10.8 Generalized Penalty Function Method 619
10.8 Generalized Penalty Function Method
The solution of an integer nonlinear programming problem, based on the concept of
penalty functions, was originally suggested by Gellatly and Marcal in 1967 [10.5].
This approach was later applied by Gisvold and Moe [10.4] and Shin et al. [10.24]
to solve some design problems that have been formulated as nonlinear mixed-integer
programming problems. The method can be considered as an extension of the interior
penalty function approach considered in Section 7.13. To see the details of the approach,
let the problem be stated as follows:
FindX=
x 1
x 2
..
.
xn
=
{
Xd
Xc
}
whichminimizesf (X)
subject to the constraints (10.72)
gj( X)≥ 0 , j= 1 , 2 ,... , m
Xc∈Sc and Xd∈Sd,
where the vector of variables (X) is composed of two vectorsXdandXc, withXd
representing the set of integer variables andXc representing the set of continuous
variables. Notice thatXcwill not be there if all the variables are constrained to take
only integer values andXdwill not be there if none of the variables is restricted to
takeonly integer values. The setsScandSddenote the feasible sets of continuous and
integer variables, respectively. To extend the interior penalty function approach to solve
the present problem, given by Eq. (10.72), we first define the following transformed
problem:
Minimizeφk( X,rk, sk)
where
φk( X,rk, sk) =f(X)+rk
∑m
j= 1
Gj[gj(X)]+skQk(Xd) (10.73)
In this equation,rkis a weighing factor (penalty parameter) and
rk
∑m
j= 1
Gj[gj(X)]
is the contribution of the constraints to theφkfunction, which can be taken as
rk
∑m
j= 1
Gj[gj( X)]=+rk
∑m
j= 1
1
gj(X)
(10.74)
It can be noted that this term is positive for allXsatisfying the relationsgj( X)> 0 and
tends to+∞if any one particular constraint tends to have a value of zero. This property
ensures that once the minimization of theφkfunction is started from a feasible point,
