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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART B: CYBERNETICS, VOL. 40, NO. 6, DECEMBER 2010

SamACO: Variable Sampling Ant ColonyOptimization Algorithm for Continuous Optimizat

Xiao-Min Hu,Student Member, IEEE , Jun Zhang,Senior Member, IEEE ,Henry Shu-Hung Chung,Senior Member, IEEE , Yun Li,Member, IEEE , and Ou Liu

Abstract An ant colony optimization (ACO) algorithm offersalgorithmic techniques for optimization by simulating the foragingbehavior of a group of ants to perform incremental solutionconstructions and to realize a pheromone laying-and-followingmechanism. Although ACO is rst designed for solving discrete(combinatorial) optimization problems, the ACO procedure isalso applicable to continuous optimization. This paper presentsa new way of extending ACO to solving continuous optimizationproblems by focusing on continuous variable sampling as a keyto transforming ACO from discrete optimization to continuousoptimization. The proposed SamACO algorithm consists of three

major steps, i.e., the generation of candidate variable values forselection, the ants solution construction, and the pheromoneupdate process. The distinct characteristics of SamACO are thecooperation of a novel sampling method for discretizing thecontinuous search space and an efcient incremental solutionconstruction method based on the sampled values. The perfor-mance of SamACO is tested using continuous numerical functionswith unimodal and multimodal features. Compared with somestate-of-the-art algorithms, including traditional ant-based algo-rithms and representative computational intelligence algorithmsfor continuous optimization, the performance of SamACO is seencompetitive and promising.

Index Terms Ant algorithm, ant colony optimization (ACO),ant colony system (ACS), continuous optimization, function opti-

mization, local search, numerical optimization.

I. INTRODUCTION

S IMULATING the foraging behavior of ants in nature,ant colony optimization (ACO) algorithms [1], [2] are aclass of swarm intelligence algorithms originally developedto solve discrete (combinatorial) optimization problems, suchas traveling salesman problems [3], [4], multiple knapsackproblems [5], network routing problems [6], [7], scheduling

Manuscript received February 20, 2009; revised July 5, 2009, November 2,2009, and January 29, 2010; accepted February 3, 2010. Date of publicationApril 5, 2010; date of current version November 17, 2010. This work wassupported in part by the National Natural Science Foundation of China JointFund with Guangdong under Key Project U0835002, by the National High-Technology Research and Development Program (863 Program) of China2009AA01Z208, and by the Sun Yat-Sen Innovative Talents Cultivation Pro-gram for Excellent Tutors 35000-3126202. This paper was recommended byAssociate Editor H. Ishibuchi.

X.-M. Hu and J. Zhang are with the Department of Computer Science,Sun Yat-Sen University, Guangzhou 510275, China, and also with the KeyLaboratory of Digital Life (Sun Yat-Sen University), Ministry of Education,Guangzhou 510275, China (e-mail: junzhang@ieee.org).

H. S.-H. Chung is with the Department of Electronic Engineering, CityUniversity of Hong Kong, Kowloon, Hong Kong.

Y. Li is with the Department of Electronics and Electrical Engineering,University of Glasgow, G12 8LT Glasgow, U.K.

O. Liu is with the School of Accounting and Finance, Hong Kong Polytech-nic University, Kowloon, Hong Kong.

Digital Object Identier 10.1109/TSMCB.2010.2043094

problems [8][10], and circuit design problems [11]. Whensolving these problems, pheromones are deposited by ants onnodes or links connecting the nodes in a construction graph [2Here, the ants in the algorithm represent stochastic constructivprocedures for building solutions. The pheromone, which iused as a metaphor for an odorous chemical substance that reaants deposit and smell while walking, has similar effects onbiasing the ants selection of nodes in the algorithm. Each nodrepresents a candidate component value, which belongs to anite setof discrete decision variables. Based on thepheromonevalues, the ants in the algorithm probabilistically select thecomponent values to construct solutions.

For continuous optimization, however, decision variables aredened in the continuous domain, and, hence, the number opossible candidate values would be innite for ACO. Thereforehow to utilize pheromones in thecontinuous domain forguidingants solution construction is an important problem to solve inextending ACO to continuous optimization. According to thuses of the pheromones, there are three types of ant-basedalgorithms for solving continuous optimization problems in thliterature.

The rst type does not use pheromones but uses other formof implicit or explicit cooperation.Forexample,API (namedaf-ter Pachycondyla APIcalis) [12] simulates the foraging behavior of Pachycondyla apicalis ants, which use visual landmarksbut not pheromones to memorize the positions and search thneighborhood of the hunting sites.

