An Extension of the Nested Partitions Method
EBERT BREA
5
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0
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0
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0
.
9
1
.
0
0
F
D
(
d
)
d
Figure 15.
Empirical CDF of the DTP for the Iceberg
problem
Figure
15
shows
the
empirical
CDF
of
the
DTP
for
the
Iceberg
problem.
As
can
be
seen
from
the
figure,
the
MINP
required
less
than
5000
objective
function
evaluations
in
the
70
%
of
replications
for
identifying
at
least
a
solution,
which
can
be
either
a
local
or
global
solution.
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0
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1
0
.
2
0
.
3
0
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4
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
.
0
0
5000
10000
15000
F
N
(
η
)
η
Figure 16.
Empirical CDF of the NE for the Iceberg
problem
Finally,
Figure
16
allows
us
to
infer
the
cost
in
function
evaluations
of
the
objective
function,
because
as
can
be
seen
from
the
figure,
the
MINP
method
needed
a
high
number
of
function
evaluations
for
globally
solving
the
Iceberg
problem.
It
is
worthwhile
pointing
out
that
the
MINP
method
was
tested
without
having
been
tuned
for
this
group
of
prob-
lems.
However,
some
setting
of
its
parameters
were
empirically
fitted
for
improvement
the
performance
of
the
algorithm,
before
running
the
numerical
experiments.
viii
.
DISCUSSION AND FUTURE RESEARCH
The
aim
of
this
article
has
been
to
propose
a
new
approach
for
globally
solving
bound
constrained
mixed
integer
nonlinear
problems
using,
for
reaching
this
tar-
get,
the
principles
of
the
NP
method
viewpoint,
namely:
i)
partitioning
into
subregions
of
the
current
promising
region;
ii)
sampling
scheme
for
obtaining
random
trial
points
from
both
each
subregions
and
surrounding
re-
gion
to
the
current
promising
region;
iii)
locating
of
where
has
came
from
the
best
sampled
trial
point
among
all
sampled
trial
points;
and
iv)
testing
of
a
stopping
rule
for
making
decision
either
executing
a
new
iteration
or
finishing
the
iterative
process
of
solving
of
the
minimiza-
tion
problem.
Nevertheless,
heretofore
this
approach
does
not
seem
to
have
been
effective
enough,
despite
the
theoretical
foundations
that
have
been
developed
in
this
research
to
reach
our
goal,
if
it
is
taken
into
account
the
results
reported
by
Brea
[25]
,
who
carried
out
a
comparative
study
among
two
implementations
of
Genetic
Algorithms
and
the
Game
of
Patterns
in
the
n-dimensional
real
field.
Although,
the
MINP
method
has
shown
to
be
a
powerful
viewpoint
for
identifying
promising
regions,
what
would
become
a
useful
algorithmic
procedure,
and
it
could
hence
be
hybridized
with
some
local
search
a
lgorithm,
e.g.,
randomized
patt
ern
search
algorithm
[10]
;
pattern
search
algorithm
[26]
,
because,
the
MINP
method
has
experimentally
shown
to
be
effective
enough
for
identi-
fying
promising
regions,
and
hence
with
information
of
the
promising
region
could
be
globally
solved
Problem
1.
Besides,
the
approach
that
has
been
used
in
the
MINP
method
could
be
easily
parallelizable
for
encoding
it
in
a
parallel
computer,
what
would
be
effective
enough
for
finding
global
solutions
to
very
large
dimension
mixed
integer
optimization
problems.
This
research
has
also
raised
several
issues
during
the
development
of
the
MINP
method.
Among
them,
one
can
remark:
i)
the
MINP
method
parameter
tuning
for
improving
it,
for
this
target,
one
could
hen
ce
use
the
viewpoint
of
Adenso-Díaz
and
Laguna
[27]
;
ii)
statisti-
cal
analysis
for
the
quality
performance
measurement
applying
non-standard
parametric
statistic,
e.g.,
Cheng
approach
[23]
;
iii)
the
optimum
quantity
of
random
trial
Revista
TEKHNÉ
N
o
25.1
Semestre
octubre-febrero
2022
ISSN
electrónico:
2790-5195
ISSN:
1316-3930
132