Journal of Applied Mathematics

Volume 2015 (2015), Article ID 245427, 8 pages

http://dx.doi.org/10.1155/2015/245427

## A Filled Function Method Dominated by Filter for Nonlinearly Global Optimization

^{1}Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China^{2}Shangnan Middle School, South Campus, Shanghai 200123, China

Received 18 August 2014; Revised 30 December 2014; Accepted 8 January 2015

Academic Editor: George Fikioris

Copyright © 2015 Wei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work presents a filled function method based on the filter technique for global optimization. Filled function method is one of the effective methods for nonlinear global optimization, since it can effectively find a better minimizer. Filter technique is applied to local optimization methods for its excellent numerical results. In order to optimize the filled function method, the filter method is employed for global optimizations in this method. A new filled function is proposed first, and then the algorithm and its properties are proved. The numerical results are listed at the end.

#### 1. Introduction

In this paper, we consider the following nonlinearly global optimization: where . Global optimization problem has been developed rapidly in recent years. One of the different significant characteristics of the global optimization from local optimization is that it has more than one minimum point. It obviously makes the problem difficult to resolve. In the search of global optimization, we will face two difficulties: the first is how to find a better minimizer from a known local one; the second is how to judge whether the current local minimizer is a global optimum or not. The filled function is one of the valuable methods for the first difficulty. The filled function was first proposed by Ge and Qin [1, 2]. A great deal of efforts has been made by successive scholars and experts [3–6], which makes filled function algorithm develop rapidly. The filled function method has been applied in many practical fields [7].

The filter method was firstly proposed by Fletcher and Leyffer [8, 9] for solving nonlinearly local optimization. Because of its excellent numerical results, many researchers show their interest in it [10, 11]. In order to optimize the filled function method, the filter method is employed for a global optimization in this paper. So we will propose a filter-filled function method for problem (1).

This paper is organized as follows. In Section 2, we first give some assumptions and marks and then some definitions of filled function and the filter method are introduced. In Section 3, a new filled function is proposed. The algorithm for problem (1) and its properties are discussed in Section 4. In the last section, we will list the numerical tests.

#### 2. Preliminaries

Our purpose is to find a global minimizer of problem (1). We make the following assumptions for the objective function throughout the paper. (A1) is continuously differentiable in . (A2) , as ; namely, is a coercive function. (A3) has only a finite number of minimal function values.

According to (A2), we just need to consider the problem where is a closed and bounded domain and contains all of the local and global minimizers of the objective .

For simplicity, we introduce some marks.

and , respectively, stand for the set of local and global minimizers.

Let be a current local minimizer of . It can be obtained by a classical algorithm, such as Newton’s method or steepest descent method.

The radius of is defined as

Consider ; we call the high level set at .

Consider ; we call the low level set at .

We let denote the interior set of and let denote the boundary set of .

For any , its neighborhood is denoted by , and its deleted neighborhood by .

The following is the definition of the filled function for solving problem (1).

*Definition 1. *Assume that is a local minimizer of the original objective function ; a function is called filled function of at , if (i) is a strict local maximizer of ,(ii) has no stationary point in the high lever set ,(iii) is not a global minimum, that is, , then there exists a point , such that is a minimizer of .

Filter technique is usually applied to local optimization methods. In order to optimize the filled function method, we employ it for global optimization in this paper because of its excellent numerical results. The filter mainly consists of two competitive objective functions and , which are denoted by . Now we borrow the concept of domination from multiobjective optimization to give a list of concept of filters.

*Definition 2. *A point is said to dominate another point if and only if both and hold.

*Definition 3. *A filter is a list of pairs such that no point dominates any other. Namely, for the two inequalities and , only one of them is true.

We use to denote the set of such that is an entry in the current filter. A point is said to be “acceptable for the filter” if and only if holds, where and are closed to zero.

We may also “update the filter,” which means that the pair is added to the list of pairs in the filter, and at the same time any pair dominated by in the filter is removed. Namely, we have

By the concepts above, we can define a filter as a criterion for accepting or rejecting a trial step. In this paper, the original objective function will replace , and the filled function will replace . Additionally, stands for the number of elements in the set .

#### 3. A New Filled Function

In this paper, we construct a new filled function with one parameter for problem (1). We suppose that a local minimizer of problem (1) has been obtained; is a parameter. The filled function is defined as

According to assumption (A1), it can be easily proved that the following conclusion is true.

Theorem 4. *The filled function is continuously differentiable in .*

Now we investigate the filled properties of the function .

Theorem 5. *If is a local minimum of problem (1), then is a strict maximizer of for any .*

*Proof. *Consider and ,

Theorem 6. *The function has no stationary point in the region .*

*Proof. *Let ; that is, and ; then
Obviously, it has no stationary point except .

Theorem 7. *Suppose , but ; then for large enough , has a local minimizer in the region .*

*Proof. *Consider ,

Since , there must exist an such that . According to assumption (A3), there is an which makes .

