Ž .Chemometrics and Intelligent Laboratory Systems 45 1999 339–346

An improved algorithm of sequential number-theoreticž /optimization SNTO based on clustering technique

Fan Gong, Hui Cui, Lin Zhang, Yizeng Liang )

Institute of Chemometrics and Chemical Sensing Technology, College of Chemistry and Chemical Engineering, Hunan UniÕersity,Changsha 410082, China

Abstract

Ž .The sequential number-theoretic optimization SNTO method for global optimization is improved by means of the clus-tering technique. In this way the improved SNTO method can easily locate potential regions for sequential search for theglobal optimum, thus overcoming the difficulty of the original SNTO needing the correct number of points uniformly scatter-ing in search space for the first search. The complexity of the studied object function can be investigated with the help of thestar discrepancy D). The identification of unimodal and multimodal objective functions seems also to be possible. Thesefindings are supported by calculations for simulated and real systems with two-way data. q 1999 Elsevier Science B.V. Allrights reserved.

Keywords: Global optimization; Sequential number-theoretic optimization method; Clustering technique

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

2. Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3402.1. Measure of uniformity of points: the star discrepancy D) . . . . . . . . . . . . . . . . . . . . . . . . . 3402.2. Calculation of the volume occupied by a number of points . . . . . . . . . . . . . . . . . . . . . . . . 3412.3. Identification of unimodal and multimodal by the star discrepancy D) . . . . . . . . . . . . . . . . . . 3412.4. Clustering the sampled points for further search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.5. The algorithm of the improved SNTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

Ž .2.6. Constrained background bilinearization CBBL for the analysis of two-way bilinear data . . . . . . . . 342

3. Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3433.1. Computer simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3433.2. Real chemical systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

) Corresponding author. E-mail: yzliang@public.cs.hn.cn

0169-7439r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-7439 98 00141-5

( )F. Gong et al.rChemometrics and Intelligent Laboratory Systems 45 1999 339–346340

4. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

1. Introduction

The sequential number-theoretic optimizationŽ . w xSNTO method 1,2 was recently introduced intochemometrics as a new global optimization ap-proach. It searches for the global optimum amongpoints uniformly distributed in the space studied, andits convergence is speeded up through sequentialcontraction of the search space. However, in eachsearch only the point with the lowest function valueis located among the uniformally scattered points.Therefore, in order to guarantee convergence to theglobal optimum one needs to choose the correct

w xnumber of points for the first search 2 . Obviously,how to determine this number is not a trivial task. Inour opinion, there is no straightforward way for theoriginal SNTO method to do so easily. Conse-quently, in complex cases it seems to be difficult notto be trapped in local optima. This paper aims at de-veloping an improved SNTO method by means of

w xclustering technique 3–8 and the concept of thew xuniformity of points within a constrained domain 1 .

Exploring clustering for the sampled points as ameans of global optimization methods is based on the

Ž .following ideas: 1 it is possible to obtain a sampleof points concentrated in the neighborhood of poten-

Ž .tial global optimum; 2 the sampled points can beclustered, identifying the neighborhood of potentialglobal optimum. Furthermore, with the help of theconcept of the uniformity of points in a constraineddomain, objective functions with variant complexi-ties can be investigated as a first step. Hence, in theunimodal case one might identify a single potentialglobal optimal point for further search, since the con-centrated points are within one cluster. On the otherhand, if the objective function is multimodal, thepoints sampled for further search are partitioned intotwo or more clusters. Almost all potential optimalpoints can be located at the first search followed by

clustering of all the sampled points. The search spaceis then contracted to the rectangle domain that eachcluster occupies, respectively. The procedures, there-fore, make the determination of the number of pointsfirst uniformly distributed less important.

The improved SNTO method by means of cluster-ing technique is developed in this paper. Compar-isons with the original SNTO method in both simu-lated systems and real two-way bilinear data are thenconducted. The results show that the performance ofthe improved SNTO method is better than the origi-nal one in several respects.

