a staged continuous tabu search algorithm for the global optimization and its applications to the...

Computational Optimization and Applications, 30, 319–335, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

A Staged Continuous Tabu Search Algorithmfor the Global Optimization and its Applicationsto the Design of Fiber Bragg Gratings

R.T. ZHENGN.Q. NGO [email protected]. SHUMS.C. TJINL.N. BINHPhotonics Research Center, School of Electrical & Electronic Engineering, Nanyang Technological University,Nanyang Avenue, Singapore 639798

Received September 23, 2003; Revised November 29, 2004; Accepted March 30, 2004

Abstract. A novel staged continuous Tabu search (SCTS) algorithm is proposed for solving global optimizationproblems of multi-minima functions with multi-variables. The proposed method comprises three stages that arebased on the continuous Tabu search (CTS) algorithm with different neighbor-search strategies, with each devotingto one task. The method searches for the global optimum thoroughly and efficiently over the space of solutionscompared to a single process of CTS. The effectiveness of the proposed SCTS algorithm is evaluated using a set ofbenchmark multimodal functions whose global and local minima are known. The numerical test results obtainedindicate that the proposed method is more efficient than an improved genetic algorithm published previously. Themethod is also applied to the optimization of fiber grating design for optical communication systems. Comparedwith two other well-known algorithms, namely, genetic algorithm (GA) and simulated annealing (SA), the proposedmethod performs better in the optimization of the fiber grating design.

Keywords: staged continuous Tabu search, global optimization, multi-variables, fiber gratings

1. Introduction

Optimization design is a scientific branch using both scientific methods and technologicalapproaches to satisfy technical, economical and social requirements in an ideal way. Usu-ally, optimization problems in engineering can be formulated as nonlinear programmingproblems. Due to the multi-modal and ill-condition character of the objective functions, itis difficult to solve these engineering problems with traditional methods. Hence the study ofglobal optimization methods has become one of the most important topics for engineeringdesigners [9].

Tabu Search (TS) is an iterative search method originally developed by Glover [4], whichhas been successfully applied to a variety of combinatorial global optimization problems[5, 8, 9]. A rudimentary form of this algorithm may be roughly summarized as follows. Itstarts from an initial solution s that is randomly selected. From this current solution s, a setof neighbors, called s ′, is generated by pre-defining such a set of ‘moves’, or perturbation

320 ZHENG ET AL.

of current solution (see details in Section 2). To avoid the endless reiterative cycle, theneighbors of the current solution, which belong to a subsequently defined ‘tabu list’, aresystematically eliminated. The objective function to be minimized is then evaluated for eachgenerated solution s ′, and the best neighborhood of s becomes the new current solution evenif it is worse than s. The ‘move’ that generates the new selected current solution will alsobe stored in the ‘tabu list’, which is circular. When it is full, it is updated by eliminating theprevious estimated solution. Then a new ‘iteration’ is performed: The previous procedure isrepeated by starting from the new current point until satisfying stopping condition. Usually,the algorithm stops after a given number of iterations has occurred without any improvementon the value of the objective function.

The more general form of the method uses more advanced recency and frequency memorythan embodied in the simple tabu list, together with associated intensification and diversi-fication strategies that exploit these memory structures (Glover and Laguna [4]). However,simpler forms of TS are sometimes used for conducting prototype studies, and in someinstances such methods perform remarkably well without resorting to more powerful formsof TS.

Comparing with analytical methods, even simple versions of the TS algorithm have asmaller probability of becoming trapped in a local optimum. The method is also organizedto take advantage of problem-specific information, in contrast to the classical forms of someother methods such as genetic algorithm (GA) and simulated annealing (SA) approaches.1

Because of this focus, the method demonstrates a highly attractive convergence velocity aswell as a high level of reliability.

