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eISSN 0454-8124
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### Article

Kyungpook Mathematical Journal 2024; 64(1): 95-111

Published online March 31, 2024

### Polynomial-Filled Function Algorithm for Unconstrained Global Optimization Problems

Salmah, Ridwan Pandiya

Department of Mathematics, Universitas Gadjah Mada, Sekip Utara BLS 21 Yogyakarta 55281, Indonesia
e-mail : syalmah@yahoo.com

Department of Informatics, Institut Teknologi Telkom Purwokerto, Jawa Tengah 53147, Indonesia
e-mail : ridwanpandiya@ittelkom-pwt.ac.id

Received: January 23, 2023; Revised: July 6, 2023; Accepted: January 22, 2024

The filled function method is useful in solving unconstrained global optimization problems. However, depending on the type of function, and parameters used, there are limitations that cause difficultiies in implemenations. Exponential and logarithmic functions lead to the overflow effect, requiring iterative adjustment of the parameters. This paper proposes a polynomial-filled function that has a general form, is non-exponential, non-logarithmic, non-parameteric, and continuously differentiable. With this newly proposed filled function, the aforementioned shortcomings of the filled function method can be overcome. To confirm the superiority of the proposed filled function algorithm, we apply it to a set of unconstrained global optimization problems. The data derived by numerical implementation shows that the proposed filled function can be used as an alternative algorithm when solving unconstrained global optimization problems.

Keywords: Global optimization, filled function method, nonlinear programming, global minimizer, auxiliary function approach

The filled function method was introduced in [6] in 1993; it was introduced to correct the limitations faced by many traditional methods, such as the tunneling function method, the covering method, and the branch and bound method. A filled function works algorithmically to move from one local minimum point to a lower local minimum point through three main steps. First, the objective function is minimized. Second, a new function, called the filled function, is built at the local minimum point found in Step 1 , and local minimum of it is found. Third, the local minimum point of the filled function found in Step 2 is used as an initial point to minimize the objective function. The capability of the filled function algorithm broke with the notion that solving unconstrained global optimization problems should always involve the use of a multi-start approach or a partitioning method in the search domain. However, along with the development of the method, the use of exponentials and parameters in the filled function [16, 17, 14, 7] led to many unexpected issues.

A filled function involving an exponential function approaches infinity rapidly; this causes the “the overflow” effect. This phenomenon makes the graph of the filled function almost the same as its tangent line. Thus, the local minimum point of the filled function is essentially a “pseudo-minimizer”. The rationale behind the use of exponential or logarithmic functions is the “stretching effect” of these two functions, especially in the region χ1x*=xχ:gxgx*\x*, where g(x) is a cost function, x* is a local minimum point of g(x) in χ, and χ is an operation domain. By this effect, the cost function g(x) has no local minimum points in χ1x*. As a result, many filled functions still employ either an exponential term or a logarithmic function.

On the other hand, the use of parameters is aimed at enabling the constructed filled function to meet three conditions stated in the definition of the filled function. The first condition states that if the filled function is built at a local minimum point x* of a cost function g(x), then the filled function attains its local maximum at x*. The second condition asserts that for all χ1x*, x is not a stationary nor a saddle point of the filled function constructed at x*. The last condition requires that if the filled function is created at x*, and x* is not a global minimum, then the set χ2x*=xχ:gx<gx* contains at least one local minimum point of the filled function.

Parametric-filled functions were studied in [15, 5, 3]. Though the parameters add some difficulty, the advantages that can provide in computational performance in the implementation phase, means that they should not be ignored. For instance, the filled function formulated by [3] utilized an inverse trigonometric function to eliminate the overflow effect produced by using exponential functions. Filled functions constructed to have a general form were studied in [7, 4, 9, 19]. Having a general form, allows a variety of functions to be used to solve optimization problems. However, it introduces the challenge of determining parameter values during the computational stage.

