Jurnal Karya Asli Lorekan Ahli Matematik Vol. 7 No.2 (2015) Page 084-097
Jurnal
Karya Asli Lorekan
Ahli Matematik
CONTINUOUS PARAMETER FREE FILLED FUNCTION METHOD
Herlina Napitupulu1, Ismail Bin Mohd2 and Ridwan Pandiya3
1,2,3School
of Informatics and Applied Mathematics,
Universiti Malaysia Terengganu,
Terengganu, Malaysia.
[email protected], [email protected], [email protected]
Abstract : In global minimization problems there are two difficulties faced by researchers, firstly is how to move
from one local minimizer to another minimizer with less function value, secondly is how to decide that the current
minima is the global. Filled function method is one of the recent known deterministic methods which widely studied
by scientists to overcome these difficulties. This method has capability in solving multidimensional problems of
multimodal function efficiently, and is considered as an easily applied method. Various kinds of parameter or
parameter free filled functions, algorithm methods as well as its modifications are proposed for the sake of
effectiveness and efficiency in solving global optimization problems. Unfortunately, until now there is no efficient
technique that can be used to compute or to suggest any appropriate parameter. Moreover, many filled function
methods need to choose more than one starting point if the involved functions have more than one global minimizer.
In this paper, we propose a new filled function method without parameter that differs from any existing parameter
free filled functions. Furthermore, we proposed an algorithm method for solving unconstrained global minimization
problems by one starting point only. The performance of the algorithm has been supported by the numerical results
presented in this paper, which show that our method is promising on solving global optimization problem.
Keywords: Filled function, Newton’s method, steepest descent, global optimization.
1. Introduction
Consider the unconstrained global minimization problem
min f ( x) : n 
x
n
.
(1.1)
One of the most worthy of note research areas in mathematics especially in numerical analysis is to
locating the global minimizer of a function of several variables. The reason is due to the existence of
multiple local minimizers, which is different from the global minimizers. There are two main obstacles
when we try to obtain the global one; how to jump from one to a lower local minimum point and how to
give the decision that the current local minimizers is the global solution. Based on these two difficulties,
almost all global optimization problems cannot be solved by classical nonlinear programming techniques
directly.
One of deterministic approach which handling the global optimization problems is known as filled
function method, the method was initially proposed for smooth optimization by Ge [1]. The Ge’s filled
function of f ( x) at isolated minimizer xk* over a domain D has the form
P

x, xk* , r , 

 x  x*
k
1

exp  

2
r  f ( x)








(1.2)
where r and  are adjustable parameters. Several filled functions have been proposed for reconsidering
the obstacles of Ge’s filled function. Some of the proposed filled function are with parameter(s) (see [419], [22], [23] and others are without parameter (see [3], [20], [21]). However, those existing filled
function methods are still have following limitations.
(i).
The existing filled functions do not give guarantee of the existence of a better local minimizer.
© 2015 Jurnal Karya Asli Lorekan Ahli Matematik
Published by Pustaka Aman Press Sdn. Bhd.
Herlina Napitupulu et. al.
(ii).
Some filled functions in the literatures require the assumption that the objective function of
global optimization problem has only a finite number of local minimizer.
(iii). Many filled function methods give an assumption that every local minimizer has the different
values, i.e. f ( x* )  f ( y* ) , if x*  y* . Almost all filled function methods require parameter(s)
to be adjusted.
2. One Dimensional Parameter Free Filled Function
In [3], Goh et al. proposed a new class of filled function which does not require any parameter to be
selected in finding the global minimizer. They used the idea of integration to build the filled function.
Following is the form of parameter free filled function proposed in [3].
  ds  x  xk* 

 x
*
  f ( s )  f xk
 xk*
P x, xk*   *
 xk
*
  f ( s )  f xk
 x


  ds  x  xk* 

(2.1)
where xk* is an isolated local minimum point of f ( x) .
Under some conditions on the function f ( x) , the function P( x, xk* ) is a filled function of f ( x) and
satisfy the definition of filled function ([3]).
3. Two Dimensional Parameter Free Filled Function
In this paper, our proposed parameter free filled function is an approach to find the global
minimizer of a multimodal function f ( x) on 2 , under the following assumptions :
(i).
f ( x) is a continuously differentiable function
(ii). f ( x) has only a finite number of minimizers, and
(iii). f ( x)   as || x || 
The assumption (iii) implies the existence of a closed bounded domain D  2 such that D contains all
minimizers of f ( x) and the value of f ( x) when x is on the boundary of D is greater than any values of
f ( x) when x is inside D .
Consider a function f : D 
D
 x , x 
T
1
2
2