Let . It is easy to see

In the following, we will show that, , , while is large enough.

Consider two cases for : (i) ; (ii) .

(i) We have and by expression (9). Therefore,
That is, , .

(ii) Since
the expression
surely holds for larger enough.

The inequality (12) is equivalent to (10). So holds.

Now we set and it is a compact set for is continuous. The function is continuous on ; thus it has a minimizer . That means

Because , , and
we can learn . But the set on the basis of ; we have got .

#### 4. Filter-Filled Function Algorithm and Its Properties

In this section the search directions and their properties will be discussed first. Then the algorithm will be presented.

With the definition of , at , so the following theorem holds.

When or , we define the search direction at : where .

Theorem 8. *Suppose , .*(1)*If and , then is a descent direction of both and while ;*(2)*if and , then is a descent direction of both and while ;*(3)*if , then is a descent direction of both and while .*

*Proof. *Let denote the angle between and .

(1) Notice that and . We have got
According to , it is obvious that
holds.

(2) The proof is similar to case (1).

(3) If , the condition is equivalent to . Consider

In the same way, we can learn .

*According to the above theorem, we have the following.*

*Corollary 9. Let ; then is a descent direction of both and if and only if .*

*In the following, we discuss the relationship between the filter and the low level set .*

*Theorem 10. Suppose the filter set . If can be accepted by the filter and dominates all of , then .*

*Proof. *If there is an , the conclusion is true since dominates . Otherwise, the point is certainly in , but dominates . According to Definition 2 and inequation (4), , that means .

*As above, we denote a global minimizer by .*

*Theorem 11. Suppose that the current filter is , . There must exist a point such that dominates all of if large enough.*

*Proof. *Two cases will be considered.

(i) , that is, . Then, , there is . Next we show
Inequality (19) is precisely equivalent to

If we set
where is defined in expression (3), there surely will be
since . We can learn that inequality (22) is exactly equivalent to inequality (20) by simple derivation, which means inequality (19) holds.

(ii) . We only need to consider those points in the set , since the situation of the point in the set is similar to case (i).

(a) If there is an , we will show that dominates all of .

Obviously, .

Because is finite, there exists to make . And, according to continuity of , there must be to make inequality
hold.

Now, , from inequality (23) we have
Thus, provided
holds, the inequality can be founded. So we have got that dominates all of if .

(b) . Let . Obviously, based on inequality (23). Similar to the proof of (a), we can get if

*Now the filled function algorithm based on the filter technique for global optimization is listed.*

*Algorithm 12. *
Consider the following.*Step** 1*. Initialization: given a tolerance , set the parameter , ; choose an upper bound of , ; the scale factor ; select an initial point . Obtain a local minimize of from by any classical method. Let the initial filter set , where is large enough. Set , .*Step** 2*. , for , do next.*Step** 3*. Choose a point .*Step** 4*. If , go to next step. Otherwise, go to Step 8.*Step** 5*. If , go to Step 5.1. Otherwise go to Step 6.*Step** 5.1*. If , let , ; go to Step 9. Otherwise, go to Step 7.*Step** 6*. If , go to Step 6.1. Otherwise go to Step 7.*Step** 6.1*. If , , ; otherwise, , ; go to Step 9.*Step** 7*. Consider ; go to Step 9.*Step** 8*. Obtain a local minimize of by taking as the initial point with any classical algorithm. Go to Step 2.*Step** 9*. Let .*Step** 9.1*. If can be accepted by , let ; go to Step 10. Otherwise, go to next step.*Step** 9.2*. Let . If , go to Step 9.1; otherwise, go to Step 11.*Step** 10*. If , update to , ; go to Step 4. Otherwise, go to next step.*Step** 11*. . If ; go to Step 3. Otherwise go to next step.*Step** 12*. If , the algorithm stops and can be treated as the optimal value. Otherwise, ; go to Step 2.

*5. Numerical Results*

*The aim of this section is to apply filter-filled function algorithm to some classical and well known minimization problems. Based on the proposed algorithm, we use matlab 2012b working on the windows 7 system with Inter3 2328 M CPU and 2 G RAM. The numerical examples investigated are the following ones.*

*Problem 1 (6-hump camel back function [4]). *
Consider

The global minimum solutions are or and .

*Problem 2 (-dimensional Sine-square function [2]). *
Consider
The function is tested for . The global minimum solution is uniformly expressed as and .

The algorithm presented can also be used to solve the nonlinear system of equations
where is a vector function. Let in the process of calculation.

*Problem 3 (see [12–14]). *
Consider
where
The known solution of this problem in [14] is

*Problem 4 (see [12–14]). *
Consider
The known solution of this problem in [14] is

*Now we list the computational results. The main iterative results are summarized in Tables 1, 2, 3, 4, and 5 for each function. The symbols used are shown as follows: : the th initial point; : the iteration number in finding the th local minimizer; : the th local minimizer; : the function value of the th local minimizer; : the only entry which dominates the others in the th basin; : the vector function values at in test Problems 3 and 4.*