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 , ζ ) for every
possible ζ :
n
q i
+ 2 γ min ( 1 ζ) b T b = 0
g 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 min , ζ ) =
+···
(5.112)
i
i
i
=
1
Neglecting the higher-order terms yields
i = 1 q i i
b T b
1
2
1 + i = 1 ( q i 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 i
= i = 1 q i 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