where D is a box defined by

| x1I  x1  x1S  x2 I  x2  x2 S   x1   x1I , x1S  , x2   x2 I , x2 S 
(3.1)
where x jI and x jS ( j  1, 2) are the infimum and supremum of the interval x j   x jI , x jS  respectively.
Assume that x*  ( x1* , x2* )T  D is a current local minimizer of f . The domain D can be divided by x*
4
into four sub domains D1, D2 , D3 , and D4 such that D   Di  D1  D2  D3  D4 , where
i 1
 [ x2* , x2 S ]

D2   x12  [ x1* , x1S ], x22  [ x2 I , x2* ]
D3   x13  [ x1I , x1* ], x23  [ x2* , x2 S ]
D4   x14  [ x1I , x1* ], x24  [ x2 I , x2* ]
D1 
x11
 [ x1* , x1S ], x21
85









(3.2)
Jurnal KALAM Vol. 7 No. 2, Page 084-097
Symbol xij denotes the interval of variable x j in sub domain Di (i  1, 2,3, 4) . Figure 3.1 shows the
illustration of domain D with its sub domain Di (i  1, 2,3, 4) .
( x1* , x2* )T
Figure 3.1 The subdomains D1 , D2 , D3 and D4 separated by ( x1* , x2* )T
One extension of parameter free filled function [3] is so-called TIHR (Two dimensional Ismail Herlina
Ridwan) function, Fi , defined on sub domain Di (i  1, 2,3, 4) , is given by

*
Fi x, x




 Fi1 x, x*     f  s1 , x2   f x1* , x2* ds1  , x1  x1i

 i


 x1





 Fi 2 x, x*     f  x1 , s2   f x1* , x2* ds2  x2  x2i
 i


 x2












(3.3)
The definition of TIHR function is given in Definition 3.1 and its validity is proved by Theorem 3.1Theorem 3.3
Definition 3.1 Fij ( x, x* ) ( i  1, 2; j  1, 2,3, 4 ) is called TIHR function of f ( x) at an isolated local
minimizer x* in D 
2
4
( D   Di ) if
i 0
(i)
x is a maximizer of Fij , Fij ( x* , x* )  Fij ( x, x* ) for all x  D
(ii)
Fij has no stationary point in the set H1  x | f ( x)  f ( x* ), x  D
(iii)
There exist a point x  H 2   x| f ( x) f ( x* ), W 0, xD \ x*    such that x is a stationary point of
*


 

Fij where



    f  s1 , x2  ds1  ( j  1)
 x2  i


 x1

W 

  
  f  x1 , s2  ds2  ( j  2)


 x1  x2i


Theorem 3.1 If x*  ( x1* , x2* )T 
2
4
is an isolated minimizer of the objective function f  x  on D   Di ,
i 0
*
then x is a maximizer of filled function Fij ( i  1, 2; j  1, 2,3, 4 ).
Proof.
86
Herlina Napitupulu et. al.


Since f  x1, x2   f x1* , x2*  0 for all sub domain Di (i  1, 2,3, 4) , we have


  f  s1 , x2   f x1* , x2* ds1  0 for x1  x1i  and
 i

 x1





  f  x1, s2   f x1* , x2* ds2  0 for x2  x2i 
 i

 x2


 

 


Then for all x  Di , Fij x, x*  0 ( i  1, 2; j  1, 2,3, 4 ). Therefore Fij x, x*  0  Fij x* , x* .
■
4
Theorem 3.2 If ( x1* , x2* )T is a local minimizer of f ( x1, x2 ) on D   Di and
H1 
 x , x  | f  x , x   f 
1
2
1
2
i 1
x1* , x2*
 ,  x1, x2   D \  x1*, x2*  ,
then FTIHR has no stationary point in the set H1 .
Proof.
Suppose that  x1, x2 T  H1 satisfies f  x1, x2   f x1* , x2* . Consider FTIHR x, x* in sub domain D1 .


