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International Congress on Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems

EUROGEN 2003 G. Bugeda, J. A- Désidéri, J. Periaux, M. Schoenauer and G. Winter (Eds)

CIMNE, Barcelona, 2003

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A MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM USING APPROXIMATE FITNESS EVALUATION

António Gaspar-Cunha* and Armando Vieira† * Department of Polymer Engineering

University of Minho Campus de Azurém, 4800-058 Guimarães, Portugal

e-mail: [email protected], web page: http://www.dep.uminho.pt

† Department of Physics Instituto Superior de Engenharia do Porto

R. S. Tomé, 4200 Porto, Portugal e-mail: [email protected] - Web page: http://www.isep.ipp.pt

Key words: Multi-Objective, Evolutionary Algorithms, Approximate Fitness Evaluation, Polymer Extrusion.

Abstract. In this work a method to accelerate the search of a MOEA using Artificial Neural Networks (ANN) to approximate the fitness functions to be optimized is proposed. The algorithm is constituted by the following steps. Initially the MOEA runs over a small number of generations. Then a neural network is trained using the evaluations obtained by the evolutionary algorithm. After the ANN is trained the MOEA runs for another set of generations but using the ANN as an approximation to the exact fitness function. As the algorithm evolves the population moves to different regions of the search space and the quality of the approximation performed by the neural network deteriorates. When the error becomes prohibitively high the evolutionary algorithm will proceed using the exact functions. A new training dataset is then collected and used to retrain the ANN. The process continues until an acceptable Pareto-front is found. This method was applied to several benchmark multi-optimization functions and to a real problem as well, namely the optimization of a polymer extrusion process. A reduction in the number of exact functions calls between 20 and 40% was achieved.

António Gaspar-Cunha and Armando Vieira.

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1 INTRODUCTION A Multi-Objective Evolutionary Algorithm (MOEA) using an approximate fitness

evaluation obtained with an artificial neural network is proposed. The objective is to reduce the number of fitness evaluations in MOEAs on computational expensive problems while maintaining their good search capabilities. We show that this approach may save considerable computational time.

One of the major difficulties in applying MOEAs to real problems is the large number of evaluations of the objective functions, of the order of thousands, necessary to obtain an acceptable solution. Often these are time-consuming evaluations obtained solving numerical codes with expensive methods like finite-differences or finite-elements. Reducing the number of evaluations necessary to reach an acceptable solution is thus of major importance [1,2]. This difficulty may be alleviated using distributed computations where each fitness evaluation in performed on a separate processor. However, this requires a large number of networked computers and an adequate parallelisation of the numerical code.

Here we investigate an alternative solution to this problem by approximating the functions to be evaluated during optimization. There are several methods that can be used to approximate the fitness evaluation. The surface response method and the Kriging statistical model are often applied in engineering and experiments design respectively.

2 NEURAL NETWORKS AS FUNCTION APPROXIMATIONS In this work Artificial Neural Networks (ANN) will be used to approximate the fitness

function. It is well known that, given sufficient training data, a neural network can approximate any function with arbitrary accuracy. ANNs are particularly well suited for non-linear regression analysis on high-dimensional data [3]. In this case the neural networks is trained using the previous function evaluations that are being performed by the evolutionary algorithm. With enough data points the training error becomes sufficiently small and the ANN is considered to be a good estimator of the fitness function.

As with any other approximation method, the performance of the neural network is closely related to the quality of the training data. If the training data does not cover all the domain range huge errors may occur due to extrapolation. Errors may also occur when the set of training points is not sufficiently dense and uniform. These problems are particularly acute for approximations to functions used in MOEA optimization. First, these fitness functions may have strong oscillations, and second, the domain where we perform the approximation varies at each iteration.

A different hybrid approach is proposed, where Neural Networks are used to estimate the functions used by a Multi-Objective Genetic Algorithm, namely the Reduced Pareto Set Genetic Algorithm (RPSGAe) [4, 5].

