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Using the monotonicity
For a given box we will compute the jacobian matrix using
interval analysis. Each row of this interval matrix enable to get some
information of the corresponding function .
 if the ith column of the jth row is an interval which is
strictly negative or strictly positive, then is monotonic with
respect to the unknowns
 if the ith column of the jth row is equal to 0, then function
does not depend on the variable
In the first case the minimal and maximal value of will be
obtained either for
or
and we are able to define the value of to get
successively the minimal and maximal value as we know the sign of the
gradient. But this procedure has to be implemented recursively. Indeed
we have previously computed the jacobian matrix for
but now
have a fixed value: hence a component of the jth row which
for
was such that
and
may now be a strictly positive or negative
intervals. Consequently the minimal and maximal value will be obtained
for some combination of in the two sets
and
. Bus as has now a fixed value
some other component of may become strictly negative or
positive...
The algorithm for computing a sharper evaluation of is:
=Evaluate(
)
 compute
 let be the number of components of such that
or
and let
be the variables for which this occur
 if
loop:
for all combination of
in the set
:
 if
 compute
 let be the number of components of such that
or
 if , then
=Evaluate(
)
 otherwise
)
 if this is the first estimation of then
 otherwise
 otherwise

)
 if this is the first estimation of then
 otherwise
 end loop:
 otherwise
 return
This procedure has to be repeated for each .
Next: Improving the evaluation using
Up: Mathematical background
Previous: Mathematical background
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JeanPierre Merlet
20121220