Page 54 - Complementarity and Variational Inequalities in Electronics

Chapter 4

A Variational Inequality Theory

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Let : R → R ∪{+∞} be a proper convex lower semicontinuous function.
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Let M ∈ R n×n and q ∈ R . We consider the following variational inequality
problem:
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VI(M,q, ):Find u ∈ R such that
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Mu + q,v − u + (v) − (u) ≥ 0, ∀v ∈ R . (4.1)

The solution set of problem VI(M,q, ) will be denoted by SOL(M,q, ),
and the resolvant set by R(M, ):
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SOL(M,q, ) ={u ∈ R : u solution of (4.1)}
and
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R(M, ) ={q ∈ R : SOL(M,q, )
= ∅}.
Remark 11. The variational inequality VI(M,q, ) is usually said to be “lin-
ear” because of the matrix M in the model. However, it does not prevent from
considering some monotone nonlinearities through the convex function .For
example, if
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(∀x ∈ R ) : (x) = (x) + K (x),
1
n
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where K ⊂ R is a nonempty closed convex set, and ∈ C (R ;R), then the
variational inequality VI(0,0, ) is equivalent to the following one:
u ∈ K : ∇ (u),v − u ≥ 0, ∀v ∈ K,
which is usually considered as a “nonlinear” variational inequality.
Remark 12. The variational inequality (4.1) is equivalent to the differential
inclusion
Mu + q ∈−∂ (u), (4.2)
and R(M, ) is nothing else that the range of the set-valued mapping x ⇒
−Mx − ∂ (x), that is,

R(M, ) = {−Mx − ∂ (x)}.
x∈R n
Complementarity and Variational Inequalities in Electronics. http://dx.doi.org/10.1016/B978-0-12-813389-7.00004-0
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