 
Since f is a twice differentiable function, then


F11 x1  ( f (x1, x2 )  f (x1*, x2* ))  0 and  F11 x2    xx x2  f (s1, x2 )  f (x1*, x2* )  ds1   0  .
1
*
1
Therefore, F11  0 , means that F11 has no stationary point in H1 .
For the gradient of F12 , also


 x 

f ( x1, s2 )  f ( x1* , x2* ) ds2   0  and F12 x2  ( f ( x1, x2 )  f ( x1* , x2* ))  0 .
 F12 x1     *2

x2 x

1



Thus F12  0 , means that F12 has no stationary point in H1 . Therefore, in sub domain D1 there is no




stationary point of FTIHR in H1 . The similar proof can be applied to F2 , F3 and F4 ■




Theorem 3.3 If H 2   x1 , x2 T | f  x1 , x2   f x1* , x*2 ,W ( x1 , x2 )  0,  x1 , x2 T  D \ (x1* , x*2 )T   where



    f  s1 , x2  ds1  ( j  1)
 x2  i


 x1

W 

  
  f  x1 , s2  ds2  ( j  2)


 x1  x2i


then there is a point x F  ( x1F , x2F )T  H 2 such that x F is a stationary point of Fij ( i  1, 2; j  1, 2,3, 4 ).
Proof.
Consider F11 in sub domain D1 . Suppose that for ( x1F , x2F )T  H 2 , f x1F , x2F  f x1* , x2* holds and

 

satisfies
  x1
 *
x2  x1
 f  s1, x2   ds1 
0
x xF
Then

F11 x, x
*
 x x
F


T

  x1

   f ( x1 , x2 )  f ( x1* , x2* ) ,
  x* f ( s1 , x2 )ds1  
x2  1


87
  0, 0 
T
x xF
Jurnal KALAM Vol. 7 No. 2, Page 084-097
Therefore, there exists a stationary point of F11 for ( x1, x2 )T  H 2 .

 

Next consider F12 in sub domain D1 . Also for ( x1F , x2F )T  H 2 , f x1F , x2F  f x1* , x2* holds and satisfies
  x2

0.
  *  f  x1 , s2   ds2 
x1  x2
 x xF
Then

F12 x, x*
 x x
F


T
   x2

* * 

  x* f ( x1 , s2 )ds2  ,  f ( x1, x2 )  f ( x1 , x2 ) 

 x1  2

  0, 0 
T
x xF
Therefore there exist a stationary point of F12 for ( x1, x2 )  H 2 . Similar proof can be applied to F2 , F3
and F4 ■
The Fij ( i  1, 2; j  1, 2,3, 4 ) in (3.3) has some special features which are described in this section.
Consider the first equation of (3.3) written as


Fi1 x, x*     f  s1 , x2   f x1* , x2* ds1  , x1  x1i


i
 x1x1


 


The gradient of Fi1 is given by
 
Fi1 x, x
*
T


f  s1 , x2 
   f  x1 , x2   f x1* , x2* ,  
ds1 


x2
x1i





There are two cases to be considered
Case 1. f ( x)  f ( x* ) . Clearly that,


f  x1 , x2 
 F

Fi1
 0 and  i1  0 if
 0  or
x1
x2
 x2

f  x1 , x2 
 F

F
(b) i1  0 and  i1  0 if
 0 .
x1

x

x
2
 2

(a)


Case 2. f ( x)  f ( x* ) . Clearly that,
f  x1 , x2 
 F

Fi1
 0 and  i1  0 if
 0  or
x1

x

x
2
 2

f  x1 , x2 
 F

F
(b) i1  0 and  i1  0 if
 0 .
x1
x2
 x2

(a)
Furthermore, the matrix of second derivatives of Fi1 is given by

 2 Fi1 x, x*

 f
  x
1


  f
 x2

f
x2




 2 f  s1 , x2 

ds1 

x22
x1i


and its determinant is

 
DFi1  Det 2 Fi1 x, x*
  f 2
 f    2 f  s1 , x2 


ds
 

1 
  x2 
x22
 x1   x1i

88
(3.4)
Herlina Napitupulu et. al.
Now suppose that xFg is a stationary (critical) point of Fi1 . According to the second derivative test and
(3.4)
(a) xFg is a local minimum of Fi1 if DFi1  0 and
f
0
x1
(b) xFg is a local maximum of Fi1 if DFi1  0 and
f
0
x1
(c) xFg is a saddle point of Fi1 if DFi1  0
Furthermore, the gradient and the matrix of second derivative of Fi 2 (i  1, 2,3, 4) are given by
 
