An Extension of the Nested Partitions Method
EBERT BREA
Table I.
Summary of the MINP method, in which just 1
sample reached 53 cumulative iterations
f
(
z
)
N
(1
,
2
,
2)
L
(1
,
2
,
2)
Q
(1
,
2
,
2)
Mean
108.65
SSD
2566.08
6.103
1.03e-2
107.49
1597.31
6.744
3.81e-2
Min
6.00
1056
0.001
9.29e-6
Q1
27.45
1056
1.058
4.44e-5
Q2
91.17
2064
3.185
1.26e-4
Q3
166.06
3528
9.107
5.34e-4
Max
696.09
8736
24.41
2.67e-1
As
is
shown
in
Table
I,
in
a
few
replications
the
MINP
method
achieved
to
identify
the
global
solution
of
the
problem.
In
fact,
according
to
the
reported
summary
from
the
table,
the
25
%
of
the
replications
reach
to
iden-
tify
solutions
to
a
distance
less
than
1.058.
Moreover,
as
can
be
seen
in
Figure
11,
the
algorithm
globally
solved
the
problem
as
much
as
16
%
of
the
samplings.
Besides,
from
Table
I
it
may
be
concluded
that
the
MINP
method
required,
in
average
2566
objective
function
evaluations
for
solving
the
problem,
what
could
be
considered
as
a
good
enough
algorithmic
method.
Nevertheless,
its
quality
performance
measurement
resulted
to
be
very
low,
yielding
a
maximum
value
equal
to
2
.
67
×
10
−
1
.
5
10
15
20
25
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
.
0
0
F
D
(
d
)
d
Figure 11.
Empirical CDF of distance to the true
point (DTP) for the Goldstein-Price problem.
As
shown
in
Figure
11,
it
is
depicted
the
empirical
cumulative
distribution
function
(CDF)
of
the
random
variable
distance
to
the
true
point
(DTP)
or
global
solution
point
as
a
function
of
d
.
As
can
be
seen
from
the
figure,
approximately
a
25
%
of
the
replications
of
the
MINP
method
yielded
good
enough
results,
i.e.,
results
less
than
1.
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
.
0
2000
4000
6000
8000
F
N
(
η
)
η
Figure 12.
Empirical CDF of number of function
evaluations (NE) for the Goldstein-Price problem.
Apart
from
that,
Figure
12
shows
the
empirical
CDF
of
the
number
of
function
evaluation
(NE)
random
variable
for
the
Goldstein-Price
problem,
which
allows
us
to
see
the
performance
of
the
algorithm
from
the
viewpoint
of
the
NE.
b
.
W
PROBLEM
Our
second
problem
is
a
minimization
problem
of
a
ob-
jective
function,
which
has
been
proposed
by
the
author
as
a
challenge
in
the
mixed
integer
programming,
and
whose
mathematical
expression
is
given
in
Appendix
A.
Table
II
presents
our
main
performance
measurements
for
the
W
problem,
namely:
N
(2
,
2
,
2)
,
L
(2
,
2
,
2)
and
Q
(2
,
2
,
2)
.
As
is
shown
in
the
table,
the
MINP
method
spent
in
average
5512
.
32
function
evaluations
for
stop-
ping
the
iterative
process,
and
the
25
%
of
replications
achieved
solutions
to
a
distance
to
global
minimum
less
than
20.84,
in
fact
Figure
14
illustrates
the
value
range
of
the
NE.
The
table
also
shows
the
low
values
of
quality
performance
Q
(2
,
2
,
2)
,
that
has
been
reached
by
the
algorithm
during
the
computational
experimentation
for
this
problem.
Revista
TEKHNÉ
N
o
25.1
Semestre
octubre-febrero
2022
ISSN
electrónico:
2790-5195
ISSN:
1316-3930
130