2. Theory

2.1. Measure of uniformity of points: the star dis-crepancy D)

The crucial step of the number-theoretic methodŽ .NTM used in the optimization procedure is how togenerate a set of points, named a NT-net, which areuniformly distributed on an s-dimensional unit cubeC s. There are mainly three kinds of NT-nets, which

Ž .are called the good lattice point set glp set , the goodŽ . Ž .point set gp set and the Halton set H set , respec-

w xtively 1 . The glp set is often used for most practicalcases if sF10. In order to address the uniformilityof the NT-net, the star discrepancy was introduced by

w x � 4the following definition 1 : Let rs x be a set ofks �wn points on C , and k be a set of rectangles; ks c,

x 4 Žw x .d : 0FcFdF1 . Further let N c, d , r be thew xnumber of points of r satisfying x g c, d . Then:k

D) n , r sD) rŽ . Ž .w x w xs sup N c , d ,r rnyV c , dŽ . Ž .

w xc , d gk

Žwis called the star discrepancy D) of r, where V c,x. s Ž . w xd sŁ d yc is the volume of c, d . Fromis1 i i

( )F. Gong et al.rChemometrics and Intelligent Laboratory Systems 45 1999 339–346 341

Ž .the definition, one can conclude that if D) n, r orŽ .D) r is smaller, the points are scattered more uni-

w xformly 1 . For the general case when D is a rectan-w xgle a, b in s-dimensional real space, a similar defi-

nition of the star discrepancy D) can be given. It isobvious that one can reach same conclusion regard-ing uniformity of the distribution.

2.2. Calculation of the Õolume occupied by a numberof points

For m points with s-dimension denoted by q sjŽ . t Ž .q , . . . , q , . . . , q , js1, . . . , m, 1s1, . . . , s ,j1 ji js

we define the volume V X, in which these m pointsoccupy:

sXV s max q ymin q .Ž . Ž .Ž .Ł ji ji

is1

It is practical that we set here a rectangle domaincomposing these m points. V X is the correspondingvolume of this rectangle domain.

2.3. Identification of unimodal and multimodal by thestar discrepancy D)

Suppose that for the first search one takes N0wpoints uniformly scattered in the studied rectangle a,

xb , and then for further search one picks up only g N0

points with the lowest values of the object functionŽ w x .0-g-1, gs0.1 4 is taken in the present work .If the selected points for further search are concen-trated and also uniformly scattered, the volume V X

which these sampled points occupy should be closeto g V. The objective function can then be regardedas a simple one because there maybe only one poten-tial global minimum or all potential global minima areclose to each other. On the other hand, if V X is farbigger than g V and the remaining points are not uni-formly scattered, it is possible that the objectivefunction has several potential global minima whichare far away from each other. The situation is illus-

Ž .trated in Fig. 1. For function f x, y , there is only1Žone minimum. The sampled points denoted by ‘o’ in

Ž ..Fig. 1 1 are concentrated, and they are almost uni-formly distributed in the domain showed by dotted

Ž . Xline in Fig. 1 1 . From the result that V rVs0.1208one may conclude that the sampled points are within

Ž .a cluster. The function f x, y can then be thought1

Fig. 1. Illustration of identification of the unimodal and multi-Ž .modal function by star discrepancy D); 1 The uniformly dis-

Ž . Žtributed points denoted by ‘.’ and the sampled points denoted by. Ž . Ž .2 Ž .2 Ž .‘o’ for an unimodal function f x, y s xy1 q yy2 . 21

Ž .The uniformly distributed points denoted by ‘.’ and the sampledŽ . Ž . Žpoints denoted by ‘o’ for a multimodal function f x, y s yy2

2 2 .2 Ž .5.1 x r4p q5xrp y6 q10 1y1r8p cos xq10.

