  Based on solving an equivalent parametric equality constrained mini-max
problem of the classic logarithmic-barrier subproblem, we present a novel
primal-dual interior-point relaxation method for nonlinear programs with
general equality and nonnegative constraints. In each iteration, our method
approximately solves the KKT system of a parametric equality constrained
mini-max subproblem, which avoids the requirement that any primal or dual
iterate is an interior-point. The method has some similarities to the
warmstarting interior-point methods in relaxing the interior-point requirement
and is easily extended for solving problems with general inequality
constraints. In particular, it has the potential to circumvent the jamming
difficulty that appears with many interior-point methods for nonlinear programs
and improve the ill conditioning of existing primal-dual interior-point methods
as the barrier parameter is small. A new smoothing approach is introduced to
develop our relaxation method and promote convergence of the method. Under
suitable conditions, it is proved that our method can be globally convergent
and locally quadratically convergent to the KKT point of the original problem.
The preliminary numerical results on a well-posed problem for which many
interior-point methods fail to find the minimizer and a set of test problems
from the CUTEr collection show that our method is efficient.
