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Conclusion 1.The statistical method presented here for the determination of a SCD allows us to accelerate significantly the RS method. One should specially mention the relative efficiency of the simplified variantA2, which does not require additional memory for the storage of the ROSV. So far, one has not proved the convergence ofA2, but numerous tests (covering hundreds of different situations, n<30) and a series of solved practical problems have not established any case of divergence.2.On sufficiently smooth functions, the algorithmsA1, A2 yield to Powell's SCD in the original realization (but exceed it in the realization of [10], using the method of the golden section for a one-dimensional search). Therefore, one can assume that the algorithmsA1 andA2 on smooth functions will yield to more powerful methods [1–3]. At the same time one should keep in mind that the algorithmA2 may turn out to be effective in nonsmooth problems, which is confirmed by the third experiment. In particular, the authors have applied it successfully to the solving of practical problems with constraints, selecting as penalty functions the simplest module functions (of the Spang type [11]).

We propose a complex learning algorithm for sigmoid Artificial Neural Networks (ANN). We introduce the concept of the working area of a neuron for sigmoid ANNs in the form of a band in the attribute space, its width and location associated with the center line of the band to a fixed point. We define of the centers and widths of the working areas of neurons by analogy to the radial ANNs. On this basis, an algorithm for selecting the initial approximation of network parameters, ensuring uniform coverage of the data area with neuron working areas was developed. Network learning is carried out using a non-smooth regularizer designed to smooth and remove non-informative neurons. The results of the computational experiment illustrate the efficiency of the proposed integrated approach.

We propose a complex learning algorithm for sigmoid Artificial Neural Networks (ANN). We introduce the concept of the working area of a neuron for sigmoid ANNs in the form of a band in the attribute space, its width and location associated with the center line of the band to a fixed point. We define of the centers and widths of the working areas of neurons by analogy to the radial ANNs. On this basis, an algorithm for selecting the initial approximation of network parameters, ensuring uniform coverage of the data area with neuron working areas was developed. Network learning is carried out using a non-smooth regularizer designed to smooth and remove non-informative neurons. The results of the computational experiment illustrate the efficiency of the proposed integrated approach.

We propose a new approach to the approximation of an arbitrary function by the sum of radial basis functions using non-smooth regularization method combined with a new algorithm in determining the initial approximation which ensures an acceptable quality of approximation uniformly over the data area.