Engineering Optimization: Theory and Practice, Fourth Edition

384 Nonlinear Programming III: Constrained Optimization Techniques

Another procedure involves constructing an unconstrained function,F (X), by

adding penalty for violating any constraint as (as described in Section 7.12):

F (X)=f (X)+a

∑m

j= 1

[Gj(X)]^2 +b

∑p

k= 1

[Hk(X)]^2 (7.4)

where

[Gj(X)]^2 = max[ ( 0 , gj( X))]^2 (7.5)

[Hk(X)]^2 =h^2 k(X) (7.6)

indicate the squares of violations of inequality and equality constraints, respectively,

andaandbare constants. Equation (7.4) indicates that while minimizing the objective

functionf (X), a positive penalty is added whenever a constraint is violated, the penalty

being proportional to the square of the amount of violation. The values of the constants

aandbcan be adjusted to change the contributions of the penalty terms relative to the

magnitude of the objective function.

Note that the random search methods are not efficient compared to the other meth-

ods described in this chapter. However, they are very simple to program and usually

are reliable in finding a nearly optimal solution with a sufficiently large number of

trial vectors. Also, these methods can find near global optimal solution even when the

feasible region is nonconvex.

7.4 Complex Method

In 1965, Box extended the simplex method of unconstrained minimization (discussed

in Section 6.7) to solve constrained minimization problems of the type [7.2]:

Minimizef (X) (7.7a)

subject to

gj(X)≤ 0 , j= 1 , 2 ,... , m (7.7b)

x(l)i ≤xi≤x(u)i , i= 1 , 2 ,... , n (7.7c)

In general, the satisfaction of the side constraints (lower and upper bounds on the

variablesxi) ay not correspond to the satisfaction of the constraintsm gj( X)≤ 0. This

method cannot handle nonlinear equality constraints. The formation of a sequence of

geometric figures each havingk=n+1 vertices in ann-dimensional space (called

thesimplex) is the basic idea in the simplex method. In the complex method also,

a sequence of geometric figures each havingk≥n+1 vertices is formed to find the

constrained minimum point. The method assumes that an initial feasible pointX 1 (which

satisfies all themconstraints) is available.

Iterative Procedure

1.Findk≥n+1 points, each of which satisfies allmconstraints. In actual prac-

tice, we start with only one feasible pointX 1 , and the remainingk− 1 points