Page 243 - Compact Numerical Methods For Computers

230 Compact numerical methods for computers
time. Perry and Soland (1975) derive analytic solutions to this problem, but both
to check their results and determine how difficult the problem might be if such a
solution were not possible, the following data were run through the Nelder-Mead
simplex algorithm 19 on a Data General NOVA operating in 23-bit binary
arithmetic.
K =3·82821 K =0·416 K =5·24263 F=8·78602
1
3
2
a=0·23047 b b=0·12 g g =0·648 d d=1·116.
T
Starting at the suggested values b =(7, 4, 300, 1621), S(b)=-77·1569, the
T
algorithm took 187 function evaluations to find S(b*)=-77·1602 at (b*) =
(6·99741, 3·99607, 300·004, 1621·11). The very slow convergence here is cause
for some concern, since the start and finish points are very close together.
T
From b =(1, 1, 300, 1621), S(b)=707·155, the Nelder-Mead procedure took
888 function evaluations to b*=(6·97865, 3·99625, 296·117, 1619·92) T and
S(b*)=-77·1578 where it was stopped manually. However, S was less than -77
after only 54 evaluations, so once again convergence appears very slow near the
T
minimum. Finally, from b =(1, 1, 1, l), S(b)=5·93981, the algorithm converged
to S(b*)=-77·1078 in 736 evaluations with b*=(11·1905, 3·99003, 481·054,
T
2593·67) . In other words, if the model of the lottery operation, in particular the
production function for the number of tickets sold, is valid, there is an alternative
solution which ‘maximises’ revenue per unit time. There may, in fact, be several
alternatives.
If we attempt the minimisation of S(b) using the variable metric algorithm 21
and analytic derivatives, we obtain the following results.

Initial point b S (b) Final point b* S(b*) efe’s
(a) 7, 4, 300, 1621 -77·1569 6·99509, 4·00185, 300, 1621 -77·1574 46
( b) 1, 1, 300, 1621 707·155 591·676, 563·079, 501·378, 1821·55 0·703075 157
(c) 1, 1, 1, 1 5·93981 6·99695, 3·99902, 318·797, 1605·5 -77·0697 252

(The efe’s are equivalent function evaluations; see §18.4 for an explanation.) In
case (b), the price per ticket (second parameter) is clearly exorbitant and the
duration of the draw (first parameter) over a year and a half. The first prize (third
parameter, measured in units 1000 times as large as the price per ticket) is
relatively small. Worse, the revenue (-S) per unit time is negative! Yet the
derivatives with respect to each parameter at this solution are small. An addi-
tional fact to be noted is that the algorithm was not able to function normally, that
is, at each step algorithm 21 attempts to update an iteration matrix. However,
under certain conditions described at the end of §15.3, it is inadvisable to do this
and the method reverts to steepest descent. In case (b) above, this occurred in 23
of the 25 attempts to perform the update, indicating that the problem is very far
from being well approximated by a quadratic form. This is hardly surprising. The
matrix of second partial derivatives of S is certain to depend very much on the
parameters due to the fractional powers (a, b, g, d) which appear. Thus it is
unlikely to be ‘approximately constant’ in most regions of the parameter space as