An Extension of the Nested Partitions Method
EBERT BREA
5
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
FD(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.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
5000
10000
15000
FN(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

algorithm,
e.g.,
randomized

pattern

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

hence

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

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
132