[MOI] Add support for nonlinear problems without Hessian

[frapac]

Solve #318

This PR adds support for MOI nonlinear problems without Hessian. This is needed to support problems with user-defined functions (whose support has been improved in the latest MOI releases). Now MadNLP switches automatically to the LBFGS algorithm if the Hessian is not available, as it is done in Ipopt.jl.

Currently, we specify to NLPModels that the problem does not have Hessian by setting nnzh=0 in NLPModelMeta, so the following code works if the Hessian callback is not defined:
https://github.com/MadNLP/MadNLP.jl/blob/fp/moi_nohessian/src/nlpmodels.jl#L360-L362
I am not sure this is the correct approach.

[franckgaga]

Thanks for this PR!
This a good compromise (clearly better than an error!) but according to JuMP stable documentation, both for the NLP Legacy syntax and the new NLP syntax, the Hessian is also automatically computed by AD with registered function/operators (with ForwardDiff.jl I think). LBFGS approximation should not required here.
Am I right @odow ?

[odow]

the Hessian is also automatically computed by AD with registered function

Only for univariate functions. We don't compute it for multivariate user-defined operators.

[franckgaga]

Oops, my bad. My interpretation of the doc was incorrect. This is indeed the best solution in this case.

[odow]

I think we kept the docs intentionally ambiguous, because it's a level of detail that shouldn't concern most people. The issue here is just a straight-up bug. You should check whether the AD backend supports :Hess before attempting to call it.

[franckgaga]

@odow I'm curious, why JuMP chose to skip the Hessian for multivariate registered functions ? For the user using the legacy syntax, would you recommend to call register with a ∇²fargument, and calling ForwardDiff.hessian! in the ∇²f method?

[sshin23]

LGTM. This reminds me of the possibility of piggybacking on Ipopt.jl/Knitro.jl...

[odow]

why JuMP chose to skip the Hessian for multivariate registered functions

Mainly because they can have many arguments, and computing the Hessian with ForwardDiff is expensive (because we'd need to compute the dense Hessian).

It would be breaking to enable this argument now.

The documentation has a tutorial on adding it manually if you're interested though:
https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/operator_ad/#Hessian-2