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effects of
γ
and
ζ
cannot be separated. The hyperbolas pass through the origin
for both
t
=−∞
and
t
=+∞
. The following interpretations for the locus are
possible:
1. If
const, the hyperbolas, which refer to couples of coordinates, rep-
resent the locus of the corresponding coordinates of the centers of the
equilevel hypersurfaces for the energy cost of the GeTLS problem at hand.
2. If
γ
=
const, the hyperbolas express the way the variability of
ζ
moves the
centers of the equilevel hypersurfaces, just giving an idea of the difficulties
of the various GeTLS problems.
3. If
∀
ζ
the zeros of
g
(γ
,
ζ)
are computed, the hyperbola is divided into loci
(
critical loci
), representing the critical points.
ζ
=
Among these interpretations, the third is the most interesting from the point
of view of the study of the domain of convergence and for the introduction of
the principle of the GeTLS scheduling. The critical loci (see Figure 5.4) are:
•The
solution locus
composed of the points having
t
≤
λ
n
and given
∀
ζ
by
2
ζγ
min
[
γ
min
is the smallest zero of
g
(γ
,
ζ)
]. This locus extends itself to
infinity only if one plane coordinate is
z
n
; if it is not the case, it represents
the part of the hyperbola branch just until
λ
n
.
•The
saddle locus
,composed of the points having
λ
j
≤
t
≤
λ
j
−
1
and given
∀
ζ
by 2
ζγ
saddle
[
γ
saddle
is the zero of
g
(γ
,
ζ)
in the corresponding interval];
it represents this saddle for every GeTLS problem. It is represented by an
entire branch of the hyperbola only in the plane
z
j
z
j
−
1
; in all other planes
it represents the part of the branch between
t
=
λ
j
and
t
=
λ
j
−
1
.
•The
maximum locus,
composed of the points having
t
≥
λ
1
and given
∀
ζ
by 2
ζγ
max
[
γ
max
is the largest zero of
g
(γ
,
ζ)
]. This locus extends itself to
infinity only if one plane coordinate is
z
1
; if it is not the case, it represents
the part of the hyperbola branch from
λ
1
to
∞
, corresponding to the origin.
This point is attained only by the DLS maximum, because in this case the
zero of
g
(γ
,1
)
is at infinity; eq. (5.50) shows that
g
(γ
,1
)
is monotonically
increasing for
t
> λ
1
and tends to
−
b
T
b
for
t
→∞
.
The position in the curve for
t
=
0 (origin of the parametric curve) coincides
with the OLS solution and is the only center for OLS equilevel hypersurfaces. The
hyperbolas can be described as a function of this point. Indeed, from eqs. (5.104)
and (5.89),
λ
q
i
λ
i
=
λ
i
λ
i
−
2
γζ
i
λ
i
−
2
γζ
z
i
z
ci
=
(5.107)
Thus,
z
z
c
(γ
,
ζ )
=
(5.108)
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