The second type of ant-based algorithms places pheromoneon the points in the search space. Each point is, in effect, a complete solution, indicating a region for the ants to perform locaneighborhood search. This type of ant-based algorithms generally hybridizes with other algorithms for maintaining diversityThe continuous ACO (CACO) [13][15] is a combinationof the ants pheromone mechanism and a genetic algorithmThe continuous interacting ant colony algorithm proposed byDro and Siarry [16] uses both pheromone information andthe ants direct communications to accelerate the diffusion oinformation. The continuous orthogonal ant colony (COACalgorithm proposed by Huet al. [17] adopts an orthogonaldesign method and a global pheromone modulation strategyto enhance the search accuracy and efciency. Other methodsuch as hybridizing a NelderMead simplex algorithm [18] anusing a discrete encoding [19] have also been proposed. Sincpheromones are associated with the entire solutions instead ocomponents in this type of algorithms, no incremental solutioconstruction is performed during the optimization process.

The third type of algorithms follows the ACO frameworki.e., the ants in the algorithms construct solutions incrementallbiased by pheromones on components. Socha [20] extended

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the traditional ACO for solving both continuous and mixeddiscretecontinuous optimization problems. Socha and Dorigo[21] later improved their algorithm and referred the resultantalgorithm to ACOR , where an archive was used to preservethe k best solutions found thus far. Each solution variablevalue in the archive is considered as the center of a Gaussianprobability density function (PDF). Pheromones in ACO

Rare

implicit in thegeneration of Gaussian PDFs. When constructinga solution, a new variable value is generated according to theGaussian distribution with a selected center and a computedvariance. The fundamental idea of ACOR is the shift fromusing a discrete probability distribution to using a continuousPDF. The sampling behavior of ACOR is a kind of probabilisticsampling, which samples a PDF [21]. Similar realizations of this type are also reported in [22][28].

Different from sampling a PDF, the SamACO algorithm pro-posed in this paper is based on the idea of sampling candidatevalues for each variable and selecting the values to form solu-tions. The motivation for this paper is that a solution of a

continuous optimization problem is, in effect, a combination of feasible variable values, which can be regarded as a solutionpath walked by an ant. The traditional ACO is good at select-ing promising candidate component values to form high-qualitysolutions. Without loss of the advantage of the traditional ACO,a means to sample promising variable values from the contin-uous domain and to use pheromones on the candidate variablevalues to guide the ants solution construction is developed inSamACO.

Distinctive characteristics of SamACO are the cooperationof a novel sampling method for discretizing the continuoussearch space and an efcient method for incremental solutionconstruction based on the sampled variable values. In

SamACO, the sampling method possesses the feature of balancing memory, exploration, and exploitation. By preservingvariable values from the best solutions constructed by theprevious ants, promising variable values are inherited from thelast iteration. Diversity of the variable values is maintainedby exploring a small number of random variable values. Highsolution accuracy is achieved by exploiting new variable valuessurrounding the best-so-far solution by a dynamic exploitationprocess. If a high-quality solution is constructed by the ants,the corresponding variable values will receive additionalpheromones, so that the latter ants can be attracted to select thevalues again.

Differences between the framework of ACO in solvingdiscrete optimization problems (DOPs) and the proposedSamACO in solving continuous optimization problems willbe detailed in Section II. The performance of SamACO insolving continuous optimization problems will be validatedby testing benchmark numerical functions with unimodal andmultimodal features. The results are compared not only withthe aforementioned ant-based algorithms but also with somerepresentative computational intelligence algorithms, e.g., com-prehensive learning particle swarm optimization (CLPSO) [29],fast evolutionary programming (FEP) [30], and evolution strat-egy with covariance matrix adaptation (CMA-ES) [31].

The rest of this paper is organized as follows. Section IIrst presents the traditional ACO framework for discrete op-timization. The SamACO framework is then introduced in amore general way. Section III describes the implementation

Fig. 1. Framework of the traditional ACO for discrete optimization.

of the proposed SamACO algorithm for continuous optimiztion. Parameter analysis of the proposed algorithm is madin Section IV. Numerical experimental results are presentein Section V for analyzing the performance of the proposealgorithm. Conclusions are drawn in Section VI.

