1034 Copyright © 2003 The Society of Chemical Engineers, Japan

Journal of Chemical Engineering of Japan, Vol. 36, No. 9, pp. 1034–1044, 2003 Research Paper

Information Directed Sampling for Combinatorial Material Synthesisand Library Design

CHIA HUANG YEN, DAVID SHAN HILL WONGAND SHI SHANG JANGDepartment of Chemical Engineering,National Tsing Hua University, Hsinchu, Taiwan 30043

Keywords: Information Theory, Combinatorial Methods, Generalized Regression, Neural Network

Combinatorial techniques have become more and more important in many areas of chemistry andchemical engineering research. It was suggested that simulated annealing can be used to improve theefficiency of sampling in combinatorial methods. However, without priori model estimates of fitnessfunction, true importance sampling cannot be performed. In this case, the efficiency of annealing is onlyas good as random search. We suggested that a simple prediction model using currently available datacan be constructed using a generalized regression neural network. An index of our uncertainty about apoint in the search space can also be established using information entropy. An information free energycombined the two indices to direct the search so that importance sampling is performed. Two benchmarkproblems were used to model the optimization problem involved in combinatorial synthesis and librarydesign. We showed that when importance sampling is performed, the combinatorial technique becamemuch more effective. The improvement in efficiency over undirected methods is especially significantwhen the size of the problem becomes very large.

Introduction

In recent years, combinatorial synthesis becomean important optimization technique in product andprocess development (Davis, 1999), e.g. material syn-thesis (Xiang et al., 1995; Szostak, 1997; Wilson andCzarnik, 1997; Danielson et al., 1998; Klein et al.,1998; van Dover et al., 1998; Cong et al. 1999;Engstrom and Weinberg, 2000), design of catalysts(Cole et al., 1996; Jandeleit et al., 1998; Schlögl, 1998;Senkan, 1998), selection of solvent (Pretel et al., 1994),drug design (Gordon et al., 1996; Gordon, 1998;Linusson et al., 2000), improvement of enzymes ac-tivity (You and Arnold 1996; Bornscheuer, 1998). Com-binatorial techniques screen a large quantity of differ-ent combinations of inputs to find the condition thatproduces the best merit function.

There are two focuses in combinatorial synthesisresearch: (1) how to create a large throughput of ex-periments (e.g. Hanak, 1970), (2) how to develop aprotocol for designing experiments, so that the inputstate-space can be sampled effectively (e.g. Gordon,1998; Linusson et al., 2000; Voigt et al., 2001). In prac-tice, the two problems may not be separable since the

Received on October 25, 2002. Correspondence concerning thisarticle should be addressed to S.-S. Jang (E-mail address:ssjang@che.nthu.edu.tw).

method of experiments may dictate under what condi-tions a batch of experiments may be done. The selec-tion of experiments is then highly correlated insteadof random. However, in this work, we are concernedonly with how previous information obtained can beused to improve sampling efficiency.

Sampling policy in combinatorial synthesis is justanother form of experimental design. In the past, wehave proposed experimental design methods based oninformation theory (Lin et al., 1995; Chen et al., 1998)for process optimization and recipe selection. The phi-losophy is as follows. Given a set of data, an empiricalmodel can be trained. This model can be used to directthe search and suggest experiments at points that havepotentially high fitness function (low information en-ergy). However, such a model is untrustworthy whendata are sparse compared to the entire search space(high information entropy). We need to explore areasthat have not yet been investigated. A temperature-annealing schedule can be used to shift from an infor-mation-entropy-based search to information-energy-based search. These methods are not efficient for com-binatorial synthesis because the modeling techniquesused are not equipped to handle problems of extremelylarge dimensional. The neural network used requireextensive training. In this work, we shall propose aninformation-based importance sampling policy thatcan be used for combinatorial synthesis. The efficiencyof this method is tested using the RPV and the N–Kmodels.

1035

1. Theory

1.1 Fitness landscape1.1.1 N–K model There are two types of opti-mization problems in combinatorial chemistry: high-dimensional “structural” search of combinations of in-teger variables and high dimensional “spatial” searchof continuous variables.

