March 2010
Intermediate to advanced
238 pages
5h 59m
English
Content preview from Portfolio Optimization
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112 A QP Solution Algorithm Chap. 6
Lemma 6.1 Let x
0
satisfy Ax
0
= b. Th en x
0
− s
0
is optimal for
min{c
′
x +
1
2
x
′
Cx | Ax = b} if and on l y if s
0
satisfies the l i n-
ear equations
C A
′
A 0
s
0
v
0
=
g
0
0
,
where g
0
= ∇f (x
0
).
Let
H =
C A
′
A 0
. (6.4)
Matrices of the form of H will play a critical role in the development of
a quadratic programming algorithm. Nonsingularity of such matrices
when A has full row rank is established in the following lemma.
Lemma 6.2 Suppose A has full row ra nk. Then H defined by (6.4) is
nonsingular.
Proof: Suppose to the cont r ary that H is singular. Then there exist
vectors s and v, not both zero such that
C A
′
A 0
s
v
=
0
0
.
Writing
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