Engineering Optimization: Theory and Practice, Fourth Edition

7.12 Basic Approach of the Penalty Function Method 431

unconstrained minimization problems. Let the basic optimization problem, with
inequality constraints, be of the form:

FindXwhich minimizesf (X)

subject to

gj( X)≤ 0 , j= 1 , 2 ,... , m (7.153)

This problem is converted into an unconstrained minimization problem by constructing
a function of the form

φk= φ(X, rk) =f(X)+rk

∑m

j= 1

Gj[gj(X)] (7.154)

whereGjis some function of the constraintgj, andrkis a positive constant known
as thepenalty parameter. The significance of the second term on the right side of
Eq. (7.154), called thepenalty term, will be seen in Sections 7.13 and 7.15. If the
unconstrained minimization of theφfunction is repeated for a sequence of values of
the penalty parameterrk(k = 1 , 2 ,.. .), the solution may be brought to converge to
that of the original problem stated in Eq. (7.153). This is the reason why the penalty
function methods are also known assequential unconstrained minimization techniques
(SUMTs).
The penalty function formulations for inequality constrained problems can be
divided into two categories: interior and exterior methods. In the interior formulations,
some popularly used forms ofGjare given by

Gj= −

1

gj(X)

(7.155)

Gj= og[l −gj(X)] (7.156)

Some commonly used forms of the functionGjin the case of exterior penalty function
formulationsare

Gj= ax[0m , gj(X)] (7.157)

Gj= { max[0, gi(X)]}^2 (7.158)

In the interior methods, the unconstrained minima ofφkall lie in the feasible region
andconverge to the solution of Eq. (7.153) asrkis varied in a particular manner. In
the exterior methods, the unconstrained minima ofφkall lie in the infeasible region
and converge to the desired solution from the outside asrkis changed in a specified
manner. The convergence of the unconstrained minima ofφkis illustrated in Fig. 7.10
for the simple problem

FindX= {x 1 } whichminimizesf (X)=αx 1

subject to (7.159)

g 1 (X)=β−x 1 ≤ 0