Ž .to be a unimodal function. For function f x, y , on2

the other hand, there are several potential global min-Žimal points. The sampled points denoted by ‘o’ in

Ž ..Fig. 1 2 are not concentrated at all and not uni-formly scattered in the domain showed by dotted line

Ž . Xin Fig. 1 2 . For this case, V rVs0.5201. It is obvi-ous that these points are not in the same cluster. There

Ž Ž ..are four clusters for this case see Fig. 1 1 . TheŽ .function f x, y may be a multimodal function.2

2.4. Clustering the sampled points for further search

The procedure how to reasonably cluster the sam-pled points for further search will be described asfollows. As the selected points are thought to bewithin a rectangle domain resulting V X Gg V even the

Ž Ž ..reduced points are concentrated see Fig. 1 1 , wesuppose that if V X F1.5g V the sampled points arestill within a cluster. For these case, it is sufficient tolocate only one minimal point among all uniformlyscattered points. The search space is then contractedaround this point. This is the same procedure as the

w xoriginal SNTO method 1 .For the case of V X

)1.5g V, we propose the fol-Ž .lowing clustering procedure: i Locate all the poten-

tial global optima. If the value of the objective func-

( )F. Gong et al.rChemometrics and Intelligent Laboratory Systems 45 1999 339–346342

tion of one point is smaller than that of any point inits neighbourhood, the point can be regarded as a po-tential global minimal point. All potential global op-timum are found similarly, or more concretely, allpoints with minimal function values relative to theirrespective neighbourhoods in the sampled points are

Ž .assigned as potential global optima. ii Set all thepoints of potential global optima as cluster centers.Ž .iii The remaining points are then clustered accord-ing to the distance between each sampled point andevery center. For instance, if the distance between theith sampled point is closer to the jth center than toany other center, then the ith point belongs to thecluster with the jth centers. This step is continued,until all the sampled points are assigned to their cor-

Ž .responding clusters. iv The subspace using for fur-ther search is then contracted to the rectangle domainthat every cluster occupies, respectively.

2.5. The algorithm of the improÕed SNTO

If one wants to find x) such that:

Ms f x) s min f xŽ . Ž .xgD

the algorithm of the improved SNTO can be con-structed as follows.

Ž Ž1..Ž0. Ž Ž1..Ž0.Step0. Set ts0, js1, D sD, a sa,Ž Ž1..Ž0. Ž Ž1..0 s Ž .b sb, V sŁ a yb .is1 i i

Step1. According to the glp set table, select aŽnumber N in general, N ) N s N . . . so as tot 0 1 2

.lighten the computational burden , to generate a setŽ t . Ž Ž j..Ž t .r with N points uniformly scattered on Dt

wŽ Ž j..Ž t . Ž Ž j..Ž t .xs a , b .Ž t . Ž t . � Ž ty1.4Step2. Find out the point x gr j x and

M Ž t . such that:Ž t . Ž t . Ž t . � Ž ty1.4M s f x F f y ,; ygr j x ,Ž . Ž .

� Žy1.4 Ž t . Ž t .where x is the empty set, x and M are thebest approximations to x) and M so far.

Ž Ž j..Ž t . ŽŽ Ž j..Ž t . Ž Ž j..Ž t ..Step3. Let c s b y a r2. IfŽ Ž j..Ž t .sum c F d , where d is a pre-assigned small

number, terminate algorithm. x Ž t . and M Ž t . are ac-ceptable. Otherwise, turn to the next step.

Step4. Determine the sampled points by takingŽ .g N gs0.1 points with minimal function values.t

Ž X Ž j . .Ž t . Ž Ž j . .Ž t .Step5. Calculate V and V . IfŽ XŽ j..Ž t . Ž Ž j..Ž t .V r V Fs 1.5g , j s 1, turn to Step7.Otherwise, continue with Step6.

Step6. Locate all k Ž t . central points accordingtheir relationship with their neighbourhood points,and then classify all sampled points with respect toall the k Ž t . central points, proceed to Step8.

