458 Nonlinear Programming III: Constrained Optimization Techniques
Another method of handling the parametric constraints is to construct theφfunction
in a different manner as follows [7.1, 7.15].Interior Penalty Function Methodφ(X, rk) =f(X)−rk∑mj= 1[∫θuθl1
gj( X,θ )dθ]
(7.235)
The idea behind using the integral in Eq. (7.235) for a parametric constraint is to
make the integral tend to infinity as the value of the constraintgj( X,θ )tends to zero
even at one value ofθin its range. If a gradient method is used for the unconstrained
minimization ofφ(X, rk) the derivatives of, φwith respect to the design variables
xi(i = 1 , 2 ,... , n)are needed. Equation (7.235) gives∂φ
∂xi( X,rk)=∂f
∂xi(X)+rk∑mj= 1[∫
θuθl1
g^2 j( X,θ )∂gj
∂xi( X,θ )dθ]
(7.236)
by assuming that the limits of integration,θl andθu, are indepdnent of the design
variablesxi. Thus it can be noticed that the computation ofφ(X,rk) ro ∂φ(X,rk)/∂xi
involves the evaluation of an integral. In most of the practic al problems, no closed-form
expression will be available forgj( X,θ), and hence we have to use some sort of a
numerical integration process to evaluateφor∂φ/∂xi. If trapezoidal rule [7.22] is used
toevaluate the integral in Eq. (7.235), we obtain†φ( X,rk) =f(X)−rk∑mr= 1
θ
2[
1
gj( X,θl)+
1
gj( X,θu)]
+ θr∑− 1p= 21
gj( X,θp)
(7.237)
†Let the interval of the parameterθbe divided intor−1 equal divisions so thatθ 1 =θl, θ 2 =θ 1 + θ, θ 3 =θ 1 + 2 . θ,... , θr=θ 1 + (r− 1 ) θ=θu,θ=
θu−θl
r− 1
If the graph of the functiongj( X,θ )looks as shown in Fig. 7.19, the integral of 1/gj( X,θ )can be found
approximately by adding the areas of all the trapeziums, likeABCD. This is the reason why the method is
known astrapezoidal rule. The sum of all the areas is given by
∫θuθldθ
gj( X,θ )
≈∑r−^1
l= 1Al=∑r−^1
p= 1[
1
gj( X,θp)
+^1
gj( X,θp+ 1 )]
θ
2=
θ
2[
1
gj( X,θl)
+
1
gj( X,θu)]
+∑r−^1
p= 2 θ
gj( X,θp)