An Extension of the Nested Partitions Method
EBERT BREA
vector,
in
this
case,
π
vector.
Proposition 6
Suppose
that
Conditions
2
and
3
hold,
and
let
π
(0)
=
Pr
{
D
(0)
=
0
}−
be
the
probability
that
the
MINPMC
visits
the
state
D
(0)
=
d
(0)˜
,
which
of
course
always
occurs
by
the
starting
of
the
MINP
method;
or
by
backtracking
operations,
which
can
be
eventually
carried
out
by
the
algorithm.
Then
π
(
ℓ
)
=
π
(0)
,
if
ℓ
∈−
{
0
,
1
}
;
π
(0)
Q
ℓ
−
1
j
=1
p
(
j,j
+1)
j
,
if
ℓ
∈−
{
2
,...,k
}
ˆ
,
(32)
where:
π
(0)
=
1
2+
P
k
ˆ
ℓ
=2
ℓ
Q
−
1
j
=1
p
(
j,j
+1)
j
;
(33)
and
p
(
j,j
+1)
j
is
given
by
(22)
.
Proof.
By
induction,
we
have
π
(
ℓ
)
=
π
(0)
ℓ
Y
−
1
j
=0
p
(
j,j
+1)
j
,
∀
ℓ
∈−
N
+
.
(34)
We
besides
know,
X
k
ˆ
ℓ
=0
π
(
ℓ
)
=
1
.
(35)
Substituting
(34)
in
(35)
,
and
knowing
that
p
because
(0
,
1)
0
=
1
of
(22)
,
we
therefore
deduce
2
π
(0)
+
X
k
ˆ
ℓ
=2
π
(0)
ℓ
Y
−
1
j
=1
p
(
j,j
+1)
j
=
1
.
(36)
Solving
(36)
for
π
(0)
,
we
effortlessly
obtain
π
(0)
=
1
2+
P
k
ˆ
ℓ
=2
ℓ
Q
−
1
j
=1
p
(
j,j
+1)
j
.
(37)
Knowing
that
p
(
j,j
+1)
j
is
estimated
by
Proposition
5,
and
using
both
(34)
and
(37)
the
proof
is
completed.
For
almost
ending
this
analysis,
we
apply
the
above
results
and
(30)
together
for
getting
our
mathematical
expression
of
the
probability
distribution
function
of
the
MINPMC
states,
yielding
π
(
k
)
=
X
1
ℓ
=0
π
(0)
δ
[
k
−
ℓ
]
+
X
k
ˆ
ℓ
=2
π
(0)
ℓ
Y
−
1
j
=1
p
(
j,j
+1)
j
δ
[
k
−
ℓ
]
,
∀
k
∈−
Z
.
(38)
The
results
heretofore
obtained
allow
us
to
describe
the
behavior
of
the
MINP
method,
which
is
a
nonho-
mogeneous
Markov
chain
with
a
conditional
geometric
distribution
of
states
given
by
(38)
.
Finally,
we
shall
make
the
follows
statements
for
justi-
fying
our
approach:
By
Proposition
4
and
taking
into
account
that
σ
(
k
)
⊆−
σ
(
k
)
,
˜
we
can
assert
that
for
each
k
∈−
N
,
V
[˜
σ
(
k
)]
≥−V
[
σ
(
k
)]
,
and
besides,
by
(21)
we
have
that
V
[˜
σ
(
k
)]
≥−V
[˜
σ
(
k
+1)]
,
what
would
allow
us
to
say
as
a
reasonable
conjecture,
that
the
MINP
method
concentrates
its
finding
better
promising
regions
during
the
progress
of
its
iterative
process,
due
to
the
fact
of
the
continuous
process
of
reduction
of
the
baggy
hull
of
the
identified
promising
region.
vi
.
OPERATING THE MINP METHOD
In
this
section,
we
shall
show
a
main
software
that
will
operate
the
MINP
method
for
taking
the
collec-
tion
of
the
performance
measurements:
η
(
ℓ
)
(
p,n,m
)
;
λ
(
ℓ
)
(
p,n,m
)
;
and
q
(
ℓ
)
(
p,n,m
)
from
each
ℓ
th
indepen-
dent
replication
of
the
MINP
method
is
depicted
in
Figure
10.
As
is
shown
in
Figure
10
on
page
129,
the
MINP
method
is
called
for
being
r-times
executed
with
different
random
seed
each,
thus
achieving
r
independent
runs
for
taking
i.d.d.
unknown
function
empirical
distribution
of
the
performance
measurements,
when
it
is
used
for
globally
solving
Problem
1.
Revista
TEKHNÉ
N
o
25.1
Semestre
octubre-febrero
2022
ISSN
electrónico:
2790-5195
ISSN:
1316-3930
128