Chapter 3 Quadratic Programming

Optimization I; Chapter 3 62
where
A −1 =
( 0 A
−1
11
A −T
11 −A −T
11 B 11 A −1
11
)
. (3.23)
We assume that Ã11 ∈ lR m 1×m 1
is an easily invertible approximation of A 11 and
define
( )
B11 Ã
à =
T 11
. (3.24)
à 11 0
We remark that the inverse of à is given as in (3.23) with A −1
11 , A −T
11 replaced
by Ã−1 11 and Ã−T 11 , respectively.
We introduce the right transform
( )
I − Ã
K R =
−1 B T
, (3.25)
0 I
which gives rise to the regular splitting
( ) ( Ã 0 (I − A
KK R =
B
} {{ ˜S
Ã
−
−1 )Ã (−I + )
AÃ−1 )B T
, (3.26)
0 0
} } {{ }
=: M 1 =: M 2 ∼ 0
where
˜S := D − BÃ−1 B T . (3.27)
Given a start iterate ψ (0) = (ψ (0)
1 , ψ (0)
2 ) T , we solve the KKT system (3.20) by
the transforming null-space iterations
ψ (k+1) = ψ (k) + K R M1 −1 (α − Kψ (k) ) = (3.28)
= (I − K R M1 −1 K)ψ (k) + K R M1 −1 α , k ≥ 0 .
3.5 Active set strategies for convex QP problems
3.5.1 Primal active set strategies
We consider the constrained QP problem
minimize f(x) := 1 2 xT Bx − x T b (3.29a)
over x ∈ lR n
subject to Cx = c (3.29b)
Ax ≤ d ,
(3.29c)