Page 189 - Compact Numerical Methods For Computers

178 Compact numerical methods for computers
Algorithm 19. A Nelder-Mead minimisation procedure (cont.)
end;{else shrink failed}
end; {if VR>=VH -- shrink}
end; {reduction}
end; {if VH>VL+...}
{STEP 29 -- end of major cycle of method}
until ((VH<=VL+convtol) or shrinkfail );
{STEP 30: if progress made, or polytope shrunk successfully, try
another major cycle from STEP 10}
end; {begin minimisation}
{STEP 31} {save best parameters and function value found}
writeln(‘Exiting from Alg19.pas Nelder Mead polytope minimiser’);
writeIn(‘‘,funcount,’ function evaluations used’);
Fmin := P[nl,L]; {save best value found}
for i := 1 to n do X[i] := P[i,L];
if shrinkfail then fail := true;
{STEP 32} {exit}
end; {alg19.pas == nmmin}

14.3. AN AXIAL SEARCH PROCEDURE
The following procedure is a device designed primarily to generate information
concerning the shape of the surface S(b) near the minimum, which will be labelled
b*. In some instances, where a minimisation algorithm has converged prema-
turely, the axial search may reveal a point having a lower function value than that
at the supposed minimum. The step taken along each axis is somewhat arbitrary.
O’Neill (1971), for instance, in applying the axial search after a Nelder-Mead
program, uses ±0·1001 times the initial steps used to generate the simplex.
However, these latter increments must be supplied by the program user. I prefer
to adjust the parameter b i by an increment
s=e( |b i | +e) (13.17)
where e is some small number such as the square root of the machine precision.
Section 18.2 gives a discussion of this choice. Its principal advantage is that the
increment always adjusts the parameter. Alternatively. I have employed the
assignment
:= b i (1±0·001) (14.18)
b i
unless these fail to change the parameter, in which case I use
:= ±0·001. (14.19)
b i
The latter axial search was used in the set of tests used to compare algorithms for
function minimisation which were included in the first edition and reported in §18.4
of both editions. What follows below reflects our current usage. including some
measures of the curvature and symmetry of the functional surface near the presumed
minimum.