From previous discussions, we tentatively concluded that parameters should be eliminated for the filled function method to be superior. Such efforts were first attempted by [2]. The idea was to select a different function on sets χ1x* and χ2x*. In χ1x*, the function is made independent from the cost function and a descent function. Thus, the local minimum points in χ1x* are all eliminated when minimizing the filled function. On the other hand, in the set χ2x*, the property of the formed filled function is influenced by the cost function. An's filled function was defined as follows:

ω1x,x*=-signgx-gx*x-x*2,

where

signl=1,l01,l<0.

From (1.1), ω1x,x* is discontinuous at points where gx=gx*. However, this property limits the kind of local minimization procedure one can use on the filled function. If a non-gradient based methodology is implemented, such a property becomes beneficial. However, such a method requires a high computational effort. Another problem arises from x-x*2. The norm causes the function value to increase uncontrollably (overflow). This undesirable effect is then reduced by changing x-x*2 into arctanx-x*2 by [10]. To increase the number of local minimization methods that can be employed, the authors in [8] offered a continuously differentiable filled function defined as follows:

ω2x,x*=-x-x*2βgx-gx*

where

βs=1s0es2+2s<0.

However, an exponential function is still involved in (1.2). Therefore, the undesired characteristic previously discussed can possibly occur during the computational stage. The author of [1] attempted to present a new type of parameter filled function as follows:

ω3x,x*=11+x-x*2βgx-gx*

where

βs=π2s0π2arctan(s2)s<0.

However, in our analysis, the following term

11+x-x*2

in (1.3) has almost the same effect as the exponential function, i.e., the change in value is as fast as that in the exponential function.

We propose a new polynomial-filled function to accommodate the need for an effective and efficient parameter-free filled function algorithm. The proposed filled function is simple, does not involve any parameters, and is continuously differentiable. The proposed filled function will be formed in a general form. The filled function was constructed with a polynomial form because polynomial functions are simpler than other filled functions, which generally use exponential, logarithmic, inverse trigonometric, or other transcendental functions. Consequently, the proposed algorithm, where one of the phases involves filled functions, is expected to become more efficient than other filled function algorithms.

This paper is organized as follows. Section 2 provides the preliminaries, assumptions, and definitions involved in this study. Section 3 introduces a new parameter-free polynomial-filled function and its analytical properties. Section 4 discusses a global minimum algorithm, where the proposed polynomial-filled function is involved in one of the algorithm steps. Section 5 presents the numerical experiments and comparisons with some recently filled function algorithms. Finally, Section 6 offers conclusions drawn from the study.

The unconstrained global optimization problems solved in this article should have a solution, i.e., the global minimum value of the cost function. As the problem is unconstrained, the global minimum point must be found in Rn. However, from the numerical point of view, yielding a global minimum value in Rn is impossible. To guarantee the existence of a global minimum point, we assumed that the cost function g(x) is coercive, that is:

limx+gx=+.

The coercive property of g(x) implies the existence of a closed bounded set χ exists, such that χ=χ1χ2x*, with

χ1=xχ:gxgx*\x*

and

χ2=xχ:gx<gx*,

where x* is a local minimum point of g(x). Therefore, the unconstrained global optimization displayed in Problem 1 could be transformed into Problem 2.

Problem 1. Minimize cost function g(x), where xRn.

Problem 2. Minimize cost function g(x), where xχ.

In conclusion, Problems (1) and (2) are equivalent, and the global solution of Problem (2) can be considered the global solution in Rn.

In this paper, g(x) possibly has infinite local minimum points but only finite local minimum points that have different values. The cost function g(x) is a first-order and continuously differentiable function. There are various definition of filled functions in the literature. We use the following.

Definition 1. [18]. A real valued function ωx,x* is a filled function of a cost function g(x) at x*, where x* is a local minimum point of g(x) in χ if it satisfies the following properties.