Tabu search also includes candidate list strategies for generating and sampling neighbors(see, e.g., chapter 3 of [4]). These strategies are extremely important, since often only arelatively small subset of neighbors is generated at any given iteration, especially whenlarge neighborhoods are used, as in the case of multi-variable problems whose neighborsare generated in a multi-dimensional space. Siarry and Berthiau have proposed a ContinuousTabu Search (CTS) approach [10] for nonlinear function optimization that employs a specialcandidate list strategy to generate neighbors. In this method, the solution space is dividedinto several regions. Neighbors are generated in these regions and the remainder of themethod consists of an elementary form of TS that uses only the simple tabu list constructionpreviously mentioned. The authors report impressive results for optimising functions of twoor three variables. But when the number of variables increases, the efficiency of the CTSalgorithm is not satisfactory and it must be improved for those problems with high dimension[10].

In this paper, a new multi-level candidate list method is introduced to give a more effectiveapproach for nonlinear function optimization called Staged Continuous Tabu search (SCTS)algorithm. The algorithm comprises three stages that are based on CTS. Each stage focuseson one task with a special neighborhood definition, and the combined stages are for globaloptimization. Section 2 gives a brief review on the CTS algorithm. Section 3 describes theSCTS algorithm. Section 4 presents experimental results of a set of benchmark functions.In Section 5, to demonstrate the effectiveness of the proposed method, it is applied to thedesign of fiber grating, which is an important component in optical communication systems.Conclusions are given in Section 6.

STAGED CONTINUOUS TABU SEARCH ALGORITHM 321

2. A brief review of the continuous Tabu search algorithm

For the following optimization problem:

mins∈�k

[�(s)], (1)

where �(s) is the objective function to be minimized, and s = [x1, x2, . . . , xk]T is definedas

s ∈ �k and �k = {s | ai ≤ xi ≤ bi }, i = 1, 2, . . . , k. (2)

ai and bi are the boundary values. The basic process of the method, which is organizedaround a simple version of tabu search is summarized as follows.

(1) Generate a random point s that belongs to the space �k as the current solution.(2) A set of neighbors s ′ ∈ �k is then generated by applying s with a series of pertur-

bations or ‘moves’. Generation of neighbors are defined by the following method: theneighborhood space �k of the current solution s is deemed as a ball B(s, r ) centeredon s with a radius r . Considering a set of concentric balls with radii h0, h1, . . . , hn , thespace is partitioned into n concentric‘crowns’. Hence n neighbors of s are obtained byselecting one point randomly inside each crown and eliminating those neighbors thatbelong to the ‘tabu list’.

(3) Evaluate these neighbors with the objective function, choose the best neighbor s∗ andreplace the starting point s even if it is worse than the current solution. Then update the‘tabu list’.

(4) Clear the ‘tabu list’: in particular, some solutions belonging to the ‘tabu list’ can releaseits tabu status if its ‘aspiration level’ is sufficiently high.

(5) Check the stopping condition and return to Step (2) if the condition is not met. Otherwise,stop the iteration procedure and report the results.

Figure 1 shows the flow chart of this algorithm, where the main stages include initial solution,generation of neighbors, selection of the solution and tabu list clearance. From the resultsreported in [10], the strategy of generating neighbors in CTS is more efficient than a naı̈vecandidate list strategy based solely on random sampling, and usually produces neighborsdistributed over the whole solution space.2 However, the method generally encountersdifficulties in finding global optima for high-dimension problems.

3. Staged Continuous Tabu Search algorithm

A new Staged Continuous Tabu Search (SCTS) algorithm is developed to improve onthe CTS algorithm. The SCTS algorithm likewise employs the same rudimentary formof tabu search embodies in CTS, but provides an enhanced candidate list strategy thatsubdivides the CTS approach into three independent processes that generate candidate

322 ZHENG ET AL.

Figure 1. General flow chart of a standard TS algorithm.

neighbors in different ways. The first stage tries to survey the whole solution space tolocalize a ‘prospective point’, which is a solution likely to produce a global optimum. Theobjective of the second stage is to find a point close to the global optimum. The third stagestarts from the solution found in the second stage, and eventually converges to the globaloptimum point. The proposed SCTS algorithm is described below.

• Generation of neighborhoods

As described in the CTS method in Section 2, the neighborhoods are generated in a ballB(s, r ) centered on s with radius r . All neighbors s ′ meet the condition: |s ′ − s| ≤ r.