Fi 2 x, x
*

f  x1 , s2 
  
ds2 ,  f  x1 , x2   f x1* , x2*
 i

x
1
 x2



T




and

 2 Fi 2 x, x*


 2 f  x1 , s2 
 
ds2
2
 xi

x
1
 2
f




x1

f 

x1 

f 

x2 

(3.5)
Now suppose that xFg is a stationary point of Fi 2 and DFi 2 is the determinant of (3.5). Then
(a) xFg is a local minimizer of Fi 2 if DFi 2  0 and 
 2 f  x1 , s2 
x12
x2i
(b) xFg is a local maximizer of Fi 2 if DFi 2  0 and 
ds2  0
 2 f  x1 , s2 
x12
x2i
ds2  0
(c) xFg is a saddle point of Fi 2 if DFi 2  0 .
By considering the first and second partial derivatives of Fij (i  1, 2,3, 4; j  1, 2) , the special features
of Fij can be explored. Suppose that x* and x** are two nearest minimizer of f , and xFg is a stationary
 
point of Fij x, x* (i  1, 2,3, 4; j  1, 2) lied between x* and x** .
Consider these two cases.


Case 1. f ( x** )  f ( x* ) . Clearly that
a)
g
xF1
is a minimizer of Fij (of one dimensional) respect to x j -axis (see Figure 3.2), since
 2 Fij
x 2j
x  xFg1

f
x j
 0 when
x  xFg1
f
x j
0
x  xFg1
b) xFg2 is a maximizer of Fij (of one dimensional) respect to x j -axis (see Figure 3.2b), since
 2 Fij
x 2j
x  xFg2

f
x j
 0 when
x  xFg2
f
x j
0.
x  xFg 2
We do not search the point xFg2 (maximizer of Fij with respect to x j -axis as explained in case 1
(b)) since this point is not the nearest zero of current minimizer x* . Consider only the point xFg1 (a
89
Jurnal KALAM Vol. 7 No. 2, Page 084-097
minimizer of Fij with respect to x j -axis as explained in case 1 (a)) which is attained while f decrease,
in other words the next minimizer, x** , does not passed by yet.
Figure 3.2 Illustration for special character of Fij


Case 2. f ( x** )  f ( x* ) . Since
 2 Fij

x 2j
x xF
f
x j
0.
x xF
for such a j then x F is an inflection point of Fij (of one dimensional) with respect to x j -axis (see
Figure 3.3b).
Figure 3.3 Illustration for special character of Fij
4. Algorithm Method and Numerical Experiment
In this section we propose the algorithm method for solving global optimization problem using
parameter free filled function, combine with radius of curvature, Newton’s method and steepest descent
method. Following are the step of the algorithm.
Data (initialization) specify initial point x0 , domain D , real number d > 0 and set, i  1 , j  1 , m  1
step 1 Specify initial step size of steepest descent  0  0, minimize f ( x) starting at x0 to obtain
minimizer xm*
step 2 Construct FTIHR function Fij ( x, xm* ) at xm*
     or 
 x   1   f ( x) / x    f ( x) / x
step 3 Compute m*  max  x1 xm* ,  x2 xm*
 x1
2 3/2
1
2
*
m
2
1
     is too small, where
 x   1   f ( x) / x    f ( x) / x
*
*
: d if max  x1 xm
,  x2 xm
,  x2
90
2 3/2
2
2
2
2
Herlina Napitupulu et. al.
step 4 If i  4
then choose vector directions ei , where (e1  (1,1)T , e2  (1, 1)T , e3  (1,1)T , e4  (1, 1)T ) and go
to step 5
else stop
step 5 Set c : 1
step 6 Set initial point x0F  xm*  cm* ei .
If x0F  Di
then go to step 7
else if j  n
then j : j  1 and go to step 5
else j  1; i : i 1; and go to step 4
step 7 If f ( x0F )  f ( xm* )
then set x0 : x0F , go to step 1
else if one of (i) or (ii) is true
(i)
(ii)