• The point x* is a strict local maximum point of ωx,x*

• If xχ1, then x is not the stationary point of ωx,x*

• If χ2, then a local minimum point x' of ωx,x* exists in χ2.

### 3. New Filled Function and Its Properties

The parameter-free, continuously differentiable filled functions considered in [8], [11], and [12], give several

idea for creating filled functions as defined in the previous section. One idea is that the filled function should be a piecewise function, allowing one to select a function,

such as -x-x*n where n1 is an integer, with a descent direction property in the region χ1. As the polynomial-filled function formed in this study is intended to be continuously differentiable, one would consider only even values of n. The next task is to find other functions. However, the selected functions ensure that the formed filled function is continuous at x, where gx=gx*. For example, -s+1, s2+1, -s2n+1+s2n+1, with n1. This rationale was used to create the polynomial-filled function

ωx,x*=1x-x*α2gx-gx*

where α is an even integer, such that α2, and 2:RR is a single real valued function with 2s=1 for s0 and 2s=λs for s<0. The condition on α will make ωx,x* continuously differentiable. Therefore, we obtained the following:

ωx,x*=1x-x*α,

for all xχ1, and

ωx,x*=1x-x*αλgx-gx*,

for all xχ2.

To build a specific general function (3.1), functions 1 and λ should satisfy some properties:

• 1s and λs are polynomial functions.

• 1 is continuously differentiable.

• 10=0.

• 1s<0 for all s0,.

• '1s0 in 0,.

• λ is continuously differentiable in -,0.

• λs>1 for all s-,0.

• λ's<0 at s-,0.

• lims0-λs=1.

We need to prove that the function in (3.1) satisfies the three properties in Definition (1).

Theorem 1. Point x*, which is the local minimum point of g(x), is a strict local maximum point of ωx,x*.

Proof. x* is a local minimum point of g(x), which implies the existence of an open ball Bx*,σ, such that gxgx* for all xBx*,σχ. Since gxgx*, then for all xBx*,σχ1, the value of the polynomial-filled function is as follows

ωx,x*=l1xx*α.

Given x-x*α>0 and from property (4), the following is obtained:

ωx,x*=l1xx*α<0.

From Property (3) of 1, we determined the following:

ωx*,x*=l1x*x*α=0.

Hence,

ωx,x*<ωx*,x*,

for all xBx*,σχ. Therefore, x* is a strict local maximum point of ωx,x* in χ.

The minimization process of the filled function in the filled function algorithm requires x* to be at the top of the basin of attraction of ωx,x*. Theorem 1 proves this property. The next two theorems are provided to show that ωx,x* has no stationary point in χ. Theorem 2 is a necessary condition for a filled function to have no stationary points.

Theorem 2. Assume that (1) x* is a local minimum point of g(x), (2) xM and xN are elements of χ1, and (3) xM-x*<xN-x*. Then,

ωxN,x*<ωxM,x*.

Proof. As xM and xN are elements of χ1, gxMgx* and gxNgx*, respectively. From the definition of the proposed filled function,

ωxM,x*=l1xMx*α

and

ωxN,x*=l1xNx*α.

The difference between the value of ω at xM and xN is as follows:

ωxN,x*ωxM,x*=l1xNx*αl1xMx*α.

From properties (4) and (5) of 1, function 1 is decreasing, and the value is negative. Moreover, from the assumption of the theorem, xM-x*<xN-x*. Therefore, the following inequality holds:

l1xNx*α<l1xMx*α.

Hence, the consequence is ωxN,x*<ωxM,x*. This condition proves the theorem.

Theorem 2 reveals that the proposed filled function is a descent territory in χ1, and it is needed because the filled function algorithm minimizes ω. However, some local minimization procedures require a zero gradient. Hence, the proposed filled function should not have any stationary points in χ1.

Theorem 3. If x* is a local minimum point of g(x), then χ1 does not contain the stationary points of ωx,x*.

