. Facchinei and J.-S. Pang [37] (Chapter 12) for more details.
n
1
Recall also that if F =∇G for some G ∈ C (R ;R), then any solution x ∗
of the optimization problem
min G(x)
x∈R n
+
∗
∗
satisfies the complementarity problem 0 ≤ x ⊥ F(x ) ≥ 0. The converse is
also true, provided that G is convex.
The complementarity mathematical theory has known important develop-
ments. Both qualitative results and numerical methods have been developed by
several authors using tools from convex analysis, optimization, and fixed point
theory. We refer the readers to the books [33], [37], [55], [56], [69], and [74],
where various results in the field are discussed.

1.4 THE COMPLEMENTARITY PROBLEM IN ELECTRONICS

Theoretical tools from complementarity theory can be used to develop a rig-
orous mathematical study of electrical networks involving devices like ideal
diodes. We present here only one example because the variational inequality
model that we will discuss in the following chapter is more general and re-
covers the complementarity model. The use of complementarity problems in
electronics originates from different papers devoted to the mathematical study
of dynamical systems in which certain variables are coupled by means of a static
piecewise linear characteristic (see e.g. [28], [30], [31], [49], [50], [51], [52],
[64], [66]).