Page 283 - MATLAB an introduction with applications

268 ——— MATLAB: An Introduction with Applications

This process is continued for i = 1, 2, …, until x is perturbed to find Y k, n .
n
4. If the point Y k, n remains same as X , reduce the step lengths ∆x (say, by a factor of 2), set i = 1 and
k
i
go to step 3. If Y k, n is different from X , obtain the new base point as
k
X k +1 = Y k, n
and go to step 5.
5. With the help of the base points X and X k + 1 , establish a pattern direction S as
k
S = X k + 1 – X k
and find a point Y k + 1, 0 as
Y k + 1,0 = X k + 1 + λS ...(5.16)
where λ is the step-length, which can be taken as 1 for simplicity. Alternatively, we can solve a one-
dimensional minimization problem in the direction S and use the optimum step length λ* in place of λ
in Eq.(5.16).
6. Set k = k + 1, f = f(Y ), i = 1, and repeat step 3. If at the end of step 3, f(Y ) < f(X ), we take the new
k0
k,n
k
k
base point as X k +1 = Y and go to step 5. On the other hand, if f (Y ) ≥ f (X ), set X k+1 ≡ Y , reduce
k,n
k,n
k
k
the step lengths ∆x , set k = k + 1, and go to step 2.
i
7. The process is assumed to have converged whenever the step lengths fall below a small quantity ε.
Thus, the process is terminated if
max(∆x ) < ε.
i
5.8 INTERIOR PENALTY FUNCTION METHOD
In the interior penalty function methods, a new function (φ function) is constructed by augmenting a penalty
term to the objective function. The penalty term is chosen such that its value will be small at points away
from the constraint boundaries and will tend to infinity as the constraint boundaries are approached. Thus,
once the unconstrained minimization of φ(X, r ) is started from any feasible point X , the subsequent points
1
k
generated will always lie within the feasible domain since the constraint boundaries act as barriers during
the minimization process. The φ function defined as
m 1
(, ) =
( ) r
φ X r k f X − k∑
()
j= 1 g X
j
Since the above equation does not allow any constraint to be violated, it requires a feasible starting point
for the search toward the optimum point. The iteration procedure of this method can be summarized as
follows:
Iterative process:
1. Start with an initial feasible point X satisfying all the constraints with strict inequality sign, that is,
1
g (X ) < 0 for j = 1, 2, …, m and an initial value for r > 0. Set k = 1.
1
j
1
2. Minimize φ(X, r ) by using any of the unconstrained minimization methods and obtain the solution
k
X* .
k
3. Test whether X* k is the optimum solution of the original problem. If X* is found to be optimum, terminate
k
the process or else, go to the next step.