In the first stage, the radius r1 is defined so that the ball B1(s, r1) contains the wholek-dimension space �k . With radii r1

1 , r21 , . . . , rn1

1 , the ball is partitioned into k concentric‘crowns’ centered on the current solution. One neighbor is produced in each crown. Thusthe i th neighbor s ′

i is generated with the condition:

r i−11 ≤ |s ′

i − s| ≤ r i1,

(r0

1 = 0). (3)


As the ball B1(s, r1) includes the whole space �k , it should be possible for all solutionswithin it to become the neighbors of the current solution s so that the process can investigatethe whole solution space. We define the ‘moves’ to generate neighbors such that someelements of the current solution are randomly replaced. The number of replaced elementsdepends on different crowns. For example, the i th neighbor s ′

i is generated by replacing anyi elements of the current solution.

The radius r2 for the generation of neighbors in the second stage is defined as the minimumradius of radii r1

1 , r21 , . . . , rn1

1 . Followed with another partition process with a set of radiir1

2 , r22 , . . . , rn2

2 , the ball B2(s, r2) is divided into n2 sections. The i th neighbor s ′1 is generated

with a condition given by

r i−12 ≤ |s ′

i − s| ≤ r i2,

(r0

2 = 0). (4)

As described above, the minimum radius defined in the first stage is propagated in onlyone dimension of the current solution. Considering the condition defined in (2), we canproportionally divide the boundary for every dimension into n2 partitions. The neighborscan then be generated by replacing the i th element of the current solution (x ′

i ) with a numbercomputed by:

x ′i = ai + ( j + µ) · (bi − ai )

n2where i = 1, 2, . . . , k; j = 1, 2, . . . , n2. (5)

where k is the dimension number of the current solution s, and µ is a random value between0 and 1. It can be seen that the number of neighbors in this stage is k × n2.

The minimum of radii r12 , r2

2 , . . . , rn22 , is set as the radius r3 to generate neighbors in the

third stage. Instead of partitioning to generate neighbors, the radius of the ball B3(s, r3)decreases with an increase in the iteration number. The generation of the i th neighbor isdefined as

x ′i = xi + µi · bi − ai

n2·(

M3 − m

M3

)(6)

where xi and x ′i are the i th elements of the current solution s and the neighbor s ′ produced,

respectively, µ is a random value between −1 and 1, m is the iteration number withoutany improvement on the current solutions, and M3 is the maximum allowable number ofiterations without any improvement.

• Tabu list

In the underlying TS algorithm, a tabu list stores some solutions that have recently beenselected. It is used to qualify the algorithm to select solutions that have not been selectedbefore so as to escape from being recycled.

Because the three stages in the SCTS algorithm are independent of each other, the tabulists in these stages are thus independent of each other. The list obtained in the first stagewill store those ‘prospective solutions’ found in recent iterations. In the second and third

324 ZHENG ET AL.

stages, the list will store the attributes of ‘moves’ or perturbations that generate the bestneighbors in recent iterations. The tabu list in each stage is always reset at the beginning ofeach stage.

• Stopping conditions

The stopping conditions for the three processes are defined as:

(1) The program will stop after a given number of iterations without any improvement onthe value of the objective function. The number of iterations differs in different stages.

(2) The results satisfy the successful conditions. This stopping condition only applies toproblems with known global optima (such as the benchmark test functions listed in theAppendix).

(3) The search procedure will stop after a pre-defined maximum number of iterations.

In the SCTS algorithm, in the first stage, if any one of the stopping conditions is reachedthen the program will move to the second stage. This also applies to the second stage. Inthe last stage, the program will stop if any one of the stopping conditions is reached.

• Description of the algorithm

Figure 2 shows a flowchart that summarizes the steps of the proposed SCTS algorithm. Thefollowing notations are used:

�k : Space of feasible solutions (k dimensions).s0: Current solution.n1: Length of neighbors generated in the first stage, which is equal to k here.n2: The section number divided within the boundary of every element of s0.n3: Length of neighbors generated in the third stage.s ′: The neighborhood of s0.s∗: The best solution in s ′.sopt: Current best solution found.Mv(i): Maximum number of iterations without improvement of sopt in the i th stage.

