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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 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: firstname.lastname@example.org, 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@example.com - Web page: http://www.isep.ipp.ptKey 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 NeuralNetworks (ANN) to approximate the fitness functions to be optimized is proposed. Thealgorithm is constituted by the following steps. Initially the MOEA runs over a small numberof generations. Then a neural network is trained using the evaluations obtained by theevolutionary algorithm. After the ANN is trained the MOEA runs for another set ofgenerations but using the ANN as an approximation to the exact fitness function. As thealgorithm evolves the population moves to different regions of the search space and thequality of the approximation performed by the neural network deteriorates. When the errorbecomes 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 continuesuntil an acceptable Pareto-front is found. This method was applied to several benchmarkmulti-optimization functions and to a real problem as well, namely the optimization of apolymer extrusion process. A reduction in the number of exact functions calls between 20 and40% was achieved. 1
António Gaspar-Cunha and Armando Vieira.1 INTRODUCTION A Multi-Objective Evolutionary Algorithm (MOEA) using an approximate fitnessevaluation obtained with an artificial neural network is proposed. The objective is to reducethe number of fitness evaluations in MOEAs on computational expensive problems whilemaintaining their good search capabilities. We show that this approach may save considerablecomputational time. One of the major difficulties in applying MOEAs to real problems is the large number ofevaluations of the objective functions, of the order of thousands, necessary to obtain anacceptable solution. Often these are time-consuming evaluations obtained solving numericalcodes with expensive methods like finite-differences or finite-elements. Reducing the numberof 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 evaluationin performed on a separate processor. However, this requires a large number of networkedcomputers and an adequate parallelisation of the numerical code. Here we investigate an alternative solution to this problem by approximating the functionsto be evaluated during optimization. There are several methods that can be used toapproximate the fitness evaluation. The surface response method and the Kriging statisticalmodel 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 fitnessfunction. It is well known that, given sufficient training data, a neural network canapproximate any function with arbitrary accuracy. ANNs are particularly well suited for non-linear regression analysis on high-dimensional data . In this case the neural networks istrained using the previous function evaluations that are being performed by the evolutionaryalgorithm. With enough data points the training error becomes sufficiently small and the ANNis considered to be a good estimator of the fitness function. As with any other approximation method, the performance of the neural network is closelyrelated to the quality of the training data. If the training data does not cover all the domainrange huge errors may occur due to extrapolation. Errors may also occur when the set oftraining points is not sufficiently dense and uniform. These problems are particularly acute forapproximations to functions used in MOEA optimization. First, these fitness functions mayhave strong oscillations, and second, the domain where we perform the approximation variesat each iteration. A different hybrid approach is proposed, where Neural Networks are used to estimate thefunctions used by a Multi-Objective Genetic Algorithm, namely the Reduced Pareto SetGenetic Algorithm (RPSGAe) [4, 5].3 ALGORITHM PROPOSED Figure 1 illustrates the method proposed. First the Genetic Algorithm runs during pgenerations to obtain the first set of evaluations necessary for the first train of the neuralnetwork. At this point two methods may be considered. First method, that will call A, is to 2
António Gaspar-Cunha and Armando Vieira.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 thepopulation, consisting of N individuals, and estimating the remaining N-M individuals usingthe output of the trained neural network. In method B the evolution of error produced by the approximate model, eNN can be directlyverified. As the algorithm evolves, points on the search space converge to the desiredsolution. Method B has the advantage that both parameters p and q are automatically determinedusing a simple criterion. In this method the error introduced by the approximations (eNN) canbe directly monitored by: M K (C NN − Ci , j ) 2 (1) ∑ ∑ j =1 i =1 i, j K e NN = Mwhere K is the number of criteria, M the number of solutions evaluated using both the exactfunction and the ANN, C iNN is the value of criteria i for solution j evaluated by ANN and Ci , j ,jis the value of criteria i for solution j evaluated by exact function. Neural Network Neural Network learning using some learning using some solutions of the p solutions of the p generations generations p generations q generations p generations q generations ... ... p generations RPSGA with RPSGA with RPSGA with RPSGA with RPSGA with exact function Neural Network exact function Neural Network exact function evaluation 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 initialtraining points where the approximation from the neural network may not be adequate. Theerror term allow us to monitor this situation and thus automatically specify the number of qgenerations in which the approximated model is used. Thus q is the number of generations forwhich the following inequality holds: e NN < e0 (2)being e0 a value that can be fixed by the user or adapted over the evolution towards thedesired Pareto-front . 3
