[Ipopt] Towards comprehension of IPOPT convergence behaviours

[mneuen]

Hi All,

For a couple of practical problems, I ran into the following phenomena, 
that however I cannot reproduce in small-dimensional problems. I seek 
small-dimensional comprehensible problems for each for the following 
phenomena, respectively:

(a) The iteration count of IPOPT for a problem can change dramatically 
when the objective function is rescaled by a factor of only 10; and it 
can also affect the local minimum that IPOPT runs into.

(b) Initialized from a feasible starting point x0, IPOPT computes 
infeasible iterates that eventually become so infeasible that it does 
not find back to a feasible solution, and then it gives in. (Typical 
error messages are: Restoration failed - or - Restoration found feasible 
point that is unacceptable to the filter.)

(c) For mu=10^-4, the solution solves the eps-KKT equations. Update of 
mu=10^-5 then causes very small step-sizes caused by the 
globalization/step-acceptance procedure, although three iterations of 
full Newton would result in interior primal-dual points (x>0,y,z>0) with 
a KKT residual in norm <10^-10 for all subsequent values of mu.

I used standard options only, and always start with high-accuracy 
feasible (but clearly non-optimal) initial guess x0. My problems do not 
contain poles, and are well-scaled.

I am thankful to any educational examples or further explanations.

Cheers,
Martin