An Extension of the Nested Partitions Method
EBERT BREA
Main software
of
the
Mixed
Integer
Nested
Partitions
Given:
a
p
th
bound
constrained
mixed
integer
nonlinear
problem
P
p
minimize
f(z),
z∈
R
n
×
Z
m
subject
to
l −
z −u,
where
l, u ∈−
R
n
×−
Z
m
;
the
global
minimum
or
considered
true
point
zˆ
p
of
the
bounded
constrained
mixed
integer
nonlinear
problem
P
p
;
a
maximum
number
of
replication,
r
;
a
set
of
S
random
generator,
namely,
Ω
=
{u(s)}
I(S)
s=I(1)
,
which
depends
on
the
s
th
random
seed
I(s)
;
the
MINP(s, p, n, m)
algorithmic
method;
Declare:
a
counter
replication,
ℓ ∈−
{1,
.
.
.
, r}
;
an
ℓ
th
zˇ
ℓ
∈−
R
n
×−
Z
m
point,
which
will
be
used
for
saving
the
best
identified
point
by
the
ℓ
th
running
of
the
MINP(s, p, n, m)
method;
an
ℓ
th
performance
measure
sample
q
(ℓ)
(p, n, m)
for
the
(n +
m)
multidimensional
p
th
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
ℓ
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
zˆ
and
the
best
point
zˇ
identified
by
the
MINP(s, p, n, m)
;
for
ℓ ←−
1
to
r
do
Choose
a
not
used
s
th
random
seed
for
getting
an
i.i.d.
random
number
generator
for
each
ℓ
replication;
Run
the
MINP(s, p, n, m)
method
for
solving
the
(n +
m)
multidimensional
p
th
problem;
Compute
the
ℓ
th
q
(ℓ)
(p, n, m)
;
Save:
the
ℓ
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
{η
(ℓ)
(p, n, m)}
rℓ=1
;
minimum,
mode,
mean,
maximum,
range,
and
deviation
of
L(p, n, m)
,
using
the
set
of
sampled
{λ
(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
2
n
+
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
N
σ
j
(
k
)
=
6
;
number
of
random
trial
points
per
surrounding
region
N
σ
M
+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
R
2
×−
Z
2
,
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
N
o
25.1
Semestre
octubre-febrero
2022
ISSN
electrónico:
2790-5195
ISSN:
1316-3930
129