3 ALGORITHM PROPOSED Figure 1 illustrates the method proposed. First the Genetic Algorithm runs during p

generations to obtain the first set of evaluations necessary for the first train of the neural network. At this point two methods may be considered. First method, that will call A, is to


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simple use the approximate model to evaluate all the solutions during the next q generations. The other method, that we call B, consists in evaluating exactly only a fraction M of the population, consisting of N individuals, and estimating the remaining N-M individuals using the output of the trained neural network.

In method B the evolution of error produced by the approximate model, eNN can be directly verified. As the algorithm evolves, points on the search space converge to the desired solution.

Method B has the advantage that both parameters p and q are automatically determined using a simple criterion. In this method the error introduced by the approximations (eNN) can be directly monitored by:

( )

MK

CC

e

M

j

K

i

jiNN

ji

NN

∑ ∑= =

−

= 1 1

2,,

(1)

where K is the number of criteria, M the number of solutions evaluated using both the exact function and the ANN, NN

jiC , is the value of criteria i for solution j evaluated by ANN and jiC , is the value of criteria i for solution j evaluated by exact function.

p generations q generations

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evaluation

Neural Network learning using some

solutions of the pgenerations

p generations q generations p generations... ...




evaluation

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evaluation


evaluation


evaluation

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p generations q generations p generations... ...




evaluation


evaluation


evaluation


evaluation Figure 1: Schematic structure of the method A algorithm

As the algorithm evolves it may drift to regions outside the domain covered by the initial training points where the approximation from the neural network may not be adequate. The error term allow us to monitor this situation and thus automatically specify the number of q generations in which the approximated model is used. Thus q is the number of generations for which the following inequality holds:

0eeNN < (2)

being e0 a value that can be fixed by the user or adapted over the evolution towards the desired Pareto-front .


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4 RESULTS AND DISCUSSION The proposed method was tested using the ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6 bi-

objective functions [6], with 30 variables each (except for ZDT4 where 10 variables were used). The aim is to cover various types of Pareto-optimal fronts, such as convex, non-convex, discrete, multimodal and non-uniform [6].

In order to achieve a clear comparison of the performance of our method the following criterion is used:

NN

NN

SSS

S−

=* (3)

where, NNS and S are the averages of the S-metric obtained with and without ANN, respectively.

Initially, the relevance of some parameters on the algorithm performance was studied, namely: number of generations evaluated by the exact function, p (5, 10 and 15 generations), number of generations evaluated by the approximate model, m (5, 10 and 15 generations), number of hidden neurons of the neural network, Nh (10, 20 and 30 neurons), learning rate of the neural network, η (0.1, 0.2, 0.3 and 0.4) and fraction individuals evaluated by the real function in each q generations, ξ (10, 30 and 50%).

During the first p generations the RPSGAe uses the exact function evaluation and a population size of 100, 300 generations, a roulette wheel selection strategy, a crossover probability of 0.8, a mutation probability of 0.05, a random seed of 0.5, a number of ranks of 30 and the limits of indifference of the clustering technique of 0.2 [4]. These data were used for the first train of the neural network obtaining a mean square error of less than 1%. This error increases as the search proceeds but never exceeding 7%.

The results obtained with this approach are compared with the ones obtained using RPSGAe alone. The comparison was quantified using the S metric proposed by Zitzler [7], which is adequate for problems with few objective dimensions [8]. Each run was performed 5 times in order to take into account the variation of the random initial population. Since the computation time required to train and test the neural network is negligible, we decided to use the number of real objective function evaluations as the significant running parameter. For each generation we calculate the average of the 5 runs of the metric as a function of the number of evaluations effectuated so far.

Figure 2 compares the results obtained with traditional RPSGAe and the results obtained with the present two methods A and B for the ZDT1 function. From this figure is possible to see that, the number of exact evaluations to reach the same S-metric is reduced to about 36%, for method A and 28% for method B. However, method B has the advantage that no parameter optimization is needed and therefore results are obtained in a single run. Similar results were achieved with different levels of allowed errors, resulting in a decrease of the number of exact evaluations necessary as the error increases.


