Page 43 - Complementarity and Variational Inequalities in Electronics

Chapter 3

The Variational Inequality

Problem

In this chapter, we show how variational inequalities can be used to develop
a suitable method for the formulation and mathematical analysis of electrical
networks involving devices like different types of diodes (not necessarily ideal)
and transistors.

3.1 THE VARIATIONAL INEQUALITY
n
n
n
Let ∈ 0 (R ;R ∪{+∞}), and let F : R → R be a given function. The
n
variational inequality problem consists in finding u ∈ R such that
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F(u),v − u + (v) − (u) ≥ 0, ∀v ∈ R . (3.1)
It is easy to see that (3.1) is equivalent to the convex subdifferential relation
F(u) ∈−∂ (u). (3.2)

Problem (3.1) is called a “variational inequality of the second kind” or “mixed
variational inequality” (see e.g. [37], [44], [60], [61], and [72]). This model
recovers the one called a “variational inequality of the first kind,” which consists
in finding u ∈ K such that
F(u),v − u ≥ 0, ∀v ∈ K, (3.3)

where K is a nonempty closed convex set. It suffices indeed to set = K to
see that in this case, (3.1) is equivalent to (3.3). Let us also recall here that if
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K = R , then (3.3) is equivalent to the complementarity problem
+
0 ≤ u ⊥ F(u) ≥ 0. (3.4)
n
n
It is well known that for each y ∈ R , there exists a unique x ∈ R such that
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x − y,v − x + (v) − (x) ≥ 0, ∀v ∈ R ,
that is,
y ∈ x + ∂ (x).

Complementarity and Variational Inequalities in Electronics. http://dx.doi.org/10.1016/B978-0-12-813389-7.00003-9
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