An Extension of the Nested Partitions Method
EBERT BREA
σ
1
(
0
)
σ
2
(
0
)
σ
3
(
0
)
σ
4
(
0
)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
x
(
1
)
x
(
2
)
Figure 3.
Sampling of each
j
th subregion
{
σ
j
(0)
}
4
j
=1
We
must
say
that
if
at
this
stage,
the
NP
method
identi-
fies
more
than
one
subregion
likewise
promising,
the
NP
method
will
arbitrarily
break
this
draw,
for
choosing
just
one
subregion,
of
course.
In
the
example
shown
from
Figure
4,
we
have
hence
assumed
that
the
promising
region
resulted
to
be
σ
3
(0)
,
which
will
therefore
be
the
next
promising
region
σ
(1)
,
and
so
starts
a
next
iteration
of
the
NP
method.
S
(
σ
(
1
))
σ
(
1
)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
x
(
1
)
x
(
2
)
Figure 4.
promising region
σ
(1)
The
partitions
of
the
promising
region
σ
(1)
,
into
a
new
set
of
four
subregions
{
σ
j
(1)
}
4
j
=1
,
and
that
had
been
denoted
by
σ
3
(0)
at
the
0th
iteration,
which
is
depicted
in
Figure
5
in
green
color,
and
also
the
surrounding
region
to
σ
(1)
,
that
is
here
denoted
by
S
(
σ
(1))
,
which
is
shown
by
the
yellow
area
of
the
figure.
Figure
6
displays
by
points
the
set
of
random
trial
points
that
have
been
taken
from
each
j
th
subregion
σ
j
(1)
,
and
from
the
current
surrounding
region
S
(
σ
(1))
.
Here,
we
can
say
that
we
have
5
subregions,
namely:
{
σ
j
(1)
}
4
j
=1
;
and
σ
5
(1)
=
S
(
σ
(1))
,
what
allows
the
NP
method
to
identify
the
best
subregion,
this
means,
if
the
best
subregion
results
ˆ
i
∈−
{
1
,
2
,
3
,
4
}−
the
algorithm
go
toward
to
next
iteration,
where
the
promising
region
is
therefore
smaller
that
the
current
promising
region,
whilst
if
ˆ
i
=
5
,
then
the
NP
method
backtracks
to
the
initial
promising
region
σ
(0)
.
S
(
σ
(
1
))
σ
1
(
1
)
σ
2
(
1
)
σ
3
(
1
)
σ
4
(
1
)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
x
(
1
)
x
(
2
)
Figure 5.
Partitions of promising region
σ
(1)
We
have
assumed
that
the
sampling
procedure
yielded
that
the
best
function
value
belongs
to
the
subregion
σ
2
(1)
,
which
has
thus
been
marked
by
a
red
cross
on
subregion
σ
2
(1)
,
and
therefore
ˆ
j
=
2
.
We
must
point
out
that
two
backtracking
rules
have
been
proposed
by
Shi
and
Ólafsson,
namely:
the
first
one
causes
a
backtracking
process
to
the
previa
promising
region;
and
the
second
one
effectuates
a
backtracking
process
to
the
entire
feasible
region
[18
]
.
S
(
σ
(
1
))
σ
1
(
1
)
σ
2
(
1
)
σ
3
(
1
)
σ
4
(
1
)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
x
(
1
)
x
(
2
)
Figure 6.
Sampling of each
j
th subregion
{
σ
j
(1)
}
4
j
=1
As
is
shown
in
Figure
7,
we
have
assumed
that
the
best
Revista
TEKHNÉ
N
o
25.1
Semestre
octubre-febrero
2022
ISSN
electrónico:
2790-5195
ISSN:
1316-3930
119