In structural search problems, components in alibrary are combined to form a particular structure.Experiments or calculations are then carried out to seeif the synthesized structure has the desired property. Abenchmark problem of such optimization is known asthe N–K problem (Kauffman and Levin, 1987). TheN–K model captures the basic physics of many phe-nomena such as genomics, protein evolution, etc.(Kauffman, 1993; Perelson and Macken, 1995). Vari-ous modified forms of the N–K model have also beenproposed (Bogarad and Deem, 1999). The simplestN–K model can be described as follows. Consider anN-dimensional array of integer variables (a

1, a

2, ···, a

N).

It represents the state of a sequence. Each entry of thesequence a

j represents a component from the library.

For example, in gene sequencing, each entry aj can be

one of the four nucleotides C, T, G, A. In protein fold-ing problems, each entry a

j can be a particular form of

amino acids. In drug design, each entry aj can be one

of the functional groups that are used to form a ligand.In the N–K problem, a

j is a binary variable that takes

values of 1 or 0. The merit function E(a1, a

2, …, a

N) is

given by:

E a a a

N Ka a a

N

j j j j K

j

N N K

1 2

1 2 1 1

1

11

1

, , ,

, , ,/

L

L

( )

=−( )[ ] ( ) ( )+ + −

=

= − +

∑σ

σj(a

j, a

j+1, ···, a

j+K–1) is the contribution to the merit func-

tion by the j-th entry. Note that the contribution de-pends not only on the state of occupancy of the j-thvariable, but also on how the next K – 1 variables areoccupied. Such a formulation is an analog of manyphysical situations. For example, a particular pheno-type is found only if a particular sequence ofnucleotides (a genotype) is found. In a protein foldingproblem, the state of a particular amino-acid in a pro-tein depends on its neighboring amino acids. A func-tional group manifests its chemical nature only if theneighboring groups provide a suitable molecular envi-ronment. In the N–K problem σ

j (a

j, a

j+1, ···, a

j+k–1) is

given by a “lookup” table of problem-defined values.Two examples of lookup tables for (N = 4, K = 2) and(N = 4, K = 3) are given in Fig. 1. Given the lookuptables, the merit function of all 16 different states canbe calculated. For example, in Fig. 1, given the lookuptable (N = 4, K = 2), the point (0, 0, 1, 0) will have a

merit function

E1000

1

1

4 2 1 0 0 0 0 0

2 0 5082 0 8424

1 5507

= −( ) ( ) + ( ) + ( )( )= ( ) +( )=

−

−

σ σ σ, , ,

. .

.

For N = 4, the changes of the merit function can beshown as two connecting Boolean cubes. Each vertexis connected to 4 different vertices, each of which dif-fers from the center point by one variable. For ex-ample, the point (1, 0, 0, 0) is connected to (0, 0, 0, 0),

Fig. 1 Local and global extremas for NK models

1036 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

(1, 1, 0, 0), (1, 0, 1, 0) and (1, 0, 0, 1). The connectionscan be shown as edges of the connecting cubes. Thearrows show the direction of decrease in the merit func-tion. The lookup table (N = 4, K = 2) generates twolocal minima, while the other with N = 4, K = 3 gener-ates three. Kauffman (1993) showed that as N and Kincrease, the number of local minima increases, andthe difference between global optimal and averagevalue of the merit function decreases. For more detailsof the N–K model and its relevance to various physi-cal situations, readers can refer to the book by Kaufman(Kauffman, 1993). The role of the N–K problem forlibrary design optimization is analogous to that of thetraveling salesman problem for network routing.1.1.2 Random phase volume model In many com-binatorial synthesis problems, experimental variablesinclude the compositions of the material and process-ing conditions. The input state-space consists of bothdiscrete and continuous variables and is typically ofvery high dimensions. The merit function is usuallyone or a set of specific properties of the material suchas superconductivity, luminescence, catalytic activity,tensile strength, etc. Such properties will depend onthe particular phase of the product material. Since thechange of physical property of the material is discon-tinuous across a phase boundary, the objective func-tion encountered in combinatorial synthesis optimiza-tion is only piecewise continuous.

