# [Ipopt] sensitivities of ipopt's solutions

Thu Apr 20 07:37:11 EDT 2017

```Hi all,

.. and what I completely forgot: DF/Dx is not available since only an
approximation of the Hessian is available. Is there still a variant of the
implicit function theorem that could be applied? I thought of interpreting
the approximation for the Lagrangian's Hessian as an preconditioner and
doing a few Krylov iterations in the formula for substantial dx/dp .(?)

Or does maybe someone know an optimization method that computes
sensitivities?

Kind regards,
Martin

2017-04-20 9:46 GMT+01:00 Martin Neuenhofen <martinneuenhofen at googlemail.com
>:

> Dear all,
>
> I want to compute sensitivities of my Ipopt's solution's (i.e.
> x) performance ( i.e. f(x) ) with respect to a real scalar perturbation p,
> where d acts on the problem by modifying cL or cR by being cL:=cL + p * v,
> where v is a constant vector (or analogously for cR), where cL and cR are
> the left and right box borders of the non-linear constraints.
>
> The math is simple:
> Solving F(x,p)=0 for p=0 after x is the way how Ipopt solves for x, where
> F is (in an ideal world) the KKT-conditions. Now, having the performance
> function f, one computes the substantial derivative df/dp =
> delta_f/delta_p - delta_f/delta_x * (DF/Dx)^{-1} * DF/Dp . df/dp is the
> quantity of interest.
>
> However, the following questions occur:
>
> 1) Since Ipopt does actually not solve F but only approximately an
> approximation of F, I want to know if I could as well take any other
> eps-KKT conditions instead of F and yield good results. Does someone have
> experience in that?
>
> 2) The F of Ipopt is quite involved due to all the substitutions made from
> the original problem formulation to the one where x>=0 and c=0. I thought
> of always changing both cL and cR by p*v since then it should be simply
> c+p*v=0. Has yet someone tried computing sensitivities from solutions of
> Ipopt?
>
> Kind regards,
> Martin
>
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