  We prove that the classic logarithmic barrier problem is equivalent to a
particular logarithmic barrier positive relaxation problem with barrier and
scaling parameters. Based on the equivalence, a line-search primal-dual
interior-point relaxation method for nonlinear programs is presented. Our
method does not require any primal or dual iterates to be interior-points,
which is prominently different from the existing interior-point methods in the
literature. A new logarithmic barrier penalty function dependent on both primal
and dual variables is used to prompt the global convergence of the method,
where the penalty parameter is updated adaptively. Without assuming any
regularity condition, it is proved that our method will terminate at an
approximate KKT point of the original problem provided the barrier parameter
tends zero. Otherwise, either an approximate infeasible stationary point or an
approximate singular stationary point of the original problem will be found.
Some preliminary numerical results are reported, including the results for a
well-posed problem for which many line-search interior-point methods were
demonstrated not to be globally convergent, a feasible problem for which the
LICQ and the MFCQ fail to hold at the solution and an infeasible problem, and
for some standard test problems of the CUTE collection. These results show that
our algorithm is not only efficient for well-posed feasible problems, but also
is applicable for some ill-posed feasible problems and some even infeasible
problems.