Proof. Assuming that xMχ1, the following can be obtained:

dTωxM,x*<0.

As xMχ1, then gxMgx*. Therefore, the value of ω at xM is as follows:

ωxM,x*=l1xMx*α.

The gradient of ω at xM is given by the following:

ωxM,x*=αl1xMx*αxMx*α2xMx*.

Given that x* is the element of interior of χ, d=xM-x* is a feasible direction. The directional derivative of ω at xM is computed as follows:

dTωxM,x*=α l 1 xM x* αxMx*α.

As α2, xM-x*α>0, and from property (5) of 1, the following is achieved:

l1 xMx*α <0.

Thus, dTωxM,x*<0. This result proves the theorem.

Thus far, Theorems 1-3 have proven that the proposed filled function satisfies the first and second axioms required by Definition 1.

Theorem 4. Assume that x* is a local minimum point of g(x). If χ2, then ωx,x* has a local minimum point in χ2.

Proof. Assume that χ¯2=xχ:gxgx*. As χ¯2χ, then χ¯2 is bounded. Hence, χ¯2 is a compact and non-empty set. From the form of the proposed filled function, Equation (3.1) is continuously differentiable. From the Weirstrass theorem, ωx,x* has a local minimum point x˜*χ¯2. The proposed filled function ωx,x* is differentiable at x˜*. Thus, x˜* is a stationary point of ωx,x*. On the other hand, ωx˜*,x*=0. From Theorems 2 - 3, ωx,x* does not have stationary points in the region, such that gx=gx*, and χ2 is non-empty. Thus, x˜*χ¯2.

Theorem 4 proves that ωx,x* satisfies the last condition of Definition 1. These theorems guarantee that the global minimum algorithm can be performed. The following theorem is given as an additional property of the proposed filled function (3.1).

Theorem 5. Assume that x* is a local minimum point of g(x). If xNχ1 and xMχ2, such that xN-x*<xM-x*, then ωxM,x*<ωxN,x*.

Proof. As xNχ1, the value of the polynomial-filled function is defined as follows:

ωxN,x*=l1xNx*α.

On the other hand, as xMχ2, based on (3.1), the value of the polynomial-filled function at xM is given as follows:

ωxM,x*=l1xMx*αλgxMgx*.

Given that xMχ2, then gxM-gx*<0. Based on the properties of λ, λgxM-gx*>1 for all xM. Properties (4) and (5) of 1 reveal that 1xM-x*α is negative and strictly decreasing. With

xN-x*<xM-x*,

the following can be obtained:

l1xMx*α<l1xNx*α.

From Equation (4), α2. Therefore, we have the following:

l1xMx*αλgxMgx*<l1xNx*α.

Thus, ωxM,x*<ωxN,x*.

### 4. Filled Function Algorithm

This section focuses on the global minimum algorithm. The following algorithm will be employed to solve the given global optimization problems in this paper.

Poly-ffm Algorithm

• This step is intended to select a certain quantity. First, the initial point x0 is selected from the feasible domain χ. Second, a small real number τ, which is usually 0<τ<1, is obtained. Third, set l=1.

• This phase minimizes the cost function g(x) by any local minimum procedure. In our study, the BFGS method was employed. This method, which is based on the literature, has a high efficiency. In this step, it yields the first local minimum point x*.

• The local minimum point x* yielded in Step 2 is employed to create the initial point to minimize the proposed filled function, i.e., xli, with i=1,2,,p, where p2n, and n is the number of dimension of the cost function. The initial points are formed as xli=x*+τei, and ei is the coordinate direction.

• The value of i starts from 1.

• In this step, if ip, then the algorithm will proceed to Step 6. If all values of i have been used to minimize g(x) using the initial points xli=x*+τei, but no better local minimum point of g(x) has been obtained, then the algorithm will be terminated, and x* will be considered as the global minimum point of g(x) in χ.

