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The netcher- Reeves Optimizer 145
convergence is obtained in exactly N linear searches. Because the objective
function is seldom quadratic in practice, (5.80) is restarted in the steepest
descent direction (/30=0) after every N iterations (linear searches).
An important program feature is the criteria for stopping the iterative
search for a minimum. This implementation by Fletcher stops when the
changes being made in every component of the x variable vector are less than
0.00001, or when 100 iterations (linear searches) have been performed. The
running time of the algorithm increases dramatically for even smaller changes;
engineering problems often allow even earlier termination. Perhaps a better
stopping criterion is the relative changes of variables. One advantage of
real-time computing is the ability of the user to manually intervene whenever
appropriate.
5.4.2. The BASIC Language Computer Program. Appendix Program B5-1
is a listing of Fletcher's program VA08A as translated into BASIC from
FORTRAN. These 114 lines require only 1849 bytes in the Commodore PET
computer, and only IS additional bytes are required for each optimization
variable. The program requires the user to define the objective function as
subroutine 1000. The particular objective function and the gradient defined in
lines 1000-1060 will be discussed in the next section. Unused BASIC mimes
are given in line 60. Each execution of the program requires the user to state
the number of variables, which should be consistent with the defined objective
function. Then the starting values of the variables are requested. That runtime
input is coded in lines 70-145.
Some program control constants are set in lines 150-170; this is less flexible
than originally provided by Fletcher (\972b). The number of iterations is
limited to 100, the absolute change in each variable must be less than 0.00001
for convergence, and the first step length in each iteration is based on an
expected 10% decrease in function value.
The flowchart in Appendix D for a single-variable linear search is very
nearly applicable to the entire B5-1 program; the reader should generalize it
by reference to the complete program listing. The initial and subsequent
setting of search direction to steepest descent is made by lines 230-240. The
FOR-NEXT loop, to accomplish N searches before resetting to steepest
descent, spans lines 260-850. These directions are calculated in lines 330-400
according to (5.80) and (5.81). Having chosen a search direction, the slope in
that direction is computed by lines 410-440 according to (5.57). The linear
search occurs as discussed in Section 5.3, except that each variable is increased
by line 535 according to (5.78), and lines 850 and 860 implement
repeated sequences of N linear searches.
5.4.3. The Rosenbrock Example. Lootsma (\972, pp. 29, 67, 68, 74-88, 101,
120, 185) gives many standard nonlinear programming (NLP) test problems,
perhaps the most popular being the so-called Rosenbrock banana function,