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

2.67×101.
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
FD(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
FN(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

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
130