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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 i-th column of the j-th row is an interval which is strictly negative or strictly positive, then is monotonic with respect to the unknowns
• if the i-th column of the j-th 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 j-th 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( )

1. compute
2. let be the number of components of such that or and let be the variables for which this occur
3. 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
• if , then
• if , then
• otherwise
• )
• if this is the first estimation of then
• otherwise
• if , then
• if , then
4. end loop:
5. otherwise
6. return
This procedure has to be repeated for each .

Next: Improving the evaluation using Up: Mathematical background Previous: Mathematical background   Contents
Jean-Pierre Merlet 2012-12-20