F
Fij x1
*
*
x  xm
 c m
ei
*
x  xm
 (c 1) m* ei
x2
ij


2
 Fij x1
2
x2
ij
 
  F
2
*
x  xm
 cm* ei
*
x  xm
 (c 1) m* ei
2
then go to step step 8
else c : c  1 and go to step 6
step 8 Solve Fij ( x, xm* )  0 , (i  1, 2,., 4; j  1, 2) using Newton’s method with initial point x0F and obtain
xF
step 9 If (i) and (ii) hold
then set x0 : x F , m : m  1 and go to step 1
else c : c  1 and go to step 6
(i)
2 Fij / x12 x x  0 or   2 Fij / x12 x  x  104 for j  1
F
F
 F
2
ij
(ii)
/ x

2
2 x  xF
 0 or
 F
2
ij
/ x

2
2 x  xF
 104 for j  2
f ( x F )  f ( xm* )  102
Some benchmark test functions are used for observing the capability of the Parameter Free Filled
Function algorithm, the Visual C++ program are used for the algorithm.
Problem 1. Six Hump Back Camel Function
1
f  x1 , x2   4 x12  2.1x14  x16  x1 x2  4 x22  4 x24 ; D   x1, x2   ([-5,5],[-5,5])
3
There are 2 global minimum of this function i.e., x*  (0.08984, 0.71265)T and x*  (0.08984, 0.71265)T
with function value f ( x* )  1.03163
Problem 2. Three Hump Back Camel Function
f  x1 , x2   2 x12  1.05 x14 
1 6
x1  x1 x2  x22 ; D   x1, x2   ([-3,3],[-3,3])
6
The global minimum of this function is x*  (0,0)T with f ( x* )  0
Problem 3. Shubert Function I
5
 5

f  x1 , x2     i cos  (i  1) x1  i    i cos (i  1) x2  i  ; D   x1, x2   ([-10,10],[-10,10])
i 1
 i 1

This function has 760 minima. There are 18 global minimum with function value f ( x* )  186.731
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Jurnal KALAM Vol. 7 No. 2, Page 084-097
Problem 4. Shubert Function II

5
 5
 1
f  x1 , x2     i cos (i  1) x1  i    i cos (i  1) x2  i   ( x1 +1.42513) 2  ( x2  0.80032) 2
i 1
 i 1
 2
D   x1, x2   ([-10,10],[-10,10])

This function has 760 minima. The global minimum is x*  (1.42513, 0.800321)T with function value
f ( x* )  186.731
Problem 5. Shubert Function III
5
 5

f  x1 , x2     i cos (i  1) x1  i    i cos (i  1) x2  i   ( x1 +1.42513) 2  ( x2  0.80032) 2
i 1
 i 1

