An Extension of the Nested Partitions Method
EBERT BREA
σ1(0)
σ2(0)
σ3(0)
σ4(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 jth subregion {σj(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

σ3(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

{σj(1)}4j=1,

and

that

had

been
denoted
by

σ3(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

jth

subregion

σj(1),

and
from
the

current

surrounding

region

S(σ(1)).

Here,

we
can
say

that

we

have

5

subregions,

namely:

{σj(1)}4j=1;
and
σ5(1)

=

S(σ(1)),

what

allows

the

NP

method

to
identify
the

best

subregion,

this

means,

if

the

best
subregion
results

ˆi ∈−

{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

ˆi =

5,
then
the

NP

method

backtracks

to

the

initial

promising
region
σ(0).
S(σ(1))
σ1(1)
σ2(1)
σ3(1)
σ4(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
σ2(1),
which

has

thus

been

marked

by

a

red

cross

on
subregion
σ2(1),

and

therefore

ˆj =

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))
σ1(1)
σ2(1)
σ3(1)
σ4(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 jth subregion {σj(1)}4
j=1
As
is

shown

in

Figure

7,

we

have

assumed

that

the

best
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
119