I've spent the last few months dealing with the following problem:
s.t. c_i(x)=0 i=1..m
where x is in R^n, f(x) and c(x) are quadratic functions, having rank-deficient gradC(x) and gradF(x). I know that at least one solution exists.
I've tried SQP, line search, trust region, incremental loading, pseudo arc-length continutation, augmented lagrangian.. all failed. Is it possible that my problem is that difficult and that there is no relevant literature on the topic?
What I usually observe with SPQ-like methods, is that the constraints are violating very little (1e-14) at each iteration, but the residual of the first order optimality condition:
R(x,s) = gradf(x) - s*gradC(x),
where s are the Lagrange multipliers, never goes below 1e-7. Also, the solution x is very close to the starting guess, and the multipliers s are very high. And clearly x is not the minimizer I'm looking for.
Thanks for the help,