Page 194 - Compact Numerical Methods For Computers

Direct search methods 183

nevertheless may need to minimise functions. I have used the Hooke and Jeeves
method in a number of forecasting courses, and published it as a step-and-
description version with a BASIC code in Nash (1982). A more advanced version
appears in Nash and Walker-Smith (1987). I caution that it may prove very slow to
find a minimum, and that it is possible to devise quite simple functions (Nash and
Walker-Smith 1989) which will defeat its heuristic search. Algorithm 27 below
presents a Pascal implementation.

Algorithm 27. Hooke and Jeeves minimiser
procedure hjmin(n: integer; {the number of parameters in the
function to be minimised}
var B,X: r-vector; {the parameter values on
input (B) and output (X) from minmeth}
var Fmin: real; {‘minimum’ function value}
Workdata: probdata; {user defined data area}
var fail: boolean; {true if method has failed}
intol: real); {user-initialized convergence
tolerance}
{alg27.pas == Hooke and Jeeves pattern search function minimisation
From Interface Age, March 1982 page 34ff.
Copyright 1988 J.C.Nash
}
var
i j: integer; {loop counters}
stepsize: real; {current step size}
fold: real; {function value at ‘old’ base point}
fval: real; {current function value}
notcomp: boolean; {set true if function not computable}
temp: real; {temporary storage value}
samepoint: boolean; {true if two points are identical}
ifn: integer; {to count the number of function evaluations}
begin
if intol<0.0 then int<1 := calceps; {set convergence tolerance if necessary}
ifn := 1; {to initialize the count of function evaluations}
fail := false; {Algorithm has not yet failed.}
{STEP HJ1: n already entered, but we need an initial stepsize. Note the
use of the stepredn constant, though possibly 0.1 is faster for
convergence. Following mechanism used to set stepsize initial value.}
stepsize := 0.0;
for i := l to n do
if stepsize < stepredn*abs(B[i]) then stepsize := stepredn*abs(B[i]);
if stepsize=0.0 then stepsize := stepredn; {for safety}
{STEP HJ2: Copy parameters into vector X}
for i := 1 to n do X[i] := B[i];
{STEP HJ3 not needed. In original code, parameters are entered into X
and copied to B}
fval := fminfn(n, B,Workdata,notcomp); {STEP HJ4}
if notcomp then
begin
writeln(‘***FAILURE*** Function not computable at initial point’);
fail := true;