An Extension of the Nested Partitions Method
EBERT BREA
Preamble of
Sampling

and

the

Measuring

of

the

Objective

Function
Given:
The
number

of

real

components,

n;
The
number

of

integer

components,

m;
Let
Mσ ←−

2n+m;
Given:
the
number

of

random

sample

for

being

taken

from

each

jth

subregion

σj(k),

Nj;
the
number

of

random

sample

for

being

taken

from

the

surrounding

region

S(σ(k))

=

σMσ+1(k),

N;
the
real

boundaries

of

each

jth

subregion

σj(k)

for

each

kth

iteration,

which

must

be

read

from

the

real
matrix
X =

[x(jℓ)]2Mσ,n

;
j=1,ℓ=1
the
integer

boundaries

of

each

jth

subregion

σj(k)

for

each

kth

iteration,

which

must

be

read

from

the
integer
matrix

Y =

[y(jℓ)]2Mσ,m ;
j=1,ℓ=1
the
real

boundaries

of

the

surrounding

region

S(σ(k))

for

each

kth

iteration,

which

must

be

read

from
the

real

matrix


=

[˘x(ij)]4,ni=1,j=1 ∈−R4×n;
the
integer

boundaries

of

the

surrounding

region

S(σ(k))

for

each

kth

iteration,

which

must

be

read
from

the

integer

matrix


=

[˘y(ij)]4,mi=1,j=1 ∈−Z4×m;
a
convertor

function

c(q),

which

converts

any

q ∈−N number,

which

is

given

by

its

decimal
representation,

to

its

binary

representation,

namely,

q =

(b(⌈lg2(q)⌉−1),

.

.

.

, b(0))2 ∈−

{0, 1}⌈lg2(q)⌉;
Declare:
the
best

current

point

t =

(ˆx(1),

.

.

.

, xˆ(n);

(n+1),

.

.

.

, yˆ(n+m))

∈−Rn ×−Zm;
the
index

of

the

best

performance

of

the

objective

function,

given

by
I(σˆ
j(k))

=
min
s∈{1,...,Nj}
f(zs,j),
where
zs,j denotes

the

sth

mixed

integer

sample

point,

which

has

been

taken

from

the

subregion

σj(k);
Choose:
an
nth

index

seed,

namely,

n ∈−N,

which

depends

on

the

sth

random

seed

I(s);
an
Ns ∈−N+ number

of

sampling

per

sector

of

surrounding

region

σ(k)

at

the

kth

iteration;
if k =
0

then Assign the

current

objective

function

value

fˆ(ˆz),

the

largest

possible

value

that

can

be
represented

in

an

x-bit

computer;
Figure 19.
Preamble of the sampling procedure
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
136