Page 234 - Compact Numerical Methods For Computers

222 Compact numerical methods for computers
the other (n-m). This yields a new problem involving fewer parameters which
automatically satisfy the constraints.
Simple inequality constraints may frequently be removed by substitutions which
satisfy them. In particular, if the constraint is simply

b > 0 (18.11)
k
then a substitution of for b suffices. If, as a further case, we have
k
v > b > u (18.12)
k
then we can replace b k by

(18.13)

It should be noted that while the constraint has obviously disappeared, these
substitutions tend to complicate the resulting unconstrained minimisation by
introducing local minima and multiple solutions. Furthermore, in many cases
suitable substitutions are very difficult to construct.
Penalty functions
The basis of the penalty function approach is simply to add to the function S( b )
some positive quantity reflecting the degree to which a constraint is violated. Such
penalties may be global, that is, added in everywhere, or partial, that is, only
added in where the constraint is violated. While there are many possibilities, here
we will consider only two very simple choices. These are, for equality constraints,
the penalty functions
(18.14)
and for inequalities, the partial penalty

(18.15)
where H is the Heaviside function
for x >0
(18.16)
for x<0.
The quantities w , j=1, 2, . . . , m, and W , k=1, 2, . . . , q, are weights which
k
j
have to be assigned to each of the constraints. The function
(18.17)

then has an unconstrained minimum which approaches the constrained minimum
of S(b) as the weights w, W become large.
The two methods outlined above, while obviously acceptable as theoretical
approaches to constrained minimisation, may nonetheless suffer serious difficulties
in the finite-length arithmetic of the computer. Consider first the example:
minimise
2
( b +b -2) (18.18a)
2
1