Falcioni and Deem (2000) proposed a “randomphase volume” (RPV) model to simulate the merit func-tion encountered in combinatorial synthesis. Essen-tially RPV is a relation between the merit function Eand a set of compositional variables

rx and non-com-

position variables rz . Composition variables are ex-

pressed as mole fractions. Therefore for a C compo-nent system, there are C – 1 independent compositionvariables. The entire composition space is divided intoα = 1, ···, M different phases by defining M phase centerpoints

rxα

0 . Each point in the composition space be-longs to the phase of the nearest phase center point.Non-composition variables may be processing condi-tions such as temperature, pressure, pH, time of reac-tion, thickness of film, etc. They may be discrete orcontinuous and normalized in the range of [–1, 1] inthe RPV model. Similarly the non-composition spaceis divided into γ = 1, ···, N different phases by definingN phase center points

rzα

0 :

E x z E

x x x x

z z z z

M

M

r r

r r r r

r r r rL

L

,

min

min

, ,

, ,

( ) =

− = −

− = − ( )′ =

′

′ =′

αγ

αα

α

γγ

γ

if

and a

0

1

0

0

1

0 2

The merit function Eαγ in this phase is represented by a

Qx-th order polynomial in composition variable and a

Qz-th order polynomial in non-composition variable:

E U f A y y y

W f B w w w

i i

i i

C

k

Qk

i ik

i i i

i i

i i

D

k

Qk

i ik

i i i

k

k

k k

k

k

k k

αγ αα

γγ

σ ξ

σ ξ

= + ×

+ + ×

≥ ≥ =

( )

≥ ≥ =

( )

∑∑

∑∑

x

=

x-

z

=

z-

x

z

1

1

1 1 2

1

1

1 1 2

11

11

1

2

L

L

L

L

L

L

L

L

( )2b

with r r ry x x= − α and

r r rw z z= − α . Uα, Wγ,

Ai ik

k1Lα( ) ,

Bi i

k

k1Lγ( ) ,

are parameters generated by Gaussian random numberwith zero mean and unit variance. The scale factors ξ

x

and ξz are chosen so that each term in the multinomial

expansion contributes roughly the same amount. Theσ

x and σ

z are chosen so that the multinomial expansion

terms contribute about 40% of Eαγ. The symmetric fac-tor is given by:

fk

oi i

ii

lk1

1

2L = ( )

=∏

!

!

c

where l is the number of distinct integer values in theset of {i

1, ···, i

k}, and o

i is the number of times that

distinct value i is repeated in the set. Note that 1 ≤ l ≤k and ∑ = =i

li ko1 . Figure 2 is a typical merit function

landscape of an RPV model with C = 3, D = 0, M = 15and Q

x = 2. The piecewise continuous and nonlinear

nature is evident. The complexity of the problem canbe increased by increasing the dimension of variableand number of phases. One can appreciate the similar-ity of RPV model landscape in Fig. 2 by comparing

Fig. 2 A typical merit function landscape of the random-phase volume model

VOL. 36 NO. 9 2003 1037

with the merit function landscape obtained in actualcombinatorial synthesis study (e.g. Cong et al., 1999).

Given the N–K model and the RPV model, thesampling algorithm for optimization in combinatorialsynthesis and library design can be benchmarked.1.2 Sampling policy

The simplest way of sampling the state space is touse grid search followed by local search. However, gridsearch becomes inefficient as the dimensionality ofstate-space increases. Another class of search methodcommonly used to solve high dimensional optimiza-tion is stochastic search methods such as simulatedannealing (SA, Kirkpatrick et al., 1983). In a typicalSA scheme applied to combinatorial synthesis, a set ofreference data points are created; a set of candidatepoints are generated. The values of the merit functionof the candidate data points are compared to those ofthe reference data points. The reference data point isthen updated by the candidate according to the transi-tion probability:

P

Y Y

Y Y

TY Y

i i

i ii i

=

<− −( )

≥

( )1

3

c r

c rc rexp

where Yic is the merit function of candidate data and Y

ir

is the merit function of reference data. The tempera-ture is reduced gradually according to a predeterminedannealing schedule to ensure that the system is nottrapped in a local minimum.

