Engineering Optimization: Theory and Practice, Fourth Edition

454 Nonlinear Programming III: Constrained Optimization Techniques

Figure 7.15 Graphs ofφk.

methods that can be used to solve a general class of problems.

Minimizef (X)

subject to

gj( X)≤ 0 , j= 1 , 2 ,... , m

lj( X)= 0 , j= 1 , 2 ,... , p

(7.227)

7.18.1 Interior Penalty Function Method

Similar to Eq. (7.154), the present problem can be 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)]+H(rk)

∑p

j= 1

l^2 j(X) (7.228)

whereGj is some function of the constraintgj tending to infinity as the constraint

boundary is approached, andH(rk) is some function of the parameterrk tending to

infinity asrktends to zero. The motivation for the third term in Eq. (7.228) is that as