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Figure 3: Evolution of the S metric and number of evaluations differences for ZDT1 test problem, using methods A and B. The following parameters were used: p = 15, q =10, Nh = 10 and η = 0.2

5 APPLICATION TO POLYMER EXTRUSION The metodology proposed (method B) was applied to the screw geometry optimization of a

single-screw polymer extruder. The extruder is characterized by an Archimedes-type screw that rotates inside a heated barrel. The extruder receives the solid pellets at the inlet and melts, mix and homogenise the material. Then, the melted polymer is pumped through the die in order to produce an extrudate with a prescribed cross-section. For modelling purposes, the process is considered as a succession of functional zones characterized by stress, mass, heat or force balances, coupled by adequate boundary conditions in the interface between the zones. The resolution of these differential equations is performed through the method of finite differences. A detailed description of these models and the required optimization can be found elsewhere [12, 18]. Recently the process was proposed as a real test problem for EMO algorithms and was made available through the internet to the EMO community [19, 20].

Method B was applied to reduce the number of exact evaluations of a MOEA applied to the problem of determining the geometry of a conventional screw extruded that simultaneously maximize the mass output and the mixing degree. Ten runs are carried out, five using the RPSGA without neural networks and five using method B with an allowed error of 3%. The ANN parameters are settled to Nh =10, η = 0.2 and α = 0.25. Figure 3 shows the evolution of S* and the difference of exact evaluations as a function of the number of generations. The improvement obtained in the number of exact function evaluations necessary is approximately 40%. This implies a reduction from 8.5 to 6.0 hours on the computation time when a PC with an AMD processor at 1666 MHz is used.

6 CONCLUSIONS The efficiency of this approach is strongly dependent not only on the difficulty of the

functions to be optimized but also on the degree of approximation chosen. Using a conservative approximation produces no relevant gain in computation time while a more aggressive approach may lead to large errors in objective functions and consequently a poor


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Pareto-front. Two methods to select an adequate approximation by the ANN have been proposed. Method A characterized by the manual adjust of the parameters that control the training of the ANN and the generalization error, and method B where these parameters were selected automatically from specifying an accuracy criterion. Method B is clearly superior since it does not require apriori selection of the parameters that control the degree of approximation used.

Both methods were applied to several benchmark problems and to a real world problem. This approach may save considerable computational time, ranging from 13% to about 40%. This is particularly relevant when the evaluation of the solutions involves the use of numerical methods having large computational costs, such the real optimization problem on polymer extrusion tested here.

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Figure 3: Evolution of the S metric and number of evaluations differences for polymer extrusion problem

REFERENCES [1] Nain, P.K.S., Deb, K., A Computationally Effective Multi-Objective Search and

Optimization Technique Using Coarse-to-Fine Grain Modeling. Kangal Report No. 2002005 (2002).

[2] Jin, Y., Olhofer, M., Sendhof, B., A Framework for Evolutionary Optimization with Approximate Fitness Functions, IEEE Trans. On Evolt. Comp., 6, pp. 481-494 (2002)

[3] Bishop, C.M., Neural Networks for Pattern Recognition, Oxford University Press, Oxford (1995).

[4] Gaspar-Cunha, A., Covas, J.A. - RPSGAe - A Multiobjective Genetic Algorithm with Elitism: Application to Polymer Extrusion, Submitted for publication in a Lecture Notes in Economics and Mathematical Systems volume, Springer (2002).

[5] Gaspar-Cunha, A.: Reduced Pareto Set Genetic Algorithm (RPSGAe): A Comparative Study, The Second Workshop on Multiobjective Problem Solving from Nature (MPSN-II), Granada, Spain (2002).

[6] Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results, Evolutionary Computation, 8, pp. 173—195 (2000).


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[7] Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications, PhD Thesis, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland (1999).

[8] Knowles, J.D., Corne, D.W., On Metrics for Comparing Non-Dominated Sets. In Proceedings of the 2002 Congress on Evolutionary Computation Conference (CEC02), pp. 711-716, IEEE Press (2002).

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