The updating procedure, given in Eq. (3), mimicsthe importance-sampling strategy, known as the Me-tropolis scheme, used in Monte Carlo simulation ofmolecular ensembles. Importance sampling is pivotalin calculating ensemble average in Monte Carlo simu-lation because it reduces the number of times that totalenergy of unimportant states are calculated and sam-pled in the ensemble average. The number of interac-tions evaluated in calculating the total energy of theensemble is proportional to n(n – 1)/2, n being thenumber of molecules. The number of interactionsevaluated in calculating the transition probability inEq. (3) is proportional to n. Thus importance samplingreduces the computing time significantly but ensuresthat states sampled in the ensemble average are dis-tributed according to the Maxwell–Boltzmann distri-bution.

The objective of the combinatorial synthesis is,however, slightly different. It is desirable that the trueglobal minimum is located, not the true ensemble av-erage. Moreover, the merit function of a data point isknown only if the actual experiment is performed.Therefore, unless some prior estimates of the meritfunction are available, importance sampling cannot beperformed. The act of updating according to the dif-

ference between the merit function of the candidateand the reference point according to the Metropolisscheme only prevents the search from being trappedin a local minimum prematurely. Without real impor-tance sampling, simulated annealing will not be anymore effective than random search.1.3 Generalized regression neural network

In order that existing data can be efficientlymodeled, the generalized regression network was used(Specht, 1991). If the joint probability density func-tion p(Y,

rX ) of a random input variable at

rX having

an output variable Y is known, then the expected valueof the output at

rX is given by:

Y X

Yp Y X dY

p Y X dY

r

r

r( ) =

( )

( )( )−∞

∞

−∞

∞

∫

∫

,

,

4

If p(Y, rX ) is unknown, it can be estimated using exist-

ing measurements ( rXi , Yi), i = 1, ···, n, as Gaussian

distribution function (Parzen, 1962):

˜ ,

exp exp

/p Y X

n

X X X X Y Y

p p

i T i

i

n i

r

r r r r

( ) =( )

⋅ −−( ) −( )

−( )

( )

+( ) +( )

=∑

1

2

1

2 2

5

1 2 1

21

2

2

π σ

σ σ

Expected value of the output at rx is given by

˜

exp

exp

Y X

X X X XY

X X X X

i T i

i

ni

i T i

i

n

r

r r r r

r r r r( ) =

−−( ) −( )

−−( ) −( )

( )=

=

∑

∑

2

2

6

21

21

σ

σ

There are several advantages of using GRNN in ourmethod.

1. GRNN requires no training. The predictionmodel of GRNN in Eq. (6) requires only (

rXi , Yi), the

location and results of data already sampled. σ is aproblem-specific smoothing factor which is preset andkept constant. Therefore, the training phase of GRNNis only a one-pass reading of all existing data.

2. Both integer and continuous inputs can beused in GRNN although different values of σ may be

1038 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

needed. Therefore N–K landscape can be modelled ef-ficiently. We compared the training and prediction ac-curacies of a GRNN model to another popular neuralnetwork, the radial basis function network (RBFN,Chen et al., 1991) for a small N(=10)–K(=5) model.The model has 1024 states. 100 states were randomlytaken as the training data, and the rest were chosen astest set data. While both models can accurately repro-duce the training data, the prediction mean square er-ror of GRNN was found to be 0.066, much less thanthat found by RBFN (0.519). Obviously, GRNN pro-duces a much better model when the amount of datasample is limited.

3. The predictions of GRNN always fall withinthe maximum and minimum of the training data set.Use of GRNN avoids unreasonable extrapolation. Thisis extremely important if we wish to model a piecewisecontinuous function such as the RPV model. Figure 3compared GRNN, RBFN and back-propagation net-work (BPN; Haykin, 1999) modeling of an RPV modelwith a limited amount of training data. Again, GRNNproduces the most reasonable fitness landscape for theRPV model.1.4 Information energy

Given a model of all the presently available ex-perimental data, we define the information energy in-dex of any candidate point in the search space

U xY X

Y X

rr

r( ) =( )

− ( )

( )

˜

˜

for minimization problems

for maximization problems7

where < Y ( rX )> indicates the predicted value of the

fitness function using a GRNN model. At any candi-date point, the better the predicted fitness function, thelower is its information energy. The more valuable isthe information at that point.1.5 Information entropy

