Line search question

[Barry Smith]

    I think the confusion comes from the fact that the Amijo condition  
is almost always in the literature for minimization; not for nonlinear  
equations.

If we minimize F(x) the condition is F(x + alpha p) <= F(x) + c alpha  
p' grad F(x) with c near 1.   Here ' denotes transpose

For the PETSc nonlinear solver F(x) = .5 f(x)' f(x). Ok, now just  
compute grad F in this case and the Jacobian pops right out. grad F(x)  
= J f

Hmm, looks like the comment is wrong where it says f_n . J . f_n, it  
should say p_n. J. f_n

    Barry


On Dec 5, 2009, at 10:24 PM, Ryan Yan wrote:

> Hi All,
> I am trying to figure out how the line search is actually working in  
> PETSc and I am looking at this link,
>
> http://www.mcs.anl.gov/petsc/petsc-as/snapshots/petsc-current/docs/manualpages/SNES/SNESLineSearchGetParams.html
> F(x
> Can anyone help to explain why there is a J in the amijo condition?
>
> alpha - The scalar such that .5*f_{n+1} . f_{n+1} <= .5*f_n . f_n -  
> alpha |f_n . J . f_n|
>
>
> Considering the fact that the newton direction p_n provide a J^{-1}  
> and the gradient of the merit function( 1/2 f . f) provide a J,  
> isn't correct to get rid of J in the definition above?
>
> Thanks for any suggestions,
>
> Yan