An Extension of the Nested Partitions Method
EBERT BREA
points
from

both

each

subregion

and

surrounding

region
as
a

function

of

the

depth

measurement,

what

could
be
solved

by

the

approach

of

sampling

budget,

pro-
posed
by

Chen

and

coworkers

[28],

and

including

others
approaches
for

improving

the

MINP

method

stopping
rule
using,

e.g.,

Berkhout

viewpoint

[29],

who,

applying
the
results

of

Chen,

et

al.,

[28],

presents

a

new

and
interesting
approach

for

accelerating

the

NP

method
stopping
rule.
On
the

other

hand,

we

believe

that

the

incorporation

of
criteria
based

on

artificial

intelligence

(AI)

for

making
decisions
on

sampling

quantity

for

taking

from

each
subregion
and

its

surrounding

region,

it

could

be

a

large
advance
in

the

mixed

integer

programming.
Finally,
as

a

future

work

we

also

propose

a

compar-
ative
study

between

the

MINP

method

and

the

Game
of
Patterns

algorithmic

method,

when

they

are

applied
for
globally

solving

bound

constrained

mixed

integer
optimization
problems.
A
LIST OF PROBLEMS
We
here

present

the

objective

functions

of

the

test
problems
used

in

our

numerical

experiments

taking

into
account
the

formulation

of

Problem

1.

Besides,

both
lower
and

upper

bound

vectors

have

also

been

specified,
and
their

respective

global

solutions.
a.
EXTENDED

GOLDSTEIN-PRICE

PROBLEM
OBJECTIVE
FUNCTION.

Let

f(z)

:

R2n ×−Z2m

→−R be
the
Extended

Goldstein-Price

function,

so

called

by

the
author,
which

is

given

by

f(z)

=

f(x)+f(y),

where
f(x) =
2Xn1
i=0

1+ (x(2i+1) +x(2i+2) +1)2
·

(1914x(2i+1)
+3x(2i+1)
2
14x(2i+2) +6x(2i+1)x
(2i+2)
+3x(2i+2)
2
)

·

30+ (2x(2i+1) 3x(2i+2))2

1832x(2i+1) +12x(2i+1)
2
+48x(2i+2) 36x(2i+1)x
(2i+2)
+27x(2i+2)
2

,
(39a)
and
f(y) =
2Xm1
i=0

1+

y(2i+1)
10
+
y(2i+2)
10
+1
2
·

1914
y(2i+1)
10
+3

y(2i+1)
10
2
14
y(2i+2)
10
+6
y(2i+1)
10
y(2i+2)
10
+3

y(2i+2)
10
2!!
·

30+

2
y(2i+1)
10
3
y(2i+2)
10
2
·

1832
y(2i+1)
10
+12

y(2i+1)
10
2
+48
y(2i+2)
10
36
y(2i+1)
10
y(2i+2)
10
+27

y(2i+2)
10
2!!
.
(39b)
BOUND
CONSTRAINTS.

Let

l,u ∈−R2n ×−Z2m

be

the
bound
constraints

given

by
l =

(2.5,...,2.5;25,...,25)t;
u =
(40a)
(40b)
(2.0,...,2.0;20,...,20)t.
OPTIMUM
SOLUTION.

The

unique

global

minimum

is
located
at
zt =
(0,1,...,0,1
|
{z
}
2n
;0,10,...,0,10
|
{z
}
2m
),
(41)
and
fz)
=

3(n+m)
(42)
b.
W

PROBLEM
OBJECTIVE
FUNCTION.

Let

f(z)

:

Rn ×−Zm →−R be

the
W
function,

so

called

by

the

author,

which

is

given

by
f(z)
Xn
i=1

x(i)
4
4


x
(i)
2
2
+
Xm
i=1

y(i)
4
4


y
(i)
2
2
(43)
BOUND
CONSTRAINTS.

Let

l,u ∈−Rn ×−Zm be

the

bound
constraints
given

by
l =

(100,...,100)t;
u =
(44a)
(100,...,100)t.
(44b)
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
133