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162 Gradient Optimization
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Frequency (Hz)
Figure 5.31. Worst.case variations for pessimization of two lowpass network designs. Lowpass
N =5 networks: a, zeros in left- and right.half planes; b, zeros only in left-half plane. [Reprinted
with permission from Manaktala, 1972,)
frequency, there must be some adverse combination of tolerances that would
produce worst-case selectivity, both maximum and minimum, This is shown in
Figure 5.31. Rather than employing the usual time-consuming Monte-Carlo
method, it was suggested that a constrained optimizer program could find the
minimum and maximum selectivity at each frequency subject to the bounding
element tolerance ranges-truly a pessimization problem. The performance of
any network would then be contained inside the envelope shown in Figure
5.31.
5.6.2. Barrier Functions for Inequality Constraints. The complete constrained
optimization problem was defined by (5.94). Tbis section considers
the vector c of inequality constraints that are generally nonlinear. It is
remarked in passing that a subset would consist of linear constraints of the
form
Ax-b:>O. (5.101)
These boundaries are lines in 2-variable space, otherwise hyperplanes. Minimization
with these constraints is like descending on the surface of Figure 5.3,
except that it has been placed in a restricting glass box; the descent should
conform to these glass walls, or hyperplanes, when encountered. The most
common means for doing this is to project linear search directions on such
constraining surfaces when encountered. This complicates linear search algorithms
and is beyond the scope of the present treatment; the interested reader
is referred to Rosen's projection method described by Hadley (l964, pp.