D   x1, x2   ([10,10],[10,10])
This function has 760 minima. The global minimum is x*  (1.42513, 0.800321)T with function value
f ( x* )  186.731 .
The computational results are summarized in tables for each example. The symbol used in the
tables are as follows. Symbols k , Fij , c , and d , denote number of iteration, evaluated Fij function in
(3.3), an integer to be multiplied with  , and positive real number which replaces  if  is too small,
respectively. The symbols x0 , x F , and xk* represents an initial point for steepest descent method, a point
which satisfies f ( x0 )  f ( xk* ) or a point for solution of F ( x, xk* )  0 which satisfies f ( x F )  f ( xk* ) , and
k-th minimizer obtained from steepest descent method respectively. The symbols f k* is the function value
of xk* . All the global minimizer of each testing functions are written in bold. The numerical results given
in Table 4.1-4.5, shows that the FTIHR algorithm method is succeed in solving two variable unconstrained
global optimization problems of given functions.
Table 4.1 Results for Six Hump Back Camel Function
Results
k
0 , Fij , c, 
0
-
x0 =(2,-1)T, f0 =5.73333
 0 = 1.0
x1* =(1.6071,-0.568652)T, f1* = 2.10425
F31 , c =4,
 =0.132952
x0F =(1.0753,-0.036843)T, f 0F = 2.36695
 0 = 0.01
x2* =(-0.0898424,-0.712657)T, f 2* = -1.03163
F11 , c =8,
 = 0.12825
x0F =(0.936161,0.313347)T, f 0F = 1.46949
 0 = 0.01
x3* =(0.0898418,0.712656)T, f 3* = -1.03163
1
2
x F =( 0.909551,-0.166528)T, f F = 2.10424
x F =(0.0882539,0.707056)T, f F = -1.03137
k
Table 4.2 Results for Three Hump Back Camel Function
Results
0 , Fij , c, 
0
-
x0 =(-2,-1)T, f0 =0.866667
 0 = 1.0
x1* =(-1.74755,-0.87378)T, f1* = 0.298638
F11 , c =3,
 =0.5
x0F =(-0.247552,0.62622)T, f 0F = 0.665833
1
x F =(-0.323583,-0.517783)T, f F = 0.298645
92
Herlina Napitupulu et. al.
 0 = 0.25
x2* =((3.5559x10-7,1.83695x10-6)T, f 2* = 2.97408x10-12
Table 4.3 Results for Shubert I Function
Results
k
0 , Fij , c, 
0
-
x0 =(-1.5, 1.5)T, f0 =-14.1995
 0 = 0.001
x1* =(-1.42513,1.32)T, f1* = -37.6811
F11 , c =2,
 =0.00106311, d=0.2
x0F =(-1.02513,1.72)T, f 0F = -15.2071
 0 = 0.001
x2* =(-0.800321,1.80566)T, f 2* = -39.5887
F11 , c =15,
 =0.00106723, d=0.2
x0F =(2.19968,4.80566)T, f 0F = -29.5833
 0 = 0.001
x3* =(0.334244,4.85806)T, f 3* = -49.5186
F11 , c =2,
 =0.000808974, d=0.2
x0F =(0.734244,5.25806)T, f 0F = -21.0711
 0 = 0.001
x4* =(0.821784,5.48286)T, f 4* = -54.4049
F11 , c =10,
 =0.000776588, d=0.2
x0F =(2.82178,7.48286)T, f 0F = -5.26261
 0 = 0.001
x5* =(5.48286,6.0878)T, f 5* = -123.577
F21 , c =3,
 =0.000341894, d=0.2
x0F =(6.08286,5.4878)T, f 0F = -123.495
 0 = 0.001
x6* =(6.0878,5.48286)T, f 6* = -123.577
F31 , c =13,
 =0.000341894, d=0.2
x0F =(3.4878,8.08286)T, f 0F = -2.73236
 0 = 0.001
x7* =(-7.08351,6.0878)T, f 7* = -123.577
F21 , c =3,
 =0.000341894, d=0.2
x0F =(-6.48351,5.4878)T, f 0F = -123.495
 0 = 0.001
x8* =(-6.47857,5.48286)T, f8* = -123.577
F21 , c =20,
 =0.000341894, d=0.2
x0F =(-2.47857,1.48286)T, f 0F = -5.13392
 0 = 0.001
x9* =(-0.195386,-0.800321)T, f 9* = -123.577
F11 , c =28,
 =0.000341894, d=0.2
x0F =(5.40461,4.79968)T, f 0F = -166.065
