问题
I'm using IPOPT within Julia. My objective function will throw an error for certain parameter values (specifically, though I assume this doesn't matter, it involves a Cholesky decomposition of a covariance matrix and so requires that the covariance matrix be positive-definite). As such, I non-linearly constrain the parameters so that they cannot produce an error. Despite this constraint, IPOPT still insists on evaluating the objective function at paramaters which cause my objective function to throw an error. This causes my script to crash, resulting in misery and pain.
I'm interested why, in general, IPOPT would evaluate a function at parameters that breach the constraints. (I've ensured that it is indeed checking the constraints before evaluating the function.) If possible, I would like to know how I can stop it doing this.
I have set IPOPT's 'bound_relax_factor' parameter to zero; this doesn't help. I understand I could ask the objective function to return NaN instead of throwing an error, but when I do IPOPT seems to get even more confused and does not end up converging. Poor thing.
I'm happy to provide some example code if it would help.
Many thanks in advance :):)
EDIT:
A commenter suggested I ask my objective function to return a bad objective value when the constraints are violated. Unfortunately this is what happens when I do:
I'm not sure why Ipopt would go from a point evaluating at 2.0016x10^2 to a point evaluating at 10^10 — I worry there's something quite fundamental about IPOPT I'm not understanding.
Setting 'constr_viol_tol' and 'acceptable_constr_viol_tol' to their minimal values doesn't noticably affect optimisation, nor does 'over-constraining' my parameters (i.e. ensuring they cannot be anywhere near an unacceptable value).
回答1:
The only constraints that Ipopt is guaranteed to satisfy at all intermediate iterations are simple upper and lower bounds on variables. Any other linear or nonlinear equality or inequality constraint will not necessarily be satisfied until the solver has finished converging at the final iteration (if it can get to a point that satisfies termination conditions). Guaranteeing that intermediate iterates are always feasible in the presence of arbitrary non-convex equality and inequality constraints is not tractable. The Newton step direction is based on local first and second order derivative information, so will be an approximation and may leave the space of feasible points if the problem has nontrivial curvature. Think about the space of points where x * y == constant as an example.
You should reformulate your problem to avoid needing to evaluate objective or constraint functions at invalid points. For example, instead of taking the Cholesky factorization of a covariance matrix constructed from your data, introduce a unit lower triangular matrix L and a diagonal matrix D. Impose lower bound constraints D[i, i] >= 0 for all i in 1:size(D,1), and nonlinear equality constraints L * D * L' == A where A is your covariance matrix. Then use L * sqrtm(D) anywhere you need to operate on the Cholesky factorization (this is a possibly semidefinite factorization, so more of a modified Cholesky representation than the classical strictly positive definite L * L' factorization).
Note that if your problem is convex, then there is likely a specialized formulation that a conic solver will be more efficient at solving than a general-purpose nonlinear solver like Ipopt.
来源：https://stackoverflow.com/questions/36907064/why-does-ipopt-evaluate-objective-function-despite-breaching-constraints