Ž Ž j..Ž tq1.Step7. Form the next search domain D swŽ Ž j..Ž tq1. Ž Ž j..Ž tq1.xa , b :

Ž . Ž . Ž .tq1 t tŽ j. Ž t . Ž j. Ž j.a smax x yh c , a ;Ž . Ž . Ž .ž /i i i i

and

Ž . Ž . Ž .tq1 t tŽ j. Ž t . Ž j. Ž j.b smin x qh c , bŽ . Ž . Ž .ž /i i i i

= is1, . . . ,sŽ .Žwhere h is a predefined contraction ratio here hs

.0.5 . Set ts tq1. Go to Step1.Step8. For js1 to k Ž t ., contract further search

domain one by one: Form the next search domainŽ Ž j..Ž tq1. wŽ Ž j..Ž tq1. Ž Ž j..Ž tq1.x Ž Ž j..Ž t .D s a , b based on n

Ž Ž j . . Ž t . ŽŽ Ž j . . Ž t .points denoted by q s q , . . . ,u u 1Ž Ž j ..Ž t . Ž Ž j ..Ž t .. Ž Ž Ž j ..Ž t .q , . . . , q u s 1, . . . , n , i s 1,u i u s

. Ž t .. . . ,s , belonging to the jth cluster as follows:

Ž . Ž .tq1 tŽ j. Ž j.a smin q ;Ž . Ž .ž /i u i

and

Ž . Ž .tq1 tŽ j. Ž j.b smax qŽ . Ž .ž /i u i

Set ts tq1. Go to Step 1.

( )2.6. Constrained background bilinearization CBBLfor the analysis of two-way bilinear data

CBBL quantitatively determines desired compo-nents in the presence of unknown interferents bymeans of an optimization procedure with the help oftwo-way bilinear data. The full theory may be found

w xin Refs. 9,10 and only a brief explanation of CBBLis given here.

The two-way bilinear response model for an ana-lytical sample with unexpected interferents can beexpressed as:

N M

Ys c X q c R qEÝ Ýi i j jis1 js1

where Y is the bilinear data matrix by measuring thesample, X and R are the bilinear data matrices fori j

N sought-for analytes and M unexpected interfer-

( )F. Gong et al.rChemometrics and Intelligent Laboratory Systems 45 1999 339–346 343

ents, respectively, c and c are corresponding con-i j

centrations, E is the noise matrix. Note that both ofM and c are unknown.j

With assumption of a bilinear response structure,the overall background responses can be decom-posed as follows:

M MTRs c R s t pÝ Ýj j j j

js1 js1

where t and p are the orthonormal score and or-j j

thogonal loading vectors, respectively.Ž .Under the constraint of rank R sM, if and only

if c are estimated correctly, the residual part of thei

background matrix, EsYyÝc X yÝt pT, shouldi i j j

be at the same level as that of the measurement noise.Hence, the sum of squares of the elements of theresidual matrix E can be an objective function of thisoptimization problem, i.e.:

5 5 2f c), R) smin f c , R smin EŽ . Ž .5 5 wwhere . denotes the Euclidean norm, c s c ,1

xTc , . . . , c is the sought-for concentration vector.2 N

Based upon the reasonable assumption of the pos-itivity of spectral intensities and analyte concentra-tions, CBBL searches c in a constrained rectangledomain which can be expressed as:

cmin FcFcmax

min w x max w max max max xwhere c s 0 . . . 0 , c s c . . . c . . . c1 j N

with cmax defined as follows:j

maxc sMin y q´ rx , js1, . . . , NŽ .j k l jk l

where ´ is the estimate for the error bound, y andk l

x are the k th row and lth column elements of Yjk l

and X , respectively.j

The objective function of CBBL is not convex inthe constrained domain and thus a global optimiza-tion method should be used here.

