Chapter 3 Quadratic Programming

Optimization I; Chapter 3 75
We will have a closer look at the relationship between a minimizer of B (β) (x)
and a point (x, µ) satisfying the KKT conditions for (3.62a)-(3.62b).
If x(β) is a minimizer of B (β) (x), we obviously have
∇ x B (β) (x(β)) = ∇ x Q(x(β)) +
p∑
i=1
β
d i − a T i x(β) a i = 0 . (3.72)
Definition 3.9
Perturbed complementarity
The vector z (β) ∈ lR p with components
z (β)
i :=
β
d i − a T i x(β) , 1 ≤ i ≤ p (3.73)
is called perturbed or approximate complementarity.
The reason for the above definition will become obvious shortly.
Indeed, in terms of the perturbed complementarity, (3.72) can be equivalently
stated as
∇ x Q(x(β)) +
p∑
i=1
z (β)
i a i = 0 . (3.74)
We have to compare (3.74) with the first of the KKT conditions for (3.62a)-
(3.62b) which is given by
∇ x L(x, µ) = ∇ x Q(x) +
p∑
µ i a i = 0 . (3.75)
i=1
Obviously, (3.75) looks very much the same as (3.74).
The other KKT conditions are as follows:
a T i x − d i ≤ 0 , 1 ≤ i ≤ p , (3.76a)
µ i ≥ 0 , 1 ≤ i ≤ p , (3.76b)
µ i (a T i x − d i ) = 0 , 1 ≤ i ≤ p . (3.76c)
Apparently, (3.76a) and (3.76b) are satisfied by x = x(β) and µ = z (β) .
However, (3.76c) does not hold true, since it follows readily from (3.73) that
z (β)
i (d i − a T i x(β)) = β > 0 , 1 ≤ i ≤ p . (3.77)
On the other hand, as β → 0 a minimizer x(β) and the associated z (β) come
closer and closer to satisfying (3.76c). This is reason why z (β) is called perturbed
(approximate) complementarity.