• The initial point xli will be examined in this step. If xli is contained in χ, then the algorithm proceeds to Step 7. However, if xli is outside the set χ, then we let i=i+1, and the algorithm proceeds to Step 5.

• In this step, the proposed filled function is constructed at x*:

ωx,x*=l1xx*αl2gxgx*.

• The minimization process of the proposed filled function ωx,x* is carried out in this step using the initial points xli=x*+τei. The local minimum of ωx,x* obtained is denoted by x'.

• This step examines the local minimum point x'. If x' is contained in χ, and gx'<gx*, then we set l=l+1, x0=x', and the algorithm will return to Step 2. However, if these two conditions do not satisfy the conditions, we set i=i+1 and proceed to Step 5.

### 5. Numerical Experiment

As the proposed filled function is in general form, in the implementation, we used one of the specific filled functions, which can be categorized as Equation (4), as follows:

ωx,x*=-x-x*22gx-gx*,

where

l2s=1s0s+1s<0.

The nine steps of Poly-ffm algorithm were implemented to solve the benchmark unconstrained global optimization problems as follows:

Probem 1: Three-hump back camel function

gx=-1.05x14+2x12+16x16-x1x2+x22.

This cost function is minimized in the interior of the box:

-3xj3,

where j=1,2. This cost function has a single global minimum point at x*=0,0. Its global minimum value is gx*=0.

Probem 2: Six-hump back camel function

gx=-2.1x14+4x12+13x16-x1x2+4x24-4x22

This cost function is minimized in the interior of the box:

-3xj3,

where j=1,2. This problem has two global minimum points, namely,

x*=0.0898,0.7126 and x*=-0.0898,-0.7126, with gx*=-1.0316.

Probem 3: Rastrigin function

gx=-cos18x1+x12-cos18x2+x22

This cost function is minimized in the interior of the box

-1xj1,

where j=1,2. Rastrigin function achieves its global minimum at x*=0,0 where gx*=-2.

Probem 4: Two-dimensional function

gx=u2+v2,

where

u=1-x1+csin4πx2-2x2

and

v=-0.5sin2πx1+x2,

with c=0.2, c=0.5, and c=0.05.

This cost function is minimized in the interior of the box

-10xj10,

where j=1,2. This problem has a global minimum value gx*=0.

Probem 5: Treccani function

gx=4x12+4x13+x14+x22

This cost function is minimized in the interior of the box:

-3xj3,

where j=1,2. This problem has a global minimum value gx*=0.

Probem 6: Shubert function

gx=u.v,

where

u= i=15icos i+1x1+i

and

v= i=15icos i+1x2+i.

This cost function is minimized in the interior of the box

0xj10,

where j=1,2. This global optimization problem has 760 local minimum points where the global minimum value is gx*=-186.7309.

Probem 7: n-Dimensional function

gx=πnu+v+w,

with

u=10sin2πx1,
v= i=1 n1 xi121+10 sin2πx i+1,

and

w=xn-12.

This cost function is minimized in the interior of the box

-10xj10,

where j=1,2,,n. This cost functions attain its global minimum value, which is gx*=0, at x*=1,,1.

Probem 8: n-Dimensional Rastrigin function

gx=10n+ i=1nxi210cos 2πxi.

This cost function is minimized in the interior of the box

-5.12xj5.12,

where j=1,2,...,n. This function achieves its global minimum point at

x*=0,,0, gx*=0.

Problems 1-8 will be solved by the Poly-ffm algorithm. The results are displayed in Tables 111. In the tables, t indicates the number of local minimum points of g(x) in χ obtained by the Poly-ffm algorithm. The last t indicates the global local minimum point. For t=1, x10 is the initial point to execute the Poly-ffm algorithm, and for (t=2,3,), xt0 is the local minimum point of the proposed filled function.

