Numerical Methods and Optimization

376 Numerical Methods and Optimization: An Introduction

Consider the i

th

component of (14.15):

y

(k+1)

i

= y

(k)

i

− α

k

d

(k)

i

=1− α

k

d

(k)

i

. (14.16)

Observe that if d

(k)

i

≤ 0, then y

(k+1)

i

> 0 for any α

k

> 0. If, on the other

hand, d

(k)

i

> 0, then

y

(k+1)

i

> 0 ⇔ 1 − α

k

d

(k)

i

> 0 ⇔ α

k

< 1/d

(k)

i

.

It is easy to see that the above inequality is satisfied for α

k

given by

α

k

= α min

i:d

(k)

i

>0

A

1/d

(k)

i

B

,

where 0 <α<1 (typically α =0.9or0.99ischosen).Notethatifd

(k)

=0

and d

(k)

≤ 0, then the problem is unbounded (there is no minimizer). Finally,

having computed y

(k+1)

i

, we need to find the corresponding feasible point of

the original problem (14.13) by applying the inverse scaling operation, ...