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The Fletcher- Reeves Optimizer 149
times. The trajectory is shown in Figure 5.24. It will not reach the minimum in
the specified loo·iteration limit; inputting GOTOl50 will cause the program
to continue the searches. Delete temporary statement 392 and add 855
GOT0260. This disables the policy of resetting to steepest descent after every
N iterations.
5.4.4. SCilling. The new user may construct a problem of his own design
only to find that it won't optimize. The difficulty is often in the scaling of the
variables, i.e., sensitivity. This is equivalent to partial derivatives; so the user
should be aware of a rule of thumb regarding units of the variables. In the
context of electrical network problems, suppose that the level curves in Figure
5.23 belong to the two variables in one of the L-section networks of Figure 4.3.
For the frequencies of interest, these network Land C design variables make
sense in units of nanohenrys and picofarads. A useful rule of thumb is: if any
variable is increased by unity, do solutions still make sense? Another symptom
is the gradient vector; the magnitude of its elements should be roughly equal
and about unity within a factor of lOOO± '. But suppose that the inductance is
specified in microhenrys; then an increase from 0.4 to 1.4 microhenrys is a
much bigger jump than from 400 to 401 nanohenrys. What is at stake is seen
in Figure 5.23; a bad choice of variable units can squeeze the curved valleys
into raror-thin slits, so that the optimizer's finite word length search is in
fundamental trouble.
An illustration of this effect is easily created using the Rosenbrock example.
One or more initial variable values input at the beginning are rescaled,
e.g., increased by a factor of 100. Then, at the beginning of subroutine 1000,
these variables are decreased by 100 and then increased again before returning
from that subroutine. Also, the corresponding derivatives must be decreased
by 100 before returning (an application of the chain rule from calculus). Upon
trying this, the effect on the gradient is immediately obvious-the number of
function evaluations is increased by about half again. The reader is urged to
try this on the Rosenbrock function to observe scaling and its effect on search
difficulty.
5.4.5. Summary of the Fletcher-Reeves Program. The Fletcher-Reeves
search strategy has been reviewed and BASIC language Program B5·1 has
been described in the context of previously discussed topics. The Rosenbrock
two-variable, nonlinear problem was described, and a number of enlightening,
temporary program modifications were suggested. Also, the subject of scaling
of variables was mentioned; it is the foremost pitfall the new user is likely to
encounter when formulating his own objective function.
In addition to scaling, an "awful warning" was issued to be sure that only
smooth functions are modeled for gradient optimization. This is especially true
when the gradient vector is obtained by finite differences. Another warning
about gradients is that evaluation of analytical expressions should be checked