Our knowledge of a candidate point can be mea-sured by the information entropy (Shannon, 1948),

S X

p Y X p Y X dY

p Y X dY

r

r r

r( ) =

( ) ( )

( )( )−∞

∞

−∞

∞

∫

∫

, ln ,

,

8

Given the distribution function in Eq. (5), the infor-mation entropy can be estimated as:

S X

X X X X X X X X

X X X X

i T i i T i

i

n

i T i

i

n

r

r r r r r r r r

r r r r

( )

=

−( ) −( )−

−( ) −( )

−−( ) −( )

( )

=

=

∑

∑

2 2

2

9

2 21

21

σ σ

σ

exp

exp

Fig. 3 Fitness function surface obtained using different neural networks for RPV models

VOL. 36 NO. 9 2003 1039

The information entropy is just the average of thesquare of the distance between a candidate point andall existing data points. The higher the informationentropy, the more we know about the candidate point.1.6 Information free energy and temperature an-

nealingA candidate is worthy of experiment if it has a

potential of having a good fitness value (low informa-tion energy), or its neighborhood has not been suffi-ciently explored (high information entropy). During theinitial stages when the number of experiment is small,the model predictions are of little value, experimentsshould be devoted to sample un-chartered search space.When sufficient information has been gathered, onlysamples that are potentially important should be tested.Chen et al. (1998) proposed an information free en-ergy index:

F = U – TS (10)

with T being an annealing temperature proportional tothe number of experiment. The proper convergence ofthe sampling procedure to the global minimum dependson the annealing schedule. In this work, the very fastre-annealing scheme proposed by Ingber (1989) wasused:

T = T0·exp(–c·k1/m) (11)

with T0, c, m being adjustable constants and k is the

total number of the experiments.1.7 Flowchart

Our proposed sampling policy can be summarizedinto a flowchart in Fig. 4 as the following steps.(i) Obtain a GRNN model for all existing data.(ii) Select the data with the best fitness function as

a reference.(iii) Use random search to produce a candidate ex-

periment. Check if the experiment has been per-formed. Generate another candidate if this ex-periment has been performed. If the experimenthas not been performed, proceed to step (iv).

(iv) Use GRNN to calculate information energy andinformation free energy at the candidate point.

(v) If the information free energy is smaller than orequal to the free energy of the existing refer-ence data, check if the batch has been filled. Ifnot go to step (iii), to generate additional ex-periments.

(vi) If the information free energy is greater than thefree energy of the existing reference data, checkif the number of trial exceeds a preset numberNT (NT should be very large). If the number oftrial is less than NT, go to step (iii). If the numberhas reached NT, a local search is used to createa new experiment.

(vii) Repeat steps (iii) to (vi) until all candidate ex-

periments of the new batch have been selected.(viii) Perform the experiments.(ix) Repeat step (i) to (viii) at a reduced tempera-

ture until some stopping criterion is met, i.e. thefitness function has met some specifications orthe number of batches reaches a preset numberNB.

We shall call the above sampling policy “free en-ergy directed simulated annealing” (Free Energy-DSA,FEDSA).

In the above algorithm, a local search is used ifdirected search cannot produce an acceptable candi-date. In the N–K model, this is done by mutating asmall number of variables of the array. In RPV model,this is done by taking a random small step away fromthe reference minimum.

Two undirected search methods: random search(RS) and undirected simulated annealing (USA) areused for comparison. In RS, a set of candidate experi-

Fig. 4 Flow chart of information directed search

1040 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

ments are selected randomly. The experiments are per-formed and the samples in the reference data set areupdated if the fitness value of the new experiments isbetter than that of the reference data. In undirectedsimulated annealing (USA), a random search gener-ates the new experiment candidates. The experimentsare performed without further screening. The Metropo-lis scheme in Eq. (3) is used to determine whether thereference data is replaced by the new data using ex-perimental results.

2. Results and Discussion

2.1 Search efficiencyTo demonstrate the advantages of our approach,

we shall use two simple examples: (i) an N–K modelwith N = 20, Boolean state inputs of 0 and 1 and K = 4,(ii) the RPV model with C = 3, D = 0, M = 15 andQ

x = 2. The small N–K problem has only 1,048,576

states that allow exhaustive search and identificationof the true global minimum and the true fitness func-tion distribution. The two-dimensional nature of theRPV problem allows visualization of the search proc-ess.

