Keywords: approximations, neural networks, radial basis function networks, global optimization, evolutionary algorithms, genetic algorithms, multi-modal problems.

This paper is directed towards the optimization of black-box functions. This includes unconstrained, real and often multi-modal functions without any knowledge of their derivatives or continuity. This is the case for many engineering problems usually connected with some form of a finite element analysis, which can cause non-linearity and discontinuity of a solved problem. We also assume that the optimized function is time demanding and therefore our goal is to minimize the number of evaluations of this function.

One of today's most promising algorithms, the radial basis function network (RBFN) is presented. This method comes from the domain of a general approximation, usually called Response Surface methods [1], Diffuse Approximations [2] or Surrogate models [3]. The methodology is based on an analogy with artificial neural networks, but differs with respect to several points: the neural net is created only with one layer of neurons, it has a specific type of transfer function and the training of this net leads to the solution of a system of linear equations.

Our particular implementation is based on the specifications presented in [4]. Particularly, the RBFN approximates a given black-box function by the sum of basis function values and neural weights for each neuron. The value of a basis function is influenced by its "distance" from the neuron's center to the vector of variables, for which the approximate function value is required. The neural weights are then derived from the condition of equality between the values of black-box function and the RBFN approximation.

The novelty in our approach is the use of the evolutionary algorithm SADE [5], and its last modification called GRADE [2], to find the global maximum of the RBFN approximation. Moreover, several scenarios for creating new points in the process of the approximation are presented. In our case, three points are added in each cycle. The first one is the maximum found by the GRADE algorithm, the second one is a randomly created point inside the definition domain. The last point added is acquired by three different methods:

First, a new point is located in the (hyper)cube with the center located at the maximum found by the GRADE algorithm and the side length is derived from the number of optima found in this (hyper)cube during previous cycles.

The second method adds a new point with respect to the standard deviation, computed for values of maxima found in previous cycles. The randomly created vector is multiplied by the standard deviation and added to the current maximum found by the GRADE algorithm.

The last method counts for a difference between the maxima found by the genetic algorithm in last two steps and adds this difference to the best of them. This method is hereafter called the "gradient method".

The results of numerical experiments shows that the gradient method is the most effective, both from the point of view of the number of function calls necessary for locating the global maximum and from the point of view of standard deviation of number of function calls. Therefore, the gradient method was used for all subsequent tests.

To show the ability of the proposed method, the suite of twenty multi-modal functions (already presented [5]) is used along with one real-world problem of optimal control of structures undergoing large displacements [2]. Preliminary results are very promising for moderate problems, for the difficult ones, possible solutions are discussed.

1

Lee, J. and Hajela, P. (2001). "Application of classifier systems in improving response surface based approximations for design optimization". Computers & Structures, 79:333-344. doi:10.1016/S0045-7949(00)00132-2

2

Ibrahimbegovic, A., Knopf-Lenoir, C., Kucerová, A., and Villon, P. (2004). "Optimal design and optimal control of elastic structures undergoing finite rotations". International Journal for Numerical Methods in Engineering, 61(14):2428-2460. doi:10.1002/nme.1150

3

Karakasis, M. K. and Giannakoglou, K. C. (2004). "On the use of surrogate evaluation models in multi-objective evolutionary algorithms". In Neittaanmäki, P., Rossi, T., Korotov, S., Oñate, E., Périaux, P., and Knörzer, D., editors (2004). European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyväskylä.

4

Nakayama, H., Inoue, K. and Yoshimori, Y. (2004). "Approximate optimization using computational intelligence and its application to reinforcement of cable-stayed bridges". In Neittaanmäki, P., Rossi, T., Korotov, S., Oñate, E., Périaux, P., and Knörzer, D., editors (2004). European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyväskylä.

5

Hrstka, O. and Kucerová, A. (2004). "Improvements of real coded genetic algorithms based on differential operators preventing the premature convergence". Advances in Engineering Software, 35(3-4):237-246. doi:10.1016/S0965-9978(03)00113-3

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