Engineering Optimization: Theory and Practice, Fourth Edition

434 Nonlinear Programming III: Constrained Optimization Techniques

4.Suitable convergence criteria have to be chosen to identify the optimum point.

5.The constraints have to be normalized so that each one of them vary between

−1 and 0 only.

All these aspects are discussed in the following paragraphs.

Starting Feasible PointX 1. n most engineering problems, it will not be very difficultI

to find an initial pointX 1 satisfying all the constraints,gj(X 1 ) < 0. As an example,

consider the problem of minimum weight design of a beam whose deflection under a

given loading condition has to remain less than or equal to a specified value. In this

case one can always choose the cross section of the beam to be very large initially so

that the constraint remains satisfied. The only problem is that the weight of the beam

(objective) corresponding to this initial design will be very large. Thus in most of the

practical problems, we will be able to find a feasible starting point at the expense of a

large value of the objective function. However, there may be some situations where the

feasible design points could not be found so easily. In such cases, the required feasible

starting points can be found by using the interior penalty function method itself as

follows:

1.Choose an arbitrary pointX 1 and evaluate the constraintsgj( X)at the point

X 1. Since the pointX 1 is arbitrary, it may not satisfy all the constraints with

strict inequality sign. Ifrout of a total ofmconstraints are violated, renumber

the constraints such that the lastrconstraints will become the violated ones,

that is,

gj(X 1 ) < 0 , j= 1 , 2 ,... , m−r

gj(X 1 ) ≥ 0 , j=m−r+ 1 , m−r+ 2 ,... , m (7.161)

2.Identify the constraint that is violated most at the pointX 1 , that is, find the

integerksuch that

gk(X 1 ) =max[gj(X 1 )]

forj=m−r+ 1 , m−r+ 2 ,... , m (7.162)

3.Now formulate a new optimization problem as

FindXwhich minimizesgk(X)

subjectto

gj( X)≤ 0 , j= 1 , 2 ,... , m−r

gj(X)−gk(X 1 ) ≤ 0 , j=m−r+ 1 , m−r+ 2 ,... ,

k− 1 , k+ 1 ,... , m (7.163)

4.Solve the optimization problem formulated in step 3 by taking the pointX 1 as

afeasible starting point using the interior penalty function method. Note that

this optimization method can be terminated whenever the value of the objective

functiongk( X)drops below zero. Thus the solution obtainedXMwill satisfy at

least one more constraint than did the original pointX 1.