 0 = 0.001
*
*
=(5.48286,4.85806)T, f10
= -186.731
x10
F31 , c =3,
 =0.000226263, d=0.2
x0F =(4.88286,5.45806)T, f 0F = -183.936
 0 = 0.001
*
*
=(4.85806,5.48286)T, f11
= -186.731
x11
F21 , c =18,
x0F =(8.45806,1.88286)T, f 0F = -4.99366
1
2
3
4
5
6
7
8
9
10
11
x F =(-0.864149,1.80566)T, f F = -37.6811
x F =(0.242784,4.85806)T, f F = -39.5887
x F =(0.817042,5.39569)T, f F = -49.5186
x F =(5.25519,6.0878)T, f F = -54.4049
x F =(6.08753,5.48286)T, f F = -123.577
x F =(-7.08309,6.0878)T, f F = -123.577
x F =(-6.47884,5.48286)T, f F = -123.577
x F =(-0.195634,-0.800474)T, f F = -123.577
x F =(5.40461,4.79968)T, f F = -166.065
x F =(4.85843,5.48286)T, f F = -186.731
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Jurnal KALAM Vol. 7 No. 2, Page 084-097
12
13
14
15
16
17
18
19
20
21
22
23
 =0.000226262, d=0.2
x F =(4.85799,-0.800545)T, f F = -186.731
 0 = 0.001
*
*
=(4.85806,-0.800321)T, f12
= -186.731
x12
F21 , c =3,
 =0.000226262, d=0.2
x0F =(5.45806,-1.40032)T, f 0F = -183.936
 0 = 0.001
*
*
=(5.48286,-1.42513)T, f13
= -186.731
x13
F31 , c =28,
 =0.000226262, d=0.2
x0F =(-0.117136,4.17487)T, f 0F = -39.6492
 0 = 0.001
*
*
=(-0.800321,4.85806)T, f14
= -186.731
x14
F31 , c =3,
 =0.000226262, d=0.2
x0F =(-1.40032,5.45806)T, f 0F = -183.936
 0 = 0.001
*
*
=(-1.42513,5.48286)T, f15
= -186.731
x15
F41 , c =15,
 =0.000226262, d=0.2
x0F =(-4.42513,2.48286)T, f 0F = -3.10727
 0 = 0.001
*
*
=(-7.08351,-1.42513)T, f16
= -186.731
x16
F21 , c =25,
 =0.000226262, d=0.2
x0F =(-2.08351,-6.42513)T, f 0F = -45.2621
 0 = 0.001
*
*
=(-1.42513,-7.08351)T, f17
= -186.731
x17
F21 , c =3,
 =0.000226262, d=0.2
x0F =(-0.825129,-7.68351)T, f 0F = -183.936
 0 = 0.001
*
*
=(-0.800321,-7.70831)T, f18
= -186.731
x18
F31 , c =34,
 =0.000226262, d=0.2
x0F =(-7.60032,-0.908314)T, f 0F = -137.703
 0 = 0.001
*
*
=(-7.70831,-0.800321)T, f19
= -186.731
x19
F11 , c =2,
 =0.000226262, d=0.2
x0F =(-7.30831,-0.400321)T, f 0F = -20.0319
 0 = 0.001
*
*
=(-7.08351,4.85806)T, f 20
= -186.731
x20
F21 , c =25,
 =0.000226262, d=0.2
x0F =(-2.08351,-0.141943)T, f 0F = -45.2621
 0 = 0.001
*
*
=(-1.42513,-0.800321)T, f 21
= -186.731
x21
F21 , c =3,
 =0.000226262, d=0.2
x0F =(-0.825129,-1.40032)T, f 0F = -183.936
 0 = 0.001
*
*
=(-0.800321,-1.42513)T, f 22
= -186.731
x22
F21 , c =25,
 =0.000226262, d=0.2
x0F =(4.19968,-6.42513)T, f 0F = -45.2621
 0 = 0.001
*
*
=(4.85806,-7.08351)T, f 23
= -186.731
x23
F21 , c =3,
x0F =(5.45806,-7.68351)T, f 0F = -183.936
x F =(5.48247,-1.42513)T, f F = -186.731
x F =(-0.800031,4.85799)T, f F = -186.731
x F =(-1.42476,5.48286)T, f F = -186.731
x F =(-7.0832,-1.42513)T, f F = -186.731
x F =(-1.42549,-7.08351)T, f F = -186.731
x F =(-0.800718,-7.70831)T, f F = -186.731
x F =(-7.70795,-0.800321)T, f F = -186.731
x F =(-7.0839,4.85806)T, f F = -186.731
x F =(-1.42549,-0.800321)T, f F = -186.731
x F =(-0.800718,-1.42513)T, f F = -186.731
x F =(4.8577,-7.08351)T, f F = -186.731
94
Herlina Napitupulu et. al.
24
25
26
 =0.000226262, d=0.2
x F =(5.48247,-7.70831)T, f F = -186.731
 0 = 0.001
*
*
=(5.48286,-7.70831)T, f 24
= -186.731
x24
F31 , c =65,
 =0.000226262, d=0.2
x0F =(-7.51714,5.29169)T, f 0F = -61.7707
