By Ya. Z. Tsypkin
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A physical interpretation of such properties of the multistage algorithms of optimization will be given in the next section. Unfortunately, we still do not have a general method for selecting the coefficients amand Tm[n]. 13 Continuous Algorithms of Optimization The continuous algorithms of optimization can be obtained by a limiting process from the difference equations describing the corresponding discrete algorithms of optimization discussed thus far. 41) with s1 = 1, we obtain the continuous algorithms of optimization after substituting the continuous time t for iz, and the derivatives for the hfferences.
2). The delay line is designated by T D in Fig. 2b. The output of the digital integrator (digrator) is always c,[n - 11 (Fig. 2). Double lines in Fig. 1 indicate vector relationships. This discrete feedback system is autonomous. Ail necessary a priori information is already present in the nonlinear transformer. Fig. 5 21 A Possible Generalization When J(c) = const has “ridges” (Fig. 3), the rate of convergence to the optimal point c* is slow. In such cases, instead of the scalar, it is better to use the matrix FCnl = l l Y v u c ~ l l l (v, p = 1, ..
Let us form the variational equation. 47) where q[n] is the deviation from the optimal vector. 48) This difference equation has a trivial solution q = 0, since by the definition of c*, we have VJ(c*) = 0. 4). As it is known, two types of stability are distinguished when all the coordinates of the vector q[n] are smali. One is the local stability, and the other is global stability (for any q[n]). In order to investigate the local stability, the gradient VJ(c* + q) must first be approximated by a linear f h c t i o n and the obtained linear difference equation is then tested for stability.
Adaptation and Learning in Automatic Systems by Ya. Z. Tsypkin