As pointed out previously, the neighbors in the first stage are generated in the largestrange so as to explore most of the space. And in the last stage, only a reduced space is usedso that the solution finally converges to the global optimum.

The sensitivity some main parameters in the CTS algorithm has already been discussedin detail in [10]. Usually, these parameters should be adjusted empirically according todifferent problems in order to achieve an efficient optimisation process. As inherited fromthe CTS algorithm, it is found that some parameters in the SCTS algorithm have similarproperties as those of the CTS algorithm. However, we have not investigated in detailthe sensitivity of the parameters. A set of empirical values of the parameters is listed inTable 1 for our experiments of the benchmark functions to test the effectiveness of the SCTSalgorithm.


Figure 2. Algorithmic description of the proposed SCTS algorithm.

326 ZHENG ET AL.

Table 1. Typical parameter values of the SCTS algorithm used for both the benchmark test functions and theoptimised design of the FBG.

Parameters used for Parameters used forList of parameters for SCTS algorithm benchmark functions optimization of FBG

Number of neighbors in the first stage (n1) Number of variables 40

Number of section in the second stage (n2) 5 5

Number of neighbors in the third stage (n3) Number of variables 40

Number of neighbors in the second stage 5 × Number of variables 5 × 40 = 200

Maximum number of iterations without any {40, 10, 10} {40, 10, 10}improvement on the objective function value (Mv)

Maximum number of iterations of SCTS algorithm 8000 8000

4. Numerical tests

To demonstrate the effectiveness of the algorithm, the important parameters to be studiedare convergence, speed and robustness.

The test for convergence employed here is the relative error between the optimum obtainedby the algorithm, Xopt, and a theoretical value of the optimum, X theo, of each function. Therelative error, Erelative, is defined as [1]

Erelative = |Xopt − X theo|X theo

(7)

If the theoretical value of the optimum is zero, the relative error in Eq. (7) becomes

Erelative = |Xopt − X theo| (8)

The criterion of speed means the time taken by the algorithm to find the global optimumof the objective function. However, the computation time also depends on the computationspeed of the computer. Thus, we define the speed criterion by determining the number ofevaluations of the objective function required till a global optimum is found.

Robustness means that the algorithm is versatile and can be applied to solving a variety offunctions. A set of commonly used benchmark test functions whose global optima are known(as listed in the Appendix) is chosen to test our algorithm. These test functions representvarious practical problems in science and engineering. To obtain a statistical comparisonof the optimization results, every test has been performed 100 times (starting from variousrandomly selected points) to ensure that the results obtained are reliable.

Table 2 shows the results obtained from the proposed SCTS algorithm for the four testfunctions, namely, Goldprice, Hartmann34, Branin and Shubert. The criterion of successis the percentage of trials (out of the 100 tests for each function) that can reach the globaloptimum with a relative error of less than 1%. Experimental data obtained from the CTSalgorithm [10] is also shown in the table. From the table, it can be seen that both the


Table 2. Experimental data of the SCTS and CTS algorithms.

Successful rate (%) Number of evaluation functions

Function CTS SCTS CTS SCTS

Goldprice 100 100 1636 696

Hartmann34 100 100 528 691

Branin 100 100 668 491

Shubert 100 100 1123 521

algorithms can successfully find the global optima of all the four test functions. Comparedwith the CTS algorithm, the SCTS algorithm reduces the number of evaluations of theGoldprice test function from 1636 to 696 and the Shubert test function from 1123 to 521.This means that the SCTS algorithm has a faster computation rate for these two particularfunctions. However, for the Hartmann34 and Branin functions, the SCTS algorithm does notshow much improvement over the CTS algorithm, showing that the algorithms are equallyfine for these two particular functions.

Table 3 shows a comparison of the experimental data of various test functions obtainedby the SCTS algorithm with an improved genetic algorithm (IGA). It is noted that the IGAalgorithm can potentially yield a complete set of optima when dealing with multimodalproblems [1]. These test functions have variables from 1 to 20 as given in the Appendix. Inthe table, the minimum found (Max) is the maximum value of the optimum found and theminimum found (Min) is the value of the minimum of optimum found over 100 tests.