II. ACO FRAMEWORK FORDISCRETE ANDCONTINUOUSOPTIMIZATION

A. Traditional ACO Framework for Discrete Optimization

Before introducing the traditional ACO framework, a dicrete minimization problem is rst dened as in [2] and [21]

Denition 1: A discrete minimization problem is denotedas (S,f, ), where

S is the search space dened over a nite set of discretdecision variables (solution components)X i with val-

uesx ( j )i D i = {x(1)i , x (2)i , . . . , x ( |D i

|)i }, i = 1 , 2, . . . , n ,withn being the number of decision variables. f : S is the objective function. Each candidate so-

lution x S has an objective function valuef (x ). Theminimization problem is to search for a solutionx S that satisesf (x ) f (x ), x S .

is the set of constraints that the solutions inS mustsatisfy.

The most signicant characteristic of ACO is the use opheromones and the incremental solution construction [2The basic framework of the traditional ACO for discreoptimization is shown in Fig. 1. Besides the initialization stethe traditional ACO is composed of the ants solution construction, an optional local search, and the pheromone updatThe three processes iterate until the termination condition satised.

When solving DOPs, solution component values or the linkbetween the values are associated with pheromones. If component values are associated with pheromones, a componenpheromone matrixM , given by (1), shown at the bottom ofthe next page, can be generated. The pheromone value ( j )ireects the desirability for adding the component valuex( j )i tothe solution,j = 1 , 2, . . . , |D i |, i = 1 , 2, . . . , n .

If the links between the values are associated with pheromones, each x( j )i , x

( l )u -tuple will be assigned a pheromone

value ( j,l )i,u , j = 1 , 2, . . . , |D i |, l = 1 , 2, . . . , |D u |, i, u =1, 2, . . . , n , i = u. Since the treatment of placing pheromones

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HU et al. : SamACO: VARIABLE SAMPLING ACO ALGORITHM FOR CONTINUOUS OPTIMIZATION 15

on links or components is similar, the rest of this paper willfocus on the case where pheromones are on components.

As the number of component values is nite in DOPs,pheromone update can be directly applied to the ( j )i in (1)for increasing or reducing the attractions of the correspondingcomponent values to the ants. The realization of the pheromone

update process is the main distinction among different ACOvariants in the literature, such as the ant system (AS) [3], therank-based AS(ASrank ) [32], the Max-Min AS [33], and theant colony system (ACS) [4].

B. Extended ACO Framework for Continuous Optimization

Different from DOPs, a continuous optimization problem isdened as follows [2], [21].

Denition 2: A continuous minimization problem is de-noted as(S,f, ), where

S is the search space dened over a nite set of con-tinuous decision variables (solution components)X i , i =1, 2, . . . , n , with valuesx i [li , u i ], li andu i representingthe lower and upper bounds of the decision variableX i ,respectively.

the denitions of f and are the same as in Denition 1.

The difference from the DOP is that, in continuous opti-mization for ACO, the decision variables are dened in thecontinuous domain. Therefore, the traditional ACO frameworkneeds to be duly modied.

In the literature, researchers such as Socha and Dorigo[21] proposed a method termed ACOR to shift the discreteprobability distribution in discrete optimization to a continuous

probability distribution for solving the continuous optimizationproblem, using probabilistic sampling. When an ant in ACORconstructs a solution, a GramSchmidt process is used for eachvariable to handle variablecorrelation. However, the calculationof the GramSchmidt process for each variable leads to a signif-icantly higher computational demand. When the dimension of the objective function increases, the time used by ACOR alsoincreases rapidly. Moreover, although the correlation betweendifferent decision variables is handled, the algorithm may stillconverge to local optima, particularly when the values in thearchive are very close to each other.

However, if a nite number of variable values are sampledfrom the continuous domain, the traditional ACO algorithms

for discrete optimization can be used. This forms the basicidea of the SamACO framework proposed in this paper. Thekey for successful optimization now becomes how to samplepromising variable values and use them to construct high-quality solutions.

Fig. 2. Framework of the proposed SamACO for continuous optimization.

III. PROPOSEDSamACO ALGORITHM FORCONTINUOUSOPTIMIZATION

Here, the detailed implementation of the proposed SamACOalgorithm for continuous optimization is presented. The success of SamACO in solving continuous optimization problemdepends on an effective variable sampling method and anefcient solution construction process. The variable samplingmethod in SamACO maintains promising variable values fothe ants to select, including variable values selected by the

previous ants, diverse variable values for avoiding trappingand variable values with high accuracy. The ants constructionprocess selects promising variable values to form high-qualitysolutions by taking advantage of the traditional ACO method iselecting discrete components.

