Sensitivities

[mzanon]

Hi guys, does HPIPM provide the possibility to compute parametric sensitivities, similar to what was possible in HPMPC? If yes, is there a minimal example on how to call that in Python?

Thanks a lot in advance!
Best,
Mario

[giaf]

Hi Mario!

Yes HPIPM provides functions to compute both the forward and the adjoint sensitivities https://github.com/giaf/hpipm/blob/master/include/hpipm_d_ocp_qp_ipm.h#L269

For the moment, they are only available in C and not part of the python interface, but they can be added if needed.
Do you have any suggestion how a convenient python interface for the sensitivities would look like?
The C one is rather low level and requires to pass the seeds in a "sol" struct, so it requires to know internal details like how variables are ordered and the like.

[mzanon]

Thanks Gian, by browsing the code I saw that in C this is possible, which is why I asked about Python. I think that this is a difficult choice and I would need to think well about it. In general, I am a medium-level guy, i.e., I do not like interfaces that are too high level, as they are often obscure and possibly not flexible enough. At the same time, interfaces that are too low level require too much work to be used. In this case I think that I would personally like an interface in which I have to pass the seed as a sol struct. In that case one simply needs to be made aware of the correct ordering of variables, but if one wants to compute the correct sensitivities one needs anyway to compute the partial derivatives of the Lagrangian gradient, so one has to anyway be aware of all involved variables. The other thing one needs to know is the convention used for the Lagrangian, so as long as these two ingredients are provided in the docs one can figure out the remaining part. In the end, if making such an interface is not too much effort, you can consider having a more high level one later, though I wonder how that would look like...

[giaf]

I finally got some time to work on this. So far, I have modified the C code to pass the seeds as a sol struct, and to get the sensitivities back as another sol struct. e119a4f ff65579
The corresponding C example is here https://github.com/giaf/hpipm/blob/master/examples/c/example_d_ocp_qp_sens.c#L447
If you are fine with that next time I'll add to the python interface in a similar way.

[mzanon]

if I get your code correctly, in your example you compute the sensitivities with respect to the first component of the initial state x_0. I guess that using the same code one could alternatively compute the sensitivities with respect to changes in the cost gradient or constraint bound value, etc. Can one as well get the sensitivities with respect to the cost Hessian or constraint Jacobian? If not, it should anyway be easy to do so by applying the chain rule, so no big issue there I'd say.

[giaf]

I added the forward solution sensitivities to the python solver (just the basic one for now, not "solver2" that also removes x0 instead of setting upper and lower bounds equal) and an example of the sensitivity wrt changes in the first component of x0. 474730d

The sensitivities are more general than just cost gradient or constraint bounds, you just need to pass the proper seeds.
For example in acados just about anything in the OCP QP formulation can be parametric (nonlinear cost, constraints, dynamic) and sensitivities are computed thanks to HPIPM.

Comments and feedback are welcome, as well as any idea for a nice example.

[giaf]

Last week I also added the sensitivities to the python OCP QP solver supporting x0 removal (i.e. "solver2"), this improves numerical accuracy.
Since I'm at it, I'll also add sensitivities to the dense QP solver.