From Table 3, the SCTS algorithm outperforms the IGA algorithm in two ways. Oneadvantage is that the SCTS algorithm can find the global optima (see, for example, theBrown1, Brown 3 and F10n functions) that the IGA algorithm fails to find. Moreover, therelative errors obtained by the SCTS algorithm for these functions can reach a satisfactorylevel cf close to zero. The other advantage is that the SCTS algorithm greatly reduces thecomputation time as indicated by the smaller number of evaluation of the test functions. Inaddition, the SCTS algorithm reduces the relative error for those functions for which theglobal optima obtained by the IGA algorithm with a successful rate of 100%. These testfunctions are F1, F3, Branin, Goldprice, Shubert1, Shubert2, Shubert, Hartmann34, F5nand F15n.

5. Application of the SCTS algorithm to the design of fiber Bragg gratings

The increasing demand for high transmission capacity requires the channel spacing of densewavelength division multiplexing (DWDM) channels to be very narrower and narrower (e.g.50 GHz and less). It is therefore important to develop optical filters with block-wall likespectral response that can separate wavelength channels of such narrow channel spacingwith minimum crosstalk. Fiber Bragg grating (FBG) technology can meet this requiredperformance. FBG can be produced by exposing a photosensitive fiber to a spatially varying

328 ZHENG ET AL.

Tabl

e3.

Exp

erim

enta

ldat

aof

the

test

func

tions

obta

ined

byth

epr

opos

edSC

TS

algo

rith

man

dth

eim

prov

edG

enet

icA

lgor

ithm

(IG

A).

Min

imum

foun

dM

inim

umfo

und

Rel

ativ

eer

ror

Num

ber

ofev

alua

tion

Succ

ess

(Min

)(M

ax)

aver

age

(%)

ofte

stfu

nctio

nra

te(%

)N

umbe

rof

The

oret

ical

Func

tion

vari

able

sm

inim

umIG

ASC

TS

IGA

SCT

SIG

ASC

TS

IGA

SCT

SIG

ASC

TS

F11

−1.1

232

−1.1

232

−1.1

232

−1.1

139

−1.1

223

0.03

078

413

410

010

0

F31

−12.

0312

−12.

0312

−12.

0312

−11.

9270

−12.

0203

0.12

074

418

110

010

0

Bra

nin

20.

3979

0.39

790.

3979

0.40

180.

3983

0.48

020

4049

210

010

0

Gol

dpri

ce2

33.

003

3.00

23.

0296

3.00

290.

430

4632

696

100

100

Shub

ert1

2−1

86.7

309

−186

.685

7−1

86.7

304

−184

.955

4−1

86.3

406

0.53

0.01

488

5321

9410

010

0

Shub

ert2

2−1

86.7

309

−186

.704

7−1

86.7

302

−184

.929

5−1

85.9

505

0.53

0.02

441

1620

5310

010

0

Shub

ert

2−1

86.7

309

−186

.728

0−1

86.7

269

−184

.875

3−1

86.5

490

0.49

0.00

0323

6452

110

010

0

Har

tman

n34

3−3

.862

8−3

.861

1−3

.862

1−3

.824

6−3

.859

10.

510.

018

1680

560

100

100

Bro

wn1

202

8.55

162.

0018

111.

2914

2.00

2026

92.6

70.

0512

8644

1114

300

100

Bro

wn3

200

0.67

460.

0006

5.91

220.

0010

2.32

40.

0009

1068

5915

142

510

0

F5n

200

0.00

220.

0001

0.59

060.

0010

0.06

70.

0004

9994

517

443

100

100

F10n

200

0.04

960.

0001

4.06

600.

0010

1.19

70.

0008

1139

2919

931

4910

0

F15n

200

0.00

340.

0003

0.73

610.

0009

0.07

50.

