difference in Armijo expression

[Petrichor]

Hello,

I was just wondering, what is the difference between these two expression, knowing that they represent the same thing: the Armijo condition (when doing manifold optimization)

(these are from Absil's book, and Nicolas Boumal's book)

Thank you

[Ronny Bergmann]

Hi,

Both do the same backtracking, just that Nicolas version is a specific case, namely where the search direction (\eta in P.-A.s book) is the negative gradient.

The more general case is important e.g. when considering CG or quasi Newton, when your direction of interest is most likely not the negative gradient direction.

You can for example also compare to Nocedal&Wright Numerical Optimization, p. 33 for the Euclidean case including good illustrations of these conditions.

Best

Ronny .

[Petrichor]

Thank you for your answer.

Is the first defintion (from Absil's book), the one used in the Manopt toolbox (line_search_hint)?

[Ronny Bergmann]

I am not so sure, because I mainly work with Manop.jl in Julia, where we implemented Absil's variant.

I think it should be Absils variant as well as far as I read the docs at 

here https://github.com/NicolasBoumal/manopt/blob/master/manopt/solvers/linesearch/linesearch_hint.m

because they work with df0 and not the gradient or own other words d is the \eta in Absil and df0 is the inner product on the right hand side.