Engineering Optimization: Theory and Practice, Fourth Edition

7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints 455

H (rk) →∞, the quantity (^) jp= 1 l^2 j( X)must tend to zero. If pj= 1 lj^2 ( X)does not tend to
zero,φkwould tend to infinity, and this cannot happen in a sequential m inimization
process if the problem has a solution. Fiacco and McCormick [7.17, 7.21] used the
following form of Eq. (7.228):
φk= φ(X, rk) =f(X)−rk
∑m
j= 1

1

gj(X)

+

1

√

rk

∑p

j= 1

l^2 j(X) (7.229)

Ifφkis minimized for a decreasing sequence of valuesrk, the following theorem proves

that the unconstrained minimaX∗k will converge to the solutionX∗of the original

problem stated in Eq. (7.227).

Theorem 7.5If the problem posed in Eq. (7.227) has a solution, the unconstrained min-

ima,X∗k , ofφ(X, rk) defined by Eq. (7.229) for a sequence of values, r 1 >r 2 >·· ·>rk,

converge to the optimal solution of the constrained problem [Eq. (7.227)] asrk→. 0

Proof: A proof similar to that of Theorem 7.1 can be given to prove this theorem.

Further, the solution obtained at the end of sequential minimization ofφkis guaranteed

to be the global minimum of the problem, Eqs. (7.227), if the following conditions are

satisfied:

(i)f (X)is convex.

(ii)gj(X),j= 1 , 2 ,... , mare convex.

(iii) (^) jp= 1 l^2 j( X)is convex in the interior feasible domain defined by the inequality
constraints.
(iv) One of the functions amongf (X),g 1 (X), g 2 ( X),... , gm( X),and pj= 1 lj^2 (X)
is strictly convex.
Note:
1.To start the sequential unconstrained minimization process, we have to start from
a pointX 1 at which the inequality constraints are satisfied and not necessarily
the equality constraints.
2.Although this method has been applied to solve a variety of practical problems,
it poses an extremely difficult minimization problem in many cases, mainly
because of the scale disparities that arise between the penalty terms
−rk
∑m
j= 1

1

gj(X)

and

1

rk^1 /^2

∑p

j= 1

lj^2 (X)

as the minimization process proceeds.

7.18.2 Exterior Penalty Function Method

To solve an optimization problem involving both equality and inequality constraints as

stated in Eqs. (7.227), the following form of Eq. (7.228) has been proposed:

φk= φ(X, rk) =f(X)+rk

∑m

j= 1

〈gj(X)〉^2 +rk

∑p

j= 1

l^2 j(X) (7.230)