0008

1024

1319

660

100

100


Figure 3. Schematic diagram of the piecewise-uniform FBG.

pattern of ultraviolet intensity. They have several unique advantages which include smallsize, low loss, low polarization sensitivity, all-fiber geometry, easy fabrication, and lowcost. A uniform FBG is the simplest type of FBG in design and fabrication but the roll-offbetween the passband and the cut-off band of its reflection spectrum is not sharp due to thepresence of the secondary maxima on both sides of the main reflectance peak. This is due tothe finite length of the uniform FBG with a constant modulation depth of the refractive indexalong the fiber length. Therefore non-uniform FBGs with square-like spectra are requiredfor the DWDM application.

The transfer matrix method (TMM) is normally used to obtain the spectra of non-uniformFBGs [2]. In TMM, the non-uniform FBGs are divided into a number of serially-connecteduniform sub-gratings or sections (as shown in figure 3). Every uniform section has an analytictransfer matrix .The transfer matrix for the entire structure can be obtained by multiplyingthe individual transfer matrices. In figure 3, E f (i) and Eb(i) are complex electric fields ofthe forward and backward propagation waves, respectively, describing, the i th section. δli ,

�i , dni and ni are the length, grating period, amplitude of index modulation and averageeffective index of the i th section, respectively. Lg is the length of whole grating . The electricfields at the input and output ports of the FBG are given by

[E f (0)

Eb(0)

]= T1 · T2 · · · TN

[E f (N )

Eb(N )

](9)

where Ti is the transfer matrix of the i th section of the FBG.Applying the boundary condition, Eb(N ) = 0 (i.e. there is no input to the right side of

the FBG), the reflection R, which is a function of wavelength, is given by

R =∣∣∣∣

Eb(0)

E f (0)

∣∣∣∣

2

(10)

Thus, the problem of optimizing the FBG with a target spectrum can be defined as

min

(∑

λ∈window

Wλ|Rλ(dn, n, �, δl) − Rλ,target|)

(11)

330 ZHENG ET AL.

where Wλ is the weight parameter used to adjust the requirements in defferent wavelength λ

range over the optimization wavelength window. Rλ,target and Rλ(dn, n, �, δl) are the targetreflectivity and calculated reflectivity with specified grating parameters at the wavelength λ.

The Genetic algorithm (GA) and Simulated Annealing algorithm (SA) have been appliedto solve such problems of optimization of FBGs [2, 11]. In this section, as an exampleto demonstrate the effectiveness of the proposed SCTS algorithm to solving an importantengineering problem, it is applied to the design of an FBG with an optimized block-walllike spectral response. Hence, the target spectrum is defined as

Rλ,target ={

1 1549.9 nm ≤ λ ≤ 1550.1 nm

0 λ < 1549.9 nm and λ > 1550.1 nm(12)

Because only the index modulation profile is optimized here, the other parameters of thegrating (i.e. grating period, average effective index and grating length) are pre-specified.The reflectance at a particular wavelength λ is a function of the index modulation profile,which can be described by the transfer matrix method. The index modulation profile is thusexpressed by dn = [dn1, dn2, . . . , dnN ]T with the condition

dnia ≤ dni ≤ dnib, (i = 1, 2, . . . , N ) (13)

where N is the number of sections, dnia and dnib are the boundaries set for the i th elementof dn.

In this design, the number of sections is chosen as N = 40, and the boundary of dn isset as [0, 0.0002], and the parameters of the SCTS algorithm are listed in Table 1.

Figure 4. Optimized index modulation profile of the designed FBG using the SCTS algorithm.


Figure 5. Spectral response of the optimized FBG design using the SCTS algorithm. (Solid line is the spectralresponse of the optimized FBG, Dotted line is the desired spectrum (target spectrum), Dashed line is the spectralresponse of the uniform FBG (non-optimized).)

Figure 4 shows the optimized index modulation profile of a 1-cm long FBG as optimizedby the SCTS algorithm. The corresponding spectral response is illustrated in figure 5. It canbe seen that the spectrum of the uniform FBG (without optimization) has the undesirablesecondary maxima or sidelobes of up to ∼30% on both sides of the main reflection peak. Thesidelobes could create crosstalk or interference in the DWDM application. These sidelobesare greatly suppressed by the optimized FBG designed by the SCTS algorithm.