The SamACO framework for continuous optimization isshown in Fig. 2. Each decision variableX i has ki sampledvaluesx(1)i , x

(2)i , . . . , x

(k i )i from the continuous domain[li , u i ],

i = 1 , 2, . . . , n . The sampled discrete variable values are thenused for optimization by a traditional ACO process as in solvingDOPs. Fig. 3 illustrates the owchart of the algorithm. Thoverall pseudocode of the proposed SamACO is shown inAppendix I.

A. Generation of the Candidate Variable Values

The generation processes of the candidate variable valuesin the initialization step and in the optimization iterations ar

M =

x(1)1 , (1)1 x(1)2 , (1)2 x(1)n , (1)nx(2)1 , (2)1 x(2)2 , (2)2 x(2)n , (2)n..

....

. ..

..

.x( |D 1 |)1 , ( |D 1 |)1 x ( |D 2 |)2 , ( |D 2 |)2 x( |D n |)n , ( |D n |)n

(1)

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Fig. 3. Flowchart of the SamACO algorithm for continuous optimization.

different. Initially, the candidate variable values are randomlysampled in the feasible domain as

x( j )i = li +u i lim +

j 1 + rand( j )i (2)

where (m + ) is the initial number of candidate values foreach variablei, and rand( j )i is a uniform random number in [0,1], i = 1 , 2, . . . , n , j = 1 , 2, . . . , m + .

During the optimization iterations, candidate variable valueshave four sources, i.e., the variable values selected by antsin the previous iteration, a dynamic exploitation, a randomexploration, and a best-so-far solution. In each iteration,mants constructm solutions, resulting inm candidate values foreach variable for the next iteration. The best-so-far solution isthen updated, representing the best solution that has ever beenfound. The dynamic exploitation is applied to the best-so-farsolution, resulting ingi new candidate variable values near the

corresponding variable values of the best-so-far solution foreach variableX i . Furthermore, a random exploration processgenerates new values for each variable by discarding theworst solutions that are constructed by the ants in the previ-ous iterations. Suppose that the worst solutions are denotedby x (m +1) , x (m +2) , . . . , x (m ) . The new variable valuesfor the solutionx ( j ) are randomly generated as

x ( j )i = li + ( u i li ) rand( j )i (3)

wherei = 1 , 2, . . . , n and j = m + 1 , . . . , m . New values,thus, can be imported in the value group. To summarize, thecomposition of the candidate variable values for the ants toselect is illustrated in Fig. 4. There are a total of (m + gi + 1)candidate values for each variableX i .

Fig. 4. Composition of the candidate variable values for the ants to select

B. Dynamic Exploitation Process

The dynamic exploitation proposed in this paper is effectivfor introducing ne-tuned variable values into the variabvalue group. We use a radiusr i to conne the neighborhoodexploitation of the variable valuex i , i = 1 , 2, . . . , n .

The dynamic exploitation is applied to the best-so-far solution x (0) = ( x (0)1 , x

(0)2 , . . . , x

(0)n ), aiming at searching the

vicinity of the variable valuex(0)i in the interval[x(0)i

r i , x(0)i + r i ], i = 1 , 2, . . . , n . The values of the variables in

the best-so-far solution are randomly selected to be increaseunchanged, or reduced as

x i =

min x (0)i + r i i , u i , 0 q < 1/ 3

x(0)i , 1/ 3 q < 2/ 3max x(0)i r i i , li , 2/ 3 q < 1

(4)

wherei (0, 1] andq [0, 1) areuniform random values,i =1, 2, . . . , n . Then, the resulting solutionx = ( x1 , x2 , . . . , xn )is evaluated. If the new solution is no worse than the recordbest-so-far solution, we replace the best-so-far solution with tnew solution. The above exploitation repeats fortimes. Thenew variable values that aregenerated by increasing or reducina random step length are recorded asx( j )

i, j = m + 1 , m +

2, . . . , m + gi , i = 1 , 2, . . . , n , wheregi counts the number of new variable values in the dynamic exploitation process.

In each iteration, the radiuses adaptively change based on thexploitation result. If the best exploitation solution is no worthan the original best-so-far solution (case 1), the radiusewill be extended. Otherwise (case 2), the radiuses will breduced, i.e.,

r i r i ve , case 1r i vr , case 2

(5)

where ve (ve 1) is the radius extension rate, andvr (0