Optimization errors with k > 7~ and poor scaling with k

[deklanw]

Testing on empirical networks I noticed that past around 7 communities the constrained optimization will fail sometimes,

 Warning: Problem status NUMERICAL_ERROR; solution may be inaccurate.

Also that the optimization succeeded is never checked. I think spurious solutions can be accepted. Tweaking it to just ignore those cases in eval_relocate is easy enough

solve!(problem, ECOS.Optimizer(verbose=false))

# println("Optval = ", -problem.optval)
return problem.status, -problem.optval

problem_status, sbm_cost = solve_convex(sol.data, m_, kappa_, omega)

if sbm_cost < sol.ll && problem_status == Convex.MOI.OPTIMAL

I'm not sure if there is some ECOS setting that can be tweaked to solve these problematic cases or not. Around the same number of communities (7~) (and above) the algorithm slows down considerably. When I flip the CONSTRAINED flag the algorithm doesn't slow down noticeably at the same point. I wonder now if these optimization errors can be worked around will the algorithm scale more closely with the unconstrained case?

Thanks for any thoughts