To verify that the optimum solution of the FBG design obtained by the SCTS algorithmis indeed the global one, two other algorithms, namely, the GA and Adaptive SA (ASA)algorithm presented in references [6, 7] are employed here for comparison. Table 4 shows theoptimum of the objective function (as given in Eq. (11)) obtained by the three algorithms.It is clear that the SCTS algorithm provides the best result with the smallest optimumvalue.

It is to be noted that we have directly used the software of the algorithms from references[6, 7] but these software packages did not give the exact number of evaluations of theobjective function. To compare the computation speed, the three software programs were runon the same computer, and it was found that the GA algorithm took the longest computationtime while the computation time of the ASA algorithm was similar to that of the SCTSalgorithm.

Table 4. The minimum value of the objective function obtained by the three algorithms.

Algorithm applied GA ASA SCTS

Minimum found 4.9583 5.3451 4.9370

332 ZHENG ET AL.

6. Conclusion

We have proposed an effective global optimization algorithm, namely, the SCTS algorithmby combining three stages of the CTS algorithm with three strategies of neighborhoodgeneration. From the test results of a number of benchmark test functions, it has been foundthat the SCTS algorithm is more effective than the original CTS and the IGA in terms of thesuccess rate and computation efficiency. It has also been shown that the SCTS algorithmis a powerful tool for solving some engineering problems such as the optimization of theFBG design. Future research works may focus on incorporating more advanced forms ofthe underlying Tabu search approach to obtain additional improvements in performance.

Appendix: List of test functions

• F1 (1 variable):

f (x) = 2(x − 0.75)2 + sin(5πx + 0.4π ) − 0.125

where 0 ≤ x ≤ 1• F3 (1 variable):

f (x) = −5∑

i=1

{i sin[(i + 1)x + i]}

where −10 ≤ x ≤ 10• Branin (2 variables):

f (x, y) = a(y − bx2 + cx − d)2 + h(1 − g) cos(x) + h

where a = 1, b = 5.14π2 , c = 5

π, d = 6, h = 10, g = 1

8π,

• Goldprice (2 variables):

f (x, y) = [1 + (x + y + 1)2(19 − 14x + 3x2 − 14y + 6xy + 3y2)]

× [30 + (2x − 3y)2(18 − 32x + 12x2 + 48y − 36xy + 27y2)]

• Hartmann1 (H3,4) (3 variables)

f (x) = −4∑

i=1

ci exp

[

−3∑

j=1

ai j (x j − pi j )2

]


where 0 < x j < 1 for j = 1–3

ai j ci pi j

1 3.0 10.0 30.0 1.0 0.3689 0.1170 0.2673

2 0.1 10.0 35.0 1.2 0.4699 0.4387 0.7470

3 3.0 10.0 30.0 3.0 0.1091 0.8732 0.5547

4 0.1 10.0 35.0 3.2 0.0382 0.5743 0.8828

• Shubert1 and 2 (2 variables):

f (x, y) ={

5∑

j=1

j · cos[( j + 1)x + j]

}

×{

5∑

j=1

j · cos[( j + 1)y + j]

}

+ β[(x + 1.42513)2 + (y + 0.80032)2]

where −10 ≤ x, y ≤ 10, β = 0.5 for Shuber1, and β = 1 for Shuber2.• Shubert (2 variables):

f (x) ={

5∑

j=1

j · cos[( j + 1)x1 + j]

}

×{

5∑

j=1

j · cos[( j + 1)x2 + j]

}

where −10 ≤ xi ≤ 10, i = 1, 2;• Brown1 (20 variables):

f (x) =[

∑

i∈J

(xi − 3)

]2

+∑

i∈J

[10−3(xi − 3)2 − (xi − xi+1) + e20(xi −xi+1)

]

where J = {1, 3, . . . , 19}−1 ≤ xi ≤ 4 for 1 ≤ i ≤ 20 and x = [x1, . . . , x20]T

