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150 Gradient Optimization
by comparison with finite differences before even trying optimization. Failure
in optimization is commonly due to incorrectly formulated or programmed
gradients, SO that the optimizer is working with bad information.
The great virtue of the Fletcher-Reeves algorithm is that its computer
memory requirements are proportional to 3N, where there are N variables.
The Fletcher-Powell and other variable metric algorithms require a memory
proportional to N 2 • They all belong to the class of conjugate gradient algorithms,
but the variable metric algorithms, being quasi-Newton, converge
more rapidly when very near a minimum. This means that Fletcher-Reeves
Program B5-l should be very satisfactory on small machines employed for
engineering applications requiring only moderate accuracy.
5.5. Network Objective Functions
The numerOUS test problems constructed by mathematicians, such as the
preceding Rosenbrock example, are enlightening and provide some measure
of effectiveness for various optimization algorithms. But what kind of objective
functions are appropriate for automatic adjustment of design variables in
electrical networks? The following methods are easy to implement and have
an interesting resemblance to weighted-sample integration techniques (Section
2.3). The optimization process can also be viewed as a curve-fitting process.
However, as mentioned in Section 2.5, nonlinear programming is often ineffective
when compared to methods that are specifically formulated for certain
problems.
On the other hand, many network design requirements cannot be solved by
existing closed-form methods, as evident by the brief exposure to network
synthesis in Chapter Three. Also, the designer may not be aware of more
appropriate methods or may not have the time or inclination to implement
them. Then optimization of networks is worth trying, especially if there is an
approximate design basis to serve as a starting point for both insight and
values.
The following sections describe several important kinds of network objective
functions and their gradients. An example using Fletcher-Reeves optimizer
Program B5-l is given.
5.5.1. Integral Error Functions. Most cases of optimization in the frequency
or time domains amount to curve fitting, as seen in Figure 5.1. The error can
be defined as the square of the area between a desired function (the rectangle)
and the approXimation function. This is expressed as
j W'
jW'
mJnE= e\x,w)dw= (R-G)'dw,
WI
WI
(5.86)
where the first integrand emphasizes its dependence on both the variables (x)