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

Surrounding

and

Promising

Region

and

the

Measuring

of

the

Objective

Function

(Part

ii)
for i ←−
0

to Mσ −−

1

do
for r ←−
1

to Ns do
for j ←−
1

to n do
if
d(j) ≤−

ε(j) then
x(j) ←−xˆ(j)
else
x(j) ←−u(x(2i+1,j), x(2i+2,j), s)
for j ←−
1

to m do
if
(j) ≤−

ε¯(j) then
y(j) ←−
(j)
else
y(j) ←−
¯u(y(2i+1,j), y(2i+2,j), s)
Let zst ←−
(x(1),

.

.

.

, x(n)
|
{z
}
n
;
y(n+1),

.

.

.

, y(n+m)
|
{z
}
m
)
Measure the
objective

function

by

f(z)|z
if f(zs)
s
< fˆ(ˆz)

then
Let fˆ(ˆz)
←−

f(zs);
Let zˆ
←−

zs;
Let ˆi ←−
i;
Let i ←−
i +

1;
switch ˆi do
case ˆi =
0

do
Backtrack to
entire

feasible

region

Θ;
Update the
depth

vector

d by

using

d(0)

=

D(0, n, m, σ(k +

1), Θ);
Let k ←−
0;
other wise do
Let σ(k +
1)

=

σˆi(k);
Update by
using

d(k +

1)

=

D(k +

1, n, m, σ(k +

1), Θ);
Let k ←−k +
1;
Figure 21.
Sampling procedure, part ii
C
PSEUDOCODE OF FUNCTIONS
Function Real
Double

Uniform(a1, b1, a2, b2, n)
Given:
a1 ∈−R:
a

lower

real

bound

of

the

first

interval

uniform

random

distribution;
b1 ∈−R:
an

upper

real

bound

of

the

first

interval

uniform

random

distribution;
a2 ∈−R:
a

lower

real

bound

of

the

second

interval

uniform

random

distribution;
b2 ∈−R:
an

upper

real

bound

of

the

second

interval

uniform

random

distribution;
n ∈−N:
a

nth

seed

from

an

available

pseudorandom

number

generator

set;
u(n)
∈−

(0, 1):

a

uniformly

distributed

random

number

between

0

and

1

from

the

nth
index

seed,

namely,

{u(s)}I(S)

s=I(1)
,

which

depends

on

the

sth

random

seed

I(s);
Output: a uniformly distributed random real number belonging
to

two disjunct

intervals,
namely: (a1, b1)

or (a2, b2)

;
Function w(a1, b1, a2, b2, n):
Calculate θ ∈−R and
x ∈−R,

namely:
θ =
b1 −−a1
b1 +
b2 −−a1 −−a2
;
x =
a1 +

(b1 +

b2 −−a1 −−a2)

u(n);
if θ ≤−u(n)
< 1

then
Let x ←−x +
a2 −−

b1;
return x;
Figure 22.
Double real uniform distribution function
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
138