download searchable PDF of Circuit Design book - IEEE Global ...

148 Gradient Optimization
The output for the Rosenbrock problem, starting at Xl = I and x,= -I, is
shown in Table 5.4 (see Figure 5.24). The data show that ITN = 21 iterations
(linear search directions) and IFN =61 function and gradient evaluations are
required to locate the global minimum at x=(I,I)T to at least 0.00001
accuracy in each variable.
The reader should run this example to observe the effects of several
changes. Several new starting points should be tried. The accuracy set in line
160 can be reduced. The number of variables can be set to 20 instead of 2 by
inputting the latter number when asked and setting all but the first two
variables to an arbitrary number, e.g., O. This will illustrate how much of the
computing time is in search overhead, because the full 20 variables will be
treated by the Fletcher-Reeves algorithm even though only the first two
determine the problem defined in subroutine 1000. It is informative to add the
statement 392 Z = O. This causes the search to be of steepest descent at all
1.0
0.9
0.8
0.7
0.6
0.5
a.'
0.3
0.2
Fletcher-Reeves
Optimum value of x~'
, ,,,,,, /
, , /
/
/.~.
, j
-0.1 \ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xl
-0.1\
-0.2
-0.5
-0.6
Steepest descent
-0.7
-0.8
Figure 5.24.
-0.9
-1.0
Starting value of x --:lIo.
Trajectories in the x plane for the Fletcher-Reeves and steepest-descent algorithms.