• Brown3 (20 variables):

f (x) =19∑

i=1

[(x2

i

)(x2i+1+1) + (

x2i+1

)(x2i +1)]

where x = [x1, . . . , x20]T and −1 ≤ xi ≤ 4 for 1 ≤ i ≤ 20• F5n (20 variables):

f (x) = (π/20)

×{

10 sin2(πy1) +∑

[(yi − 1)2 × (1 + 10 sin2(πyi+1))] + (y20 − 1)2}

where x = [x1, . . . , x20]T , −10 ≤ xi ≤ 10 and yi = 1 + 0.25(xi − 1)

334 ZHENG ET AL.

• F10n (20 variables):

f (x) =(

π

20

)

×{

10 sin2(πx1) +19∑

i=1

[(xi − 1)2 × (1 + 10 sin2(πxi+1))] + (x20 − 1)2

}

where x = [x1, x2, . . . , x20]T and −10 ≤ xi ≤ 10• F15n (20 variables):

f (x) = (1/10)

{sin2(3πx1)

+19∑

i=1

[(xi − 1)2(1 + sin2(3πxi+1))] + (1/10)(x20 − 1)2[1+sin2(2πx20)]

}

where x = [x1, x2, . . . , x20]T and − 10 ≤ xi ≤ 10.

Acknowledgment

We would like to thank the anonymous reviewers for their constructive comments andsuggestions on the paper.

Notes

1. In recent years, “hybrid variants” of GA and SA methods have emerged that seek to incorporate problem-specific information in a better manner than the classical versions. Some of the more effective instances ofthese hybrid approaches make use of TS strategies.

2. Other ways of applying tabu search to continuous nonlinear optimization are described in [4], but were nottested in [10].

References

1. J. Andre, P. Siarry, and T. Dognon, “An improvement of the standard genetic algorithm fighting prematureconvergence in continuous optimization,” Advances in Engineering software, vol. 32, pp. 49–60, 2001.

2. J. Bae and J. Chun, “Design of fiber bragg gratings using the simulated annealing technique for an ideal WDMfilter bank,” IEEE MILCOM 2000. 21st Century Military Communications Conference Proceedings, 2000,vol. 2, pp. 892–896.

3. T. Erdogan, “Fiber grating spectra,” IEEE J. of Lightwave Technol, vol. 15, no. 8, pp. 1277–1294, 1997.4. F. Glover and M. Laguna, Tabu Search, Kluwer Academic Publishers, 1998.5. S.L. Ho, S.Y. Yang, G.Z. Ni, and H.C. Wong, “An improved Tabu search for the global optimizations of

electromagnetic devices,” IEEE Transactions on Magnetics, vol. 37, pp. 3570–3574, 2001.6. C. Houck, J. Joines, and M. Kay, A Genetic Algorithm for Function Optimization: A Matlab Implementation,

NCSU-IE TR 95-09, 1995. http://www.ie.ncsu.edu/mirage/GAToolBox/gaot/.


7. L. Ingber, “Adaptive simulated annealing (ASA): Lessons learned,” Control and Cybernetics, vol. 25, pp. 33–54, 1996. http://www.ingber.com.

8. D. Karaboga, D.H. Horrocks, N. Karaboga, and A. Kalinli, “Designing digital FIR filters using tabu search al-gorithm,” in IEEE International Symposium on Circuits and Systems, Hong Kong, June 9–12, 1997, pp. 2236–2239.

9. J.M. Machado, S. Yang, S.L. Ho, and P. Ni, “A common Tabu search algorithm for the global optimization ofengineering problems,” Comput. Methods Appl. Mech. Engrg., vol. 190, pp. 3501–3510, 2001.

10. P. Siarry and G. Berthiau, “Fitting of Tabu search to optimize functions of continuous variables,” InternationalJ. for Numerical Methods in Engineering, vol. 40, pp. 2449–2457, 1997.

11. J. Skaar and K.M. Risvik, “A genetic Algorithm for the inverse problem in synthesis of fiber gratings,” IEEEJ. of Lightwave Technol., vol. 16, no. 10, pp. 1928–1932, 1998.

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