The Complementarity Problem

Chapter 1

The Complementarity Problem In this chapter, we show how complementarity problems can be used to develop a suitable approach for the formulation and mathematical analysis of electrical networks involving devices like ideal diodes. For U, V ∈ Rn , we use the notation U, V  =

n 

Ui Vi

i=1

√ for the euclidean scalar product on Rn and U  = U, U  to denote the corresponding norm. The identity matrix of order n is denoted by In×n , whereas idRn stands for the identity mapping on Rn . We set Rn+ = [0, +∞[n and denote by “≤” the partial order induced by Rn+ , that is, U ≤ V ⇔ V − U ∈ Rn+ . We will also use the notations ⎛ min{U1 , V1 } ⎜ min{U , V } 2 2 ⎜ min{U, V } = ⎜ .. ⎜ ⎝ . min{Un , Vn }

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎛ and

⎜ ⎜ max{U, V } = ⎜ ⎜ ⎝

max{U1 , V1 } max{U2 , V2 } .. . max{Un , Vn }

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

1.1 THE COMPLEMENTARITY RELATION We say that two vectors U, V ∈ Rn satisfy the complementarity relation if U ≥ 0, V ≥ 0 and U, V  = 0. The equation U, V  = 0 being an orthogonality condition, we also present the complementarity relation as 0≤U ⊥V ≥0 Complementarity and Variational Inequalities in Electronics. http://dx.doi.org/10.1016/B978-0-12-813389-7.00001-5 Copyright © 2017 Elsevier Inc. All rights reserved.

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2 Complementarity and Variational Inequalities in Electronics

or Rn+ U ⊥ V ∈ Rn+ . Example 1. We have ⎞ ⎞ ⎛ 2 0 ⎟ ⎟ ⎜ ⎜ 0 ≤ ⎝ 1 ⎠ ⊥ ⎝ 0 ⎠ ≥ 0. 0 4 ⎛

It is easy to check that the complementarity relation is equivalent to the following set of relations (∀ i ∈ {1, 2, . . . , n}): ⎧ Ui ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎨ V ≥0 i ⎪ Ui > 0 =⇒ Vi = 0 ⎪ ⎪ ⎪ ⎩ Vi > 0 =⇒ Ui = 0, which is also equivalent to the equation min{U, V } = 0. Indeed, we have U, V  =

n 

Ui Vi ,

i=1

and for U, V ≥ 0, the equation U, V  = 0 is equivalent to the system ⎧ ⎪ U1 V1 = 0 ⎪ ⎪ ⎪ ⎪ ⎨ U2 V2 = 0 .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ Un Vn = 0.

1.2 THE COMPLEMENTARITY RELATION IN ELECTRONICS The diode is a device that constitutes a rectifier that permits the easy flow of charges in one direction but restrains the flow in the opposite direction. Diodes are used in power electronics applications like rectifier circuits, switching inverter and converter circuits. Fig. 1.1 illustrates the ampere–volt characteristic of an ideal diode. This kind of diode is a simple switch. Denoting by i the diode current and by V the voltage across the diode, if V < 0, then i = 0 and the diode is blocking, whereas if i > 0, then V = 0 and the diode is conducting.

The Complementarity Problem Chapter | 1

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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 Let F : Rn → Rn be a given function. The complementarity problem consists in finding x ∈ Rn such that x and F (x) satisfy the complementarity relation ⎧ ⎪ ⎪ ⎨ x≥0 F (x) ≥ 0 ⎪ ⎪ ⎩ x, F (x) = 0 ⇔ 0 ≤ x ⊥ F (x) ≥ 0 ⇔ Rn+

x ⊥ F (x) ∈ Rn+ .

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 ⇔

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 method to solve the complementarity problem. Let x0 ∈ Rn+ be given. We may consider the recurrence: xk+1 = max{0, xk − αF (xk )}. This simple iteration is a prototype that has been used to develop more advanced numerical methods and algorithms. We refer the reader to the book of F. Facchinei and J.-S. Pang [37] (Chapter 12) for more details. Recall also that if F = ∇G for some G ∈ C 1 (Rn ; R), then any solution x ∗ of the optimization problem min G(x)

x∈Rn+

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 developments. 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 rigorous 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 recovers 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]).

The Complementarity Problem Chapter | 1

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FIGURE 1.2 Clipping circuit 1: Diode as shunt element.

Let us consider the clipping circuit of Fig. 1.2 involving a load resistance R > 0, an input signal source u and the corresponding instantaneous current i, an ideal diode as a shunt element, and a supply voltage E. Kirchoff’s voltage law gives u = UR + V + E, where UR = Ri denotes the difference of potential across resistor, and V is the difference of potential across diode. Thus 0 ≤ i ⊥ −V ≥ 0 ⇔ 0 ≤ i ⊥ E + Ri − u ≥ 0 ⇔ min{i, E − u + Ri} = 0 E−u E−u + i} = 0 ⇔ i + min{0, }=0 R R 1 E−u } = max{0, u − E}. ⇔ i = − min{0, R R ⇔ min{i,

If u ≤ E, then the diode is blocking (i = 0), whereas if u > E, then the diode is conducting (i = R1 (u − E)). Let us now consider a driven time-dependent input t → u(t) and define the output signal t → Vo (t) as Vo (t) = E + V (t). The time-dependent current t → i(t) is given by i(t) =

1 max{0, u(t) − E}, R

(1.1)

6 Complementarity and Variational Inequalities in Electronics

FIGURE 1.3 Clipping circuit 1: Ideal diode as shunt element, E = 1.

and thus Vo (t) = V (t) + E = u(t) − Ri(t) = u(t) + min{0, E − u(t)} = min{u(t), E}.

(1.2)

This shows that the circuit in Fig. 1.2 can be used to transmit the part of a given input signal u that lies below some given reference level E (see Fig. 1.3).