Information Technology Reference
In-Depth Information
and
n
q
i
λ
i
−
t
l
(
t
,
ζ)
=
(5.110)
i
=
1
Note that
h
(
0,
ζ)
≥
l
(
0,
ζ)
[see inequality (5.68)]. Figure 5.6 illustrates the three-
dimensional case. The straight line
h
(
t
,0
)
(OLS) is vertical and has only a unique
intersection at
t
=
0. The straight line
h
(
t
,
ζ)
rotates counterclockwise around
0,
b
T
b
as
ζ
increases. As a consequence of the fact that
l
(
t
,
ζ)
is always
strictly increasing, the saddles and maximum intersections have bigger and bigger
parameters
t
as
ζ
increases.
The first part of the theorem is also demonstrated as follows.
The solution locus is given by the first zero (minimum) of
g
(γ
,
ζ )
for every
possible
ζ
:
n
q
i
+
2
γ
min
(
1
−
ζ)
−
b
T
b
=
0
g
(γ
min
,
ζ )
=
(5.111)
λ
i
−
2
γ
ζ
min
i
=
1
For the points of the solution locus,
t
min
=
2
γ
min
ζ<λ
1
.Hence,
1
+
n
q
i
λ
2
γ
min
ζ
λ
+
2
γ
min
(
1
−
ζ)
−
b
T
b
=
0
g
(γ
min
,
ζ )
=
+···
(5.112)
i
i
i
=
1
Neglecting the higher-order terms yields
−
i
=
1
q
i
/λ
i
b
T
b
1
2
1
+
i
=
1
(
q
i
/λ
i
)
−
1
ζ
γ
min
=
=
γ
min
(ζ )
(5.113)
Correspondingly,
A
ζ
t
min
=
(5.114)
1
+
(
B
−
1
) ζ
2
b
T
b
−
i
=
1
q
i
/λ
i
≥
=
i
=
1
q
i
/λ
i
>
0and
A
1
where
B
0 [see inequality
(5.68)]. Figure 5.6 shows that the solution locus parameter
t
min
is always an
increasing function for every GeTLS problem
=
ζ
∈
[0, 1]. In particular:
•
ζ
=
0
⇒
t
min
=
0.
•
ζ
=
0
.
5
⇒
t
min
=
A
/ (
B
+
1
)
.
•
ζ
=
1
⇒
t
min
=
A
/
B
.
Hence,
t
DLS
≥
t
TLS
≥
t
OLS
=
0.
Figure 5.5 also shows the DLS line (dashed), which is horizontal. This has
two consequences:
Search WWH ::
Custom Search