3. Experimental

3.1. Computer simulation

ŽTwo mathematical functions called F1 and F2 see.Table 1 which both have many local minima and a

unique global minimizer are tested with the originaland improved SNTO methods.

3.2. Real chemical systems

A four-component system consisting of o-anilinobenzoic acid, 1,3-benzenediol, benzoic acidand p-hydroxybenzoic acid is analyzed by HPLC-DAD for this paper. o-anilinobenzoic acid, benzoicacid and p-hydroxybenzoic acid are taken as ex-pected analytes and 1,3-benzenediol is regarded asunknown interferent. Standard solutions of o-anilinobenzoic acid, 1,3-benzenediol, benzoic acidand p-hydroxybenzoic acid and two mixtures of fourcomponents are prepared by dissolving correspond-ing chemicals in the solvent. The solvent is preparedby diluting glacial acetic acid to 0.25 molrl with amixture of 85% of methylene dichloride and 15% ofanhydrous ethanol. All the regents are of analytical-grade purity. Benzoic acid and p-hydroxybenzoic arerecrystallized with doubly distilled water.

Standard solutions of o-anilinobenzoic acid, ben-zoic acid and p-hydroxybenzoic acid and two mix-tures are measured on a Shimadzu Liquid Chro-matography 4A equipped with a Shimadzu DAD withthe same solvent as mobile phase at a linear flow-rate

w Žof 1.2 mlrmin. A ZORBOX-SIL column 250 mm.=4.6 mm inner diameter is used at 208C. Spectrum

is recorded in the range 246–406 nm.

Table 1Two mathematical functions F1 and F2

Search domain Global optimal point Global response2 2 2 2� w Ž . x w Ž . x w Ž . x w Ž . x4F1sy 3 exp y xy1.5 r0.2 q5 exp y yy4.8 r0.3 q4 exp y zy3.5 r0.4 q2 exp y uy5 r0.8

Ž .y10Fx, y, z, uF10 1.5000, 4.8000, 3.5000, 5.0000 y14.00002 ny1 2 2 2Ž .� Ž . wŽ . Ž Ž ..x Ž . 4 Ž .F2s prn 10 sin p x qÝ x y1 1q10 sin p x q x y1 ns31 is1 i iq1 n

Ž .y10Fx , x , x F10 1.0000, 1.0000, 1.0000 0.00001 2 3

()

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Table 2Optimization results of functions F1 and F2 by the original and improved SNTO methods, respectively

Improved SNTO method Original SNTO method

Ž . Ž .N N is1, . . . Optimal response Optimal corridate N N is1, . . . Optimal response Optimal corridate0 i 0 i

F1Ž . Ž .307 307 y14.0000 1.5001, 4.8001, 3.4999, 5.0002 307 307 y12.0000 1.5000, 4.8000, 3.5000, 9.3121Ž . Ž .562 307 y14.0000 1.4989, 4.8002, 3.5003, 5.0019 562 307 y14.0000 1.5000, 4.8000, 3.5000, 5.0000Ž . Ž .701 307 y14.0000 1.4989, 4.8000, 3.4999, 5.0001 701 307 y11.0000 3.4480, 4.8000, 3.5000, 5.0000

F2Ž . Ž .266 101 0.0000 1.0000, 1.0000, 0.9990 266 101 0.0328 1.0000, 0.8257, 1.0150Ž . Ž .418 101 0.0000 1.0000, 1.0002, 0.9999 418 101 0.0000 1.0000, 1.0008, 1.0009Ž . Ž .597 101 0.0000 0.9999, 1.0043, 0.9998 597 101 0.3332 1.0000, 0.9999, 1.5641

()

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Table 3Optimization results of the CBBL for sample 1 and 2 by the improved and original SNTO method

Improved SNTO method Original SNTO method

N N Optimal Optimal corridate Relative error N N Optimal Optimal corridate Relative error0 i 0 iŽ . Ž . Ž . Ž .is1, . . . response % is1, . . . response %

