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
M
σ
←−
2
n+m
;
Given:
the
number
of
random
sample
for
being
taken
from
each
jth
subregion
σ
j
(k),
N
j
;
the
number
of
random
sample
for
being
taken
from
the
surrounding
region
S(σ(k))
=
σ
M
σ
+1
(k),
N;
the
real
boundaries
of
each
jth
subregion
σ
j
(k)
for
each
kth
iteration,
which
must
be
read
from
the
real
matrix
X =
[x
(jℓ)
]
2M
σ
,n
;
j=1,ℓ=1
the
integer
boundaries
of
each
jth
subregion
σ
j
(k)
for
each
kth
iteration,
which
must
be
read
from
the
integer
matrix
Y =
[y
(jℓ)
]
2M
σ
,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˘
=
[˘x
(ij)
]
4,ni=1,j=1
∈−
R
4×n
;
the
integer
boundaries
of
the
surrounding
region
S(σ(k))
for
each
kth
iteration,
which
must
be
read
from
the
integer
matrix
Y˘
=
[˘y
(ij)
]
4,mi=1,j=1
∈−
Z
4×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
(⌈lg
2
(q)⌉−1)
,
.
.
.
, b
(0)
)
2
∈−
{0, 1}
⌈lg
2
(q)⌉
;
Declare:
the
best
current
point
zˆ
t
=
(ˆx
(1)
,
.
.
.
, xˆ
(n)
;
yˆ
(n+1)
,
.
.
.
, yˆ
(n+m)
)
∈−
R
n
×−
Z
m
;
the
index
of
the
best
performance
of
the
objective
function,
given
by
I(σˆ
j
(k))
=
min
s∈{1,...,N
j
}
−
f(z
s,j
),
where
z
s,j
denotes
the
sth
mixed
integer
sample
point,
which
has
been
taken
from
the
subregion
σ
j
(k);
Choose:
an
nth
index
seed,
namely,
n ∈−N,
which
depends
on
the
sth
random
seed
I(s);
an
N
s
∈−
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
N
o
25.1
Semestre
octubre-febrero
2022
ISSN
electrónico:
2790-5195
ISSN:
1316-3930
136