Figures 5 and 6 illustrate the decrease in the ob-jective function with the number of batches of experi-ments for the small N–K model and the RPV modelrespectively. For a pure random search, the fitness func-tion initially decreases rapidly but improvement be-comes more and more difficult. An undirected simu-lated annealing search causes the system to deviatefrom the current minimum so that the search processwill avoid being trapped in a local minimum. Withoutimportance sampling, generation of states with betterobjective fitness is by pure chance. The rate of findingthe best fitness function can only be as fast as randomsearch. If a model of the fitness function is constructedusing previous data and importance sampling (directed

Fig. 5 Changes in the best fitness function located usingdifferent search strategies for a simple N–K prob-lem (N = 20, K = 5)

Fig. 6 Changes in the best fitness function located usingdifferent search strategies for a simple RPV prob-lem (C = 3, M = 15, Q

x = 2)

VOL. 36 NO. 9 2003 1041

annealing) are performed based on information free en-ergy, the rate of finding the best fitness function canbe much improved. In Fig. 5, we found that the FEDSAlocated true global minimum even though the landscapeof the N–K model is very rugged. Random search andundirected simulated annealing fails to locate the trueminimum. Similarly, FEDSA is able to find the approxi-mate region of the true optimum at around the 10thbatch for the RPV model (Fig. 6). At this point, thevalues of the fitness function obtained by a randomsearch procedure and simulated annealing are muchhigher. Random search and undirected simulated an-nealing fail to locate the true minimum.2.2 Importance sampling

Figures 7 and 8 illustrate the distribution of thedata sampled at different stages in optimization. Forrandom search the distribution is similar to simulatedannealing and therefore not shown in the figures.

Figure 7 shows the distribution of the data sam-pled from the 1st to 10th batches, 21st to 30th batchesand 51st to 60th batches. It is obvious that the distri-bution of the data sampled shifted towards low energyend. Such changes were not found for simulated an-nealing. FEDSA allows us to perform importance sam-pling in the early stages of the search. After the 50thbatch, mutation becomes dominant. There is not muchshift in the energy distribution of the data sampled.During the later stages of the search, mutation fre-quently became the sampling technique rather thanimportance sampling. While the data sampled differ inonly a few bits, the rugged Nature of the fitness land-scape results in a broad distribution of fitness valuesfor the sampled data.

For the RPV model, local search is important evenwhen temperature annealing virtually stopped. Impor-tance sampling allows us to locate the correct phase.Figure 8 illustrates the distributions of the data duringdifferent stages of the optimization. In FEDSA, dataare scattered across the state space initially. Then dataare selected around the projected local minima. Finallyall data are concentrated in the correct phase with theglobal minimum. This again illustrates the efficiencyof importance sampling. On the other hand, the dataare scattered around the entire search space during dif-ferent stages of the optimization for undirected simu-lated annealing. No importance sampling was per-formed. The backgrounds of the figures in the columnunder undirected simulated annealing in Fig. 8 are thetrue contours of the RPV model. The backgrounds ofthe figures under free energy DSA are contours of theGRNN model. Note that the GRNN model capturedthe general feature of the RPV model, but the detailsare far from the global optimum. When we attempt tocreate a model of a large search space it is importantthat efforts are not wasted in reproducing the exactdetails in regions that are not relevant in an optimiza-tion sense.

2.3 Effect of problem sizeIf the size of the N–K model increases, exhaus-

tive search becomes difficult. Similarly as the size ofthe RPV model increases visualization becomes im-possible. Table 1 presents the optimal fitness functionlocated for a given number of experiments (NB = 100).Since these are stochastic searches, the optimal valuelocated will vary from run to run as the size of the prob-lem increases. Therefore the averages and standarddeviations of 10 different runs are listed. It was foundthat Free-Energy-DSA always has lower averages andsmaller standard deviations than simulated annealing.The difference between FEDSA and USA/RS becomeslarger as the size of the problem increases. This em-phasizes the need of true importance sampling in solu-tion of complex problems.2.4 Computation Effort

