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Constraints 157
transislors could be optimized, even accounting for gain slope versus frequency.
Table 5.5 summarized some results that showed a 10: I reduction in
squared error as well as the tendency toward minimax behavior for P= to and
P=30 minimizations. It is suggested that the stopping criterion in line 160
(0.00001) is probably smaller than need be. Engineering design usually does
not require this kind of accuracy, and the Fletcher-Reeves algorithm is known
to converge slowly near a minimum. Users might consider a value of E=O.OI
or use of a 0.1% relative change-stopping criterion instead of the absolute
change criterion presently incorporated.
A common rule of thumb is that the number of samples should be at least
twice the number of variables. If there are too few samples, the function may
oscillate wildly between frequency samples while giving the illusion of a very
good fit of sampled response to goals. Which samples to take, how they are
weighted, which multiple response types are not conflicting, and many other
aspects of network optimization are more a matter of experience and insight
than science. This is also true of questions concerning how close to a
minimum must one start the variables and whether the minimum is global as
opposed to inferior local minima, which trap the search prematurely. In the
latter case, the usual advice is to try starting at a variety of points in the
variable space. As for starting reasonably near a solution, that is what the rest
of this book is all about. The main virtue of an optimizer is its ability to treat
significant second-order effects that are too difficult or inconvenient to treat
otherwise.
5.6. Constraints
The subject of constraints deals with the explicit or implicit relationships
among optimization variables (x). The most elementary constraints are upper
and/or lower bounds and linear dependence, such as
and
xj>O,
k 1