Results of Problem 1..

txt0xt*g(xt*)
11.8883,2.43481.7476,0.87380.2986
2-0.3211,-0.39200.0060e-11,-0.2869e-118.4103e-24

Results of Problem 2..

txt0xt*g(xt*)
1-2.3651,1.5669-1.6071,0.56872.1043
20.2332,-0.7941-0.0898,-0.7127-1.0316

Results of Problem 3..

txt0xt*g(xt*)
10.3897,-0.36580.3469,-0.3469-1.7578
20.3469,0.00380.3469,-0.0000-1.8789
3-0.0038,0.0000-0.1428e-18,-0.0157e-18-2

Results of Problem 4, with c=0.2..

txt0xt*g(xt*)
17.5774,-8.23468.7341,-3.33558.8414
20.0756,0.58761.0175,0.05481.7660e-17

Results of Problem 4, with c=0.5..

txt0xt*g(xt*)
17.6552,-6.55107.8000,-6.585072.5124
2-3.8865,0.40911.0000,-0.00001.4348e-19

Results of Problem 5..

txt0xt*g(xt*)
11.1690,-1.09740.0038e-08,0.1341e-081.8033e-18

Results of Problem 6..

txt0xt*g(xt*)
16.1165,-3.47126.6174,-2.5109-13.8031
26.0535,-3.03026.0878,-3.0032-30.7808
34.8338,-1.99424.8581,-2.0072-79.4109
45.5216,-1.39185.4829,-1.4251-186.7309

Results of Problem 7 with n=2..

txt0xt*g(xt*)
15.3103,5.90404.9594,5.996864.1238
24.9594,0.99674.9594,1.000024.8793
30.9587,1.00001.0000,1.00008.2195e-16

Results of Problem 7 with n=3..

txt0xt*g(xt*)
1-2.4363,4.0868,4.5903-1.9697,2.9977,4.989930.2267
21.0135,-0.7409,4.98671.0000,1.0000,1.00006.7045e-20

Results of Problem 8 with n=2..

txt0xt*g(xt*)
11.5648,2.07991.9899,0.99504.9748
2-0.0120,-0.0060-0.2408e-10,0.5000e-100

Results of Problem 8 with n=3..

txt0xt*g(xt*)
1-0.6573,-2.3235,2.5430-1.9899,0.0000,2.984912.9344
20.0125,0.0000,2.9849-0.0000,0.0000,2.98498.9546
30.0000,0.0000,-0.01670.0000,0.0000,0.2489e-090

Tables 111 illustrate some of the results obtained by the Poly-ffm algorithm. The findings indicate that the proposed filled function is reliable to solve unconstrained global optimization problems. Comparison should be performed to examine the competitiveness of the Poly-ffm algorithm. The accuracy of the global minimum value of the cost function should be considered in the comparison stage. The recent filled function algorithm offered by [13] (we call it the ffm algorithm) was selected. The comparison is given as follows.

From Table 12, the Poly-ffm algorithm yields more accurate results than the algorithm given in [13].

Comparison of the results..

ProblemPoly-ffm algorithmffm algorithm
gx*gx*
18.4103e-241.3537e-15
4 (c=0.2)1.7660e-176.4583e-16
4 (c=0.5)1.4348e-192.3665e-15
51.8033e-182.3139e-16
7 (n=2)8.2195e-161.4720e-14
7 (n=3)6.7045e-205.7060e-14

This paper proposed a general form of polynomial-filled functions, where neither exponential nor logarithmic functions are involved. These filled functions were employed in the global optimization algorithm called Poly-ffm algorithm. Eights cost functions, which are commonly used as test functions to examine the effectiveness of an algorithm, were solved by the Poly-ffm algorithm. The numerical data yielded from the experimental computation showed that our study successfully obtained the global minimum values of the given cost functions. Further, comparison was performed to reveal the accuracy of the global minimum value obtained by the Poly-ffm algorithm with another typical filled function algorithm. The comparison results revealed the that global minimum values yielded by Poly-ffm algorithm were more accurate.

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