Tune the default linear solver

[klamike]

@sshin23 @frapac perhaps the Mumps defaults should be re-evaluated? Especially since Mumps is the default linear solver now.

Originally posted by @klamike in #602

Maybe we can even have multiple presets per linear solver, like a "fast" config and a "robust" config..

[frapac]

That's a legit question. By default, we solve a regularized augmented KKT system with a SQD structure, so I think we should use signed Cholesky (LDL) for performance. We can tune the options in Mumps to avoid numerical pivoting, and make it behave like a classical LDL factorization.

[apozharski]

To put in my 0.02€, we have observed over several generations of SYSCOP PhD students (and more recently through some industry collaboration) that the default mumps_pivtol=1e-6 is tantamount to sabotage when it comes to the performance of mumps on problems we solve here (mostly OCPs). In my experience many problems fail to converge reasonably unless this is adjusted to at least 1e-3 and often higher than that.

[frapac]

@apozharski That's a very legit concern.

I would suggest the following plan: we run a benchmark on CUTEst, testing the different parameters we can use in Mumps. And identifies which ones work best and which ones are the most robust.

According to your comment, I think we would have to increase mumps_pivtol in our default.

[michel2323]

@amontoison This also fits with your suggestion on the Schur KKT default solver.