An Extension of the Nested Partitions Method
EBERT BREA
Main software of
the

Mixed

Integer

Nested

Partitions
Given:
a
pth

bound

constrained

mixed

integer

nonlinear

problem

Pp
minimize
f(z),
z∈Rn×Zm
subject
to

l −

z −u,
where
l, u ∈−Rn ×−Zm;
the
global

minimum

or

considered

true

point

p of

the

bounded

constrained

mixed

integer

nonlinear

problem

Pp;
a
maximum

number

of

replication,

r;
a
set

of

S

random

generator,

namely,


=

{u(s)}I(S)
s=I(1)
,
which

depends

on

the

sth

random

seed

I(s);
the
MINP(s, p, n, m)

algorithmic

method;
Declare:
a
counter

replication,

ℓ ∈−

{1,

.

.

.

, r};
an
th

∈−Rn ×−Zm point,

which

will

be

used

for

saving

the

best

identified

point

by

the

th

running

of

the
MINP(s, p, n, m)

method;
an
th

performance

measure

sample

q(ℓ)(p, n, m)

for

the

(n +

m)

multidimensional

pth

problem,

which

is

given

by
q(ℓ)(p, n, m)
=
1
1
+

η(ℓ)(p, n, m)

·−λ(ℓ)(p, n, m)
,
∀ℓ ∈−N+,
where:
η(ℓ)(p, n, m)

∈−N

is

the

number

of

times

that

has

been

evaluated

the

objective

function

during

the

th

running
of
the

MINP(s, p, n, m)

method;

and

λ(ℓ)(p, n, m)

=

||ˆz −−

z||ˇ

∈−R

is

the

distance

or

norm

between

the

true

point

or
global
optimum

point


and

the

best

point


identified

by

the

MINP(s, p, n, m);
for ℓ ←−
1

to r do
Choose a
not

used

sth

random

seed

for

getting

an

i.i.d.

random

number

generator

for

each


replication;
Run the
MINP(s, p, n, m)

method

for

solving

the

(n +

m)

multidimensional

pth

problem;
Compute the
th

q(ℓ)(p, n, m);
Save:
the
th

of

η(ℓ)(p, n, m),

λ(ℓ)(p, n, m)

and

q(ℓ)(p, n, m);
Estimate:
minimum,
mode,

mean,

maximum,

range,

and

deviation

of

N(p, n, m),

using

the

set

of

sampled

(ℓ)(p, n, m)}rℓ=1;
minimum,
mode,

mean,

maximum,

range,

and

deviation

of

L(p, n, m),

using

the

set

of

sampled

(ℓ)(p, n, m)}rℓ=1;
minimum,
mode,

mean,

maximum,

range,

and

deviation

of

Q(p, n, m),

using

the

set

of

sampled

{q(ℓ)(p, n, m)}rℓ=1;
Figure 10.
Main software of the MINP for taking sampling
vii.
NUMERICAL EXPERIMENTS
In
this

section,

we

shall

summarize

from

a

set

of

three
numerical
examples,

which

are

described

in

Appendix
A.
The

performance

of

the

MINP

method,

and

whose
analysis
will

then

be

discussed

latter

for

illustrating

the
eventual
usefulness

of

the

MINP

method.
One
of

main

noteworthy

features

of

each

one

of

following
problems
is

the

existence

of

2n+m

local

minima

within
its
correspondent

feasible

region,

and

only

one

of

them
is
a

global

minimum,

whereby

could

result

a

challenge,
because
these

problems

are

relatively

difficult

to

identify
their
respective

global

minimum.
The
experiments

were

conducted

for:

a

number

of
random
trial

points

per

subregion

Nσj(k)

=

6;

number

of
random
trial

points

per

surrounding

region

NσM+1(k)

=
96;
and

an

expected

maximum

depth

vector

ε

=
(ǫ,...,ǫ;ǫ,¯
...,ǫ

t,

where

ǫ =

0.1

and

¯ǫ =

0.
a.
GOLDSTEIN-PRICE

PROBLEM
In
this

first

example,

we

have

taken

into

account

an
extension
of

Goldstein-Price

function

to

the

mixed

inte-
ger
Euclidean

field

R2

×−Z2,

which

has

explicitly

been
defined
in

Appendix

A.

For

this

numerical

experiment,
we
run

r =

100

independent

replications

of

the

MINP
method,
using

the

software

of

Figure

10.
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
129