1Ž . Ž . Ž . Ž .266 101 0.0294 1.0381, 0.9389, 0.9974 3.81, y6.11, y0.26 266 101 0.0294 1.0379, 0.9389, 0.9977 3.79, y6.11, y0.23Ž . Ž . Ž . Ž .418 101 0.0295 1.0375, 0.9879, 0.9976 3.75, y1.21, y0.24 418 101 0.0295 1.0374, 0.9878, 0.9976 3.74, y1.22, y0.24Ž . Ž . Ž . Ž .597 101 0.0294 1.0382, 0.9495, 0.9977 3.82, y5.05, y0.23 597 101 0.0294 1.0376, 0.9636, 0.9977 3.76, y3.64, y0.23

2Ž . Ž . Ž . Ž .266 101 0.0990 1.4687, 1.4342, 1.0053 y2.09, y4.39, 0.53 266 101 0.0990 1.4689, 1.4341, 1.0052 y2.07, y4.39,0.52Ž . Ž . Ž . Ž .418 101 0.0990 1.4689, 1.4412, 1.0050 y2.07, y3.92, 0.50 418 101 0.0990 1.4688, 1.4412, 1.0052 y2.08, y3.92, 0.52Ž . Ž . Ž . Ž .597 101 0.0990 1.4690, 1.4297, 1.0053 y2.07, y4.67, 0.53 597 101 0.0990 1.4689, 1.4297, 1.0052 y2.07, y4.69, 0.52

Ž . Ž .The relative standard concentrations of sample 1 and 2 are 1.0000, 1.0000, 1.0000 and 1.5000, 1.5000, 1.0000 , respectively.

( )F. Gong et al.rChemometrics and Intelligent Laboratory Systems 45 1999 339–346346

4. Results and discussion

Both the F1 and F2 have many local minima. Forthese two functions, both the original SNTO methodand improved SNTO method are run three times withdifferent values of N . The results are shown in Table0

2. This table shows that it is possible for the originalSNTO method to be trapped in local optimal pointsif N is not correctly selected, and as for different0

values of N , the local minima to be obtained are also0

possibly different. It seems difficult to select N in0

order to find the global optimal point with the origi-nal SNTO method. However, the improved SNTOmethod locates the unique global optimal point irre-spective of the starting value of N . The good per-0

formance of the improved SNTO method is notclosely relative to N as long as the value of N se-0 0

lected from the glp set table is not too small. There-fore, the difficulty of how to select N in original0

SNTO is elegantly overcome with the help of clus-tering technique.

Table 3 lists the predicted results of CBBL forsamples 1 and 2 with three expected analytes and oneunknown interferent with the help of the original andimproved SNTO methods. For both samples, threedifferent values of N are selected. In all the cases0

good predicted results are obtained. Hence, it seemsas if these systems are less complicated than the F1and F2 functions. They may have only a single opti-mal point or all potential minima are not far fromeach other causing no difference between the twomethods.

5. Conclusions

In this paper, an improved SNTO method bymeans of clustering technique is developed. Two sig-nificant improvements have been achieved upon theoriginal SNTO method. First, the complexity of thestudied object function can be investigated by theproposed method with the help of the star discrep-ancy D). Identification of unimodal or multimodal

of the objective function becomes possible. Sec-ondly, with the help of the clustering technique, thedifficulty of the original SNTO needing correctchoice of the number of the points uniformly dis-tributed in the search space can be overcome ele-gantly. For a simple system, it directly use the origi-nal SNTO method to conduct optimization proce-dure. While, if the studied system is rather complex,say the multimodal function, the parallel searchingswill then be conducted for locating the global opti-mization.

Acknowledgements

The authors are thankful to the National NaturalScience Foundation of China and Fok Yin TungFoundation of Education Commission of China forfinancial support. Several valuable comments from

ŽProfessor Rolf Manne University of Bergen, Nor-.way are also gratefully acknowledged.

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