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Subsections
In other words, if is small, then the reduction in
that occurs at the point is close to the greatest
reduction that is allowed by the trust region
constraint.
Proof
for any , we have the identity:
If we choose such that
, we have:

(4.24) 
From 4.27, using the 2 hypothesis:
Combining 4.28 and 4.29, we obtain finally
4.26.
Lemma
From the hypothesis:
Combining 4.31 and 4.27 when reveals that:
The required inequality 4.30 is immediate from
4.28 and 4.32.
We will use this lemma with
.
We will choose as (see paragraph containing Equation
4.15 for the meaning of and ):

(4.28) 
Thus, the condition for
ending the trust region calculation simplifies to the inequality:

(4.29) 
We will choose
.
Next: An estimation of the
Up: The TrustRegion subproblem
Previous: The Rayleigh quotient trick
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Frank Vanden Berghen
20040419