António Gaspar-Cunha and Armando Vieira.4 RESULTS AND DISCUSSION The proposed method was tested using the ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6 bi-objective functions , with 30 variables each (except for ZDT4 where 10 variables wereused). The aim is to cover various types of Pareto-optimal fronts, such as convex, non-convex, discrete, multimodal and non-uniform . In order to achieve a clear comparison of the performance of our method the followingcriterion is used: S NN − S (3) S* = S NNwhere, S NN 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 ofthe neural network, η (0.1, 0.2, 0.3 and 0.4) and fraction individuals evaluated by the realfunction in each q generations, ξ (10, 30 and 50%). During the first p generations the RPSGAe uses the exact function evaluation and apopulation size of 100, 300 generations, a roulette wheel selection strategy, a crossoverprobability of 0.8, a mutation probability of 0.05, a random seed of 0.5, a number of ranks of30 and the limits of indifference of the clustering technique of 0.2 . These data were usedfor the first train of the neural network obtaining a mean square error of less than 1%. Thiserror increases as the search proceeds but never exceeding 7%. The results obtained with this approach are compared with the ones obtained usingRPSGAe alone. The comparison was quantified using the S metric proposed by Zitzler ,which is adequate for problems with few objective dimensions . Each run was performed 5times in order to take into account the variation of the random initial population. Since thecomputation time required to train and test the neural network is negligible, we decided to usethe number of real objective function evaluations as the significant running parameter. Foreach generation we calculate the average of the 5 runs of the metric as a function of thenumber of evaluations effectuated so far. Figure 2 compares the results obtained with traditional RPSGAe and the results obtainedwith the present two methods A and B for the ZDT1 function. From this figure is possible tosee 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 noparameter optimization is needed and therefore results are obtained in a single run. Similarresults were achieved with different levels of allowed errors, resulting in a decrease of thenumber of exact evaluations necessary as the error increases. 4
António Gaspar-Cunha and Armando Vieira. S*(%) METHOD A Eval*(%) S*(%) METHOD B Eval*(%) 0 0 25 25 -10 -10 15 15 -20 -20 5 5 -5 -30 -5 -30 -15 -40 -15 -40 0 100 200 300 0 100 200 300 Generations Generations S* Eval*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 = 0.25 APPLICATION TO POLYMER EXTRUSION The metodology proposed (method B) was applied to the screw geometry optimization of asingle-screw polymer extruder. The extruder is characterized by an Archimedes-type screwthat 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 inorder to produce an extrudate with a prescribed cross-section. For modelling purposes, theprocess is considered as a succession of functional zones characterized by stress, mass, heat orforce balances, coupled by adequate boundary conditions in the interface between the zones.The resolution of these differential equations is performed through the method of finitedifferences. A detailed description of these models and the required optimization can be foundelsewhere [12, 18]. Recently the process was proposed as a real test problem for EMOalgorithms 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 tothe problem of determining the geometry of a conventional screw extruded thatsimultaneously 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 allowederror of 3%. The ANN parameters are settled to Nh =10, η = 0.2 and α = 0.25. Figure 3 showsthe evolution of S* and the difference of exact evaluations as a function of the number ofgenerations. The improvement obtained in the number of exact function evaluations necessaryis approximately 40%. This implies a reduction from 8.5 to 6.0 hours on the computation timewhen 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 thefunctions to be optimized but also on the degree of approximation chosen. Using aconservative approximation produces no relevant gain in computation time while a moreaggressive approach may lead to large errors in objective functions and consequently a poor 5
António Gaspar-Cunha and Armando Vieira.Pareto-front. Two methods to select an adequate approximation by the ANN have beenproposed. Method A characterized by the manual adjust of the parameters that control thetraining of the ANN and the generalization error, and method B where these parameters wereselected automatically from specifying an accuracy criterion. Method B is clearly superiorsince it does not require apriori selection of the parameters that control the degree ofapproximation 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 numericalmethods having large computational costs, such the real optimization problem on polymerextrusion tested here. 0 0 -20 -10 -20 Eval* (%) -40 S* (%) -30 -60 S* -40 -80 -50 Eval* -100 -60 0 10 20 30 40 50 Generations Figure 3: Evolution of the S metric and number of evaluations differences for polymer extrusion problemREFERENCES 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). 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) Bishop, C.M., Neural Networks for Pattern Recognition, Oxford University Press, Oxford (1995). 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). 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). Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results, Evolutionary Computation, 8, pp. 173—195 (2000). 6
António Gaspar-Cunha and Armando Vieira. Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications, PhD Thesis, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland (1999). 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). 7