 0 = 0.001
*
*
=(-7.70831,5.48286)T, f 25
= -186.731
x25
F21 , c =23,
 =0.000226262, d=0.2
x0F =(-3.10831,0.882864)T, f 0F = -10.1604
 0 = 0.001
*
*
=(-7.08351,-7.70831)T, f 26
= -186.731
x26
F31 , c =3,
 =0.000226262, d=0.2
x0F =(-7.68351,-7.10831)T, f 0F = -183.936
 0 = 0.001
*
*
=(-7.70831,-7.08351)T, f 27
= -186.731
x27
x F =(-7.70805,5.48286)T, f F = -186.731
x F =(-7.08383,-7.70831)T, f F = -186.731
x F =(-7.70794,-7.08351)T, f F = -186.731
Table 4.4 Results for Shubert II Function
Results
k
0 , Fij , c, 
0
-
x0 =(0.5,0.9)T, f0 =-2.49691
 0 = 0.001
x1* =(0.331661, 0.818834)T, f1* = -11.5688
F11 , c =15,
 =0.00181691, d=0.2
x0F =(3.33166,3.81883)T, f 0F = 10.2742
 0 = 0.001
x2* =(5.4805,6.08599)T, f 2* = -76.0079
F41 , c =2,
 =0.000341475, d=0.2
x0F =(5.0805,5.68599)T, f 0F = -2.01863
 0 = 0.001
x3* =(-7.08223,-1.42499)T, f 3* = -170.531
F11 , c =23,
 =0.000226341, d=0.2
x0F =(-2.48223,3.17501)T, f 0F = -1.79234
 0 = 0.001
x4* =(-1.42513,-0.800321)T, f 4* = -186.731
1
2
3
x F =(5.26512,6.132)T, f F = -11.5687
x F =(-7.07115,-1.64595)T, f F = -76.0079
x F =(-1.51032,-0.800321)T, f F = -170.531
Table 4.5 Results for Shubert III Function
Results
k
0 , Fij , c, 
0
-
x0 =(-1,1)T, f0 =-11.0314
 0 = 0.001
x1* =(0.819615,-0.800321)T, f1* = -49.3611
F11 , c =29,
 =0.000775453, d=0.2
x0F =(6.61961,4.99968)T, f 0F = 61.1614
 0 = 0.001
x2* =(4.85536,-0.800321)T, f 2* = -147.269
F42 , c =31,
 =0.000226181, d=0.2
x0F =(-1.34464,-7.00032)T, f 0F = -119.407
 0 = 0.001
x3* =(-1.42513,-7.08066)T, f 3* = -147.27
F11 , c =47,
 =0.000226463, d=0.2
x0F =(7.97487,2.31934)T, f 0F = 92.2591
1
2
3
x F =(4.77877,-0.581157)T, f F = -49.3609
x F =(-1.42506,-7.07993)T, f F = -147.269
x F =(-0.831108,-1.29877)T, f F = -147.27
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Jurnal KALAM Vol. 7 No. 2, Page 084-097
4
 0 = 0.001
x4* =(-0.800604,-1.42486)T, f 4* = -185.95
F31 , c =3,
 =0.000226133, d=0.2
x0F =(-1.4006,-0.82486)T, f 0F = -183.996
 0 = 0.001
x5* =(-1.42513,-0.800321)T, f 5* = -186.731
x F =(-1.40687,-0.800321)T, f F = -185.95
5. Conclusion
From the numerical results in previous section, it is clear that TIHR algorithm which incorporate
TIHR function, steepest descent method, radius of curvature and Newton’s method succeeds in solving
unconstrained global optimization of given problems by using one starting point only. Summary of
advantages of our proposed algorithm method are as follows:
(i).
(ii).
(iii).
(iv).
(v).
FTIHR has no parameter to be adjusted
FTIHR continuous everywhere
FTIHR has no exponential nor logarithmic term,
FTIHR algorithm method has a simple stopping criteria
FTIHR method succeeds in obtaining/exploring global minimizers of objective function which
has more than one global minimizer with only one starting point.
We conclude that our proposed method is a promising method that effective and efficient compared
with another existing filled function method in solving global optimization problems.
Acknowledgment
This research was supported by Fundamental Research Grant Scheme of Malaysia Vot. No. 59255
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