Stochastic Minimization with Adaptive Memory

Fig. 12. Comparison between the number of functional evaluations with gaussian and exponential damping functions. The results on a complete sweep on 24 values of m in 0,1] are shown. Finally, we should note that the choice of gaussian damping for the memory mechanism is not critical. We experimented with a pure exponential damping getting similar results. This suggests that the dynamic mechanism used to maintain the necessary depth is suuciently strong to cope with the eeects deriving from the choice of diierent damping functions (see Fig. 12). A random search algorithm with adaptive memory has been presented which is characterized by the use of an adap-tive gaussian memory for biasing the exploration. The algorithm has been compared to its variant without memory and to traditional techniques in several minimization tasks. Of particular interest is the application to the computation of the minimum eigenvalue of a singular diierential operator on a Hilbert space for which traditional techniques perform badly. Acknowledgements The authors would like to thank Prof. T. Poggio and B. Caprile for their suggestions. G. T. would like to thank warmly E. Onofri and V. Fateev for stimulating discussions. 7 (b) we can eeciently compute the values of at points xj = 2 n j 4K(m) (21) using a Fast Fourier transform algorithm 14], (c) the derivative terms contained in H1 and H2 can be computed directly in the Fourier Space, because, from Eq. 20, we have d dx (x) = n X j=1 wj 2 4K(m) jj(x) (22) This allows the fast computation of H1 and H2 without evaluating the matrix representations of H1 and H2, 2. Given the set fmig such that 0 < m1 < m2; ::::: < mk 1, compute (x;mk) with algorithm of Fig. 1.