Page 29 - Complementarity and Variational Inequalities in Electronics

The Convex Subdifferential Relation Chapter | 2 19

FIGURE 2.11 Electrical Device.

obtain
V,U ≥ 0. Then setting h = 0, we get
V,U 0. Thus V ⊥ U. More-
over, for any H ∈ K,wehave H +U ∈ K, and we may set h = H +U to see that
n

V,H ≥ 0. This results in V ≥ 0. Thus U,V ∈ R satisfy the complementarity
relation.

2.3 THE CONVEX SUBDIFFERENTIAL RELATION IN
ELECTRONICS

An electrical device like a diode is usually described by means of some ampere–
volt characteristic (i,V ), which is a graph expressing the difference of potential
V across the device as a function of current i through the device.
The schematic symbol of a circuit element is given in Fig. 2.11. The con-
ventional current flow i will be depicted on the conductor in the direction of the
arrow, and the potential V = V A − V B across the device will be denoted along-
side the device. Here V A (resp. V B ) denotes the potential of point A (resp. B).
Experimental measures and empirical and physical models lead to a variety of
monotone graphs that may present vertical branches. The reader can find general
descriptions of devices and ampere–volt characteristics either in the appropriate
electronics literature (see e.g. [14], [65]) or in the various electronics society
catalogs available on the web.
Let us suppose here that we may write

(∀i ∈ R) : V ∈ F(i)
for some set-valued function F : R ⇒ R. The domain D(F) of F is defined by

D(F) ={x ∈ R : F(x) = ∅}.

We assume that F is maximal monotone. This means that F is monotone, that
is,
∀ x 1 ,x 2 ∈ D(F),z 1 ∈ F(x 1 ),z 2 ∈ F(x 2 ) : (z 1 − z 2 )(x 1 − x 2 ) ≥ 0,

and the graph G(F) of F,

G(F) ={(x,y) ∈ R × R : x ∈ D(F), y ∈ F(x)},
is not properly included in any other monotone subset of R × R.