Emit warnings on invalid problems

[ericphanson]

It would be nice to emit a warning when e.g. P is not positive semi-definite. When the problem is put together through JuMP for example, it's not immediately clear how P is assembled, so you might not know whether or not P is PSD. From my JuMP model model I eventually could find P via

model.moi_backend.optimizer.model.optimizer.inner.p.P

and then check it's eigenvalues for example, but it would be nice if COSMO just emitted a warning when optimize! is called in such a case. In my example with my non-PSD P, it quickly reports it found an "optimal" solution and gives a valid feasible point as the solution, but since I know the true optimum in this case, I can see it's way off. Of course, COSMO isn't at fault since my problem doesn't fit the assumptions, but since optimal is such a strong word it would be nice if there was a warning to counter that.

[migarstka]

I think this is a good idea. I would like to keep the test outside the solver and have the check in the MOI-wrapper when the COSMO model is assembled.

One thing that I am unsure of is how to efficiently test if P is positive semidefinite symmetric. Julia only has a isposdef()function that doesn't account for a matrix P with zero-eigenvalues which COSMO allows.

Since P is handled as a sparse matrix there is currently no Julia Base function available that computes the eigenvalues of P without converting it to a full matrix. I would like to avoid the conversion since that could be potentially slow for very large problems.

eigs() could be used but has been moved to a separate package and I am not sure if it's worth adding an extra dependency just for that.

Edit: Until we find a better option: aeab91c

[ericphanson]

Thanks for the quick response! That seems helpful. Another idea would be to keep it entirely opt-in and provide some function

check_valid(::JuMP.AbstractModel)

and then at the start or end of the optimization (if in verbose mode) print something to tell the user that they can call check_valid on their model to verify the assumptions. That could get annoying though, since it would be printing every time.

edit: or instead of printing every time, maybe it would be enough to mention it in the docs

[goulart-paul]

One thing that I am unsure of is how to efficiently test if P is positive semidefinite symmetric. Julia only has a isposdef()function that doesn't account for a matrix P with zero-eigenvalues which COSMO allows.

Since P is handled as a sparse matrix there is currently no Julia Base function available that computes the eigenvalues of P without converting it to a full matrix. I would like to avoid the conversion since that could be potentially slow for very large problems.

It is not necessary to do this, since semidefiniteness of P can be inferred from the inertia of the KKT matrix. The KKT matrix is quasidefinite and will have a number of positive pivots equal to the dimension of P as long as P is semidefinite (strictly speaking, the number of positive pivots should equal the number of positive eigenvalues in P+sigma*I). For an LDL factorisation, checking for convexity therefore amounts to counting the positive elements in D.

Both QDLDL and Pardiso report information about matrix inertia as a side effect of factorization. This is used to throw an error about problem non-convexity when using either solver in the pg/lin_solvers branch.

[goulart-paul]

Edit: Until we find a better option: aeab91c

I think that this is not a good long-term solution, since P might be very large (or even just a Linear Map object). In cases where that's not true, we can catch indefiniteness as a side effect of the linear solves as explained above.

[migarstka]

Since QDLDL is now the default KKT system solver, we get the semidefiniteness check of P for free. This wasn't the case before.

Relevant: #85