The main objective of our approach is to reducethe number of actual sampling. In this benchmarkingstudy, an actual sampling is equivalent to a functionevaluation. Figures 5 and 6 compared the values of ob-jective function obtained at different number of func-

Fig. 7 Distribution of data sampled by different searchstrategies at different stages of optimization for theN–K model

1042 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

tion evaluations. It is obvious that our information di-rected approach is more efficient than an undirectedsampling method such as random search or simulatedannealing in terms of finding an optimum landscapewith a least number of samplings. Since GRNN requiresno training, only a little extra computation effort is re-quired to determine which sample is worth testing. Inthe simulation study using N–K and RPV, since evalu-ation of the objective function is relatively easy, thenet saving of computation time is negligible. However,an actual library design and combinatorial synthesis,

an actual sampling usually involves the performanceof an experiment, or running an elaborate molecularsimulation. Actual sampling are time consuming andexpensive. The advantage of information-directed sam-pling will be substantial.

Conclusions

In this work, we have demonstrated the importanceof true importance sampling in solving optimizationproblems of high dimension. Our results show that

Fig. 8 Distribution of sampling points at different stages of optimization for the RPV model(C = 3, M = 15, Qx = 2)

VOL. 36 NO. 9 2003 1043

brute force combinatorial techniques may be power-ful, but the technique becomes much more effective ifwe can organize the present knowledge periodically todirect the search. We have demonstrated that the or-ganization of knowledge need not be done in a theo-retical manner. It can be done by constructing a simpleempirical model with sufficient flexibility. Due to thelarge amount of data involved, this model should re-quire a minimal regression computation but must alsobe statistically sound. A generalized regression neuralnetwork was selected to perform the task of modeling.Furthermore, during early stages of the search, we mustnot put too much emphasis on model predictions sincea large fraction of the search space remains unexplored.An information entropy index allows us to direct thesearch to unexplored regions of the search space. Aninformation free energy index was used to balance theneed to confirm the model predictions of regions ofoptimality and the need of chartering unexplored searchspace. True importance sampling can be achieved us-ing this information free energy index and effective-ness of combinatorial can be substantially improved.

AcknowledgmentThe authors thank the financial support provided by the Na-

tional Science Council, Taiwan, for this work through the grantNSC90-2622-E007-003.

Nomenclature

Ai i

k

k1L

α( ) = parameters in the PRV modela

j= j-th entry of the N-dimensional array in the NK

model

Bi i

k

k1L

γ( ) = parameters in the PRV modelC = dimensions of the composition variables in the RPV

modelc = parameters in very fast simulated re-annealingD = dimensions of the non-composition variables in the

RPV modelE = merit function of the RPV modelEαγ = merit function of the NK modelF = information free energyf

i= symmetry factor of the polynomials

K = interactive loci in the NK modelk = number of the experiments

M = phase numbers of the composition variablesm = parameters in very fast simulated re-annealingN = loci length in the NK modelp = joint probability density functionp : = joint probability density function estimator

Qx, Q

z= polynomial order of the composition and non-com-

position variablesS = information entropyT = annealing temperatureT

0= inital annealing temperature

U = information EnergyUα = parameters in the RPV modelWγ = parameters in the RPV model

rw = non-composition variable vectors relative to the

phase center

rX = input variable vectors in a general gression neural

network

rx = composition variable vectors mole fraction

Y = output variables in a general gression neural net-work or merit function

Y = predict value of output in a general gression neu-ral network

ry = composition variable vectors relative to the phase

center

rz = non-composition variable vectors mole fraction

α = composition phase indexγ = non-composition phase indexξ

x, ξ

z= parameters in the RPV model

σ = smooth factor in a general regression neural net-work

σa

= parameters in the NK modelσ

x, σ

z= parameters in the RPV model

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RPV modelM = 15, C = 3, D = 0, Qx = 2 –1.71 0.08 –1.827 0.006M = 15, C = 3, D = 0, Qx = 6 –1.24 0.16 –1.697 0.004M = 37, C = 6, D = 0, Qx = 2 –1.01 0.21 –1.987 0.052M = 37, C = 6, D = 0, Qx = 6 –1.02 0.12 –1.618 0.022

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