An Extension of the Nested Partitions Method
EBERT BREA
SOLUTION.
The

unique

global

minimum

is

located

at
zˆ
=

xt;yˆt)t,

where
xˆt =

(12.2054969669241,...,12.2054969669241
|
{z
}
n
);
(45a)
yˆt =
(12,...,12
|
{z
}
m
),
(45b)
and
fz)
=

115.1035690056n115m.
(46)
c.
ICEBERG

PROBLEM
OBJECTIVE
FUNCTION.

Let

f(z)

:

Rn ×−Zm →−R be

the
Iceberg
function,

so

called

by

the

author,

which

is

given
by
f(z)
=
Xn
i=1

x(i)4
α sin(x(i))

+
Xm
i=1

y(i)4
β sin(y(i))

,
(47)
where
α =

β =

1000.
BOUND
CONSTRAINTS.

Let

l,u ∈−Rn×Zm

be

the

bound
constraints
given

by
l =
(10,...,10)t;
(48a)
u =

(10,...,10)t.

(48b)
SOLUTION.
The

unique

global

minimum

is

located

at
zˆt =
(1.55573432449541;|

...;1.55573432449541
{z
}
n
;2,...,2
|
{z
}
m
),
and
fz)

=

994.028673136238n893.297426825682m.
B
PSEUDOCODE OF PROCEDURES
Procedure Surrounding(n, m, Θ, σ)
Given:
The
number

of

real

components,

n;
The
number

of

integer

components,

m;
The
bounded

feasible

region

Θ,

namely:
l(ℓ) ≤−x(ℓ) ≤−u(ℓ),
∀ℓ ∈−

{1,

.

.

.

, n},
¯l(ℓ) ≤−
y(ℓ) ≤−u¯(ℓ),

∀ℓ ∈−

{1,

.

.

.

, m};
A
promising

region

σ ⊆−Θ,

defined

by
lσ(ℓ) ≤−x(ℓ) ≤−u(ℓσ ),
∀ℓ ∈−

{1,

.

.

.

, n},
¯lσ(ℓ) ≤−
y(ℓ) ≤−u¯(ℓσ ),

∀ℓ ∈−

{1,

.

.

.

, m};
Declare:
A
matrix


=

[˘x(ij)]4,ni=1,j=1 ∈−R4×nfor

saving

real

boundary

components

of

the

set

S(σ(k));
A
matrix


=

[˘y(ij)]4,mi=1,j=1 ∈−Z4×mfor

saving

integer

boundary

components

of

the

set

S(σ(k));
for ℓ ←−
1

to n do
(1,ℓ) ←−
l(ℓ);
(2,ℓ) ←−
lσ(ℓ);
(3,ℓ) ←−uσ(ℓ);
(4,ℓ) ←−u(ℓ);
for ℓ ←−n +
1

to n +

m do
(1,ℓ−n) ←−
¯l(ℓ);
(2,ℓ−n) ←−
¯lσ(ℓ);
(3,ℓ−n) ←−u¯σ(ℓ);
(4,ℓ−n) ←−u¯(ℓ);
Figure 17.
Surrounding procedure
Revista
TEKHNÉ

No

25.1
Semestre
octubre-febrero

2022
ISSN
electrónico:

2790-5195
ISSN:
1316-3930
134