This paper concerns the formulation and analysis of a new interior method
for general nonlinearly constrained optimization that combines a shifted
primal-dual interior method with a projected-search method for
bound-constrained optimization. The method involves the computation of an
approximate Newton direction for a primal-dual penalty-barrier function
that incorporates shifts on both the primal and dual variables. Shifts on
the dual variables allow the method to be safely ``warm started'' from a
good approximate solution and eliminates the ill-conditioning of the
associated linear equations that may occur when the dual variables are
close to zero. The approximate Newton direction is used in conjunction
with a new projected-search line-search algorithm that employs a flexible
non-monotone quasi-Armijo line search for the minimization of each
penalty-barrier function. Numerical results show that the proposed method
requires fewer iterations than a conventional interior method, thereby
reducing the number of times that a search direction must be computed. In
particular, results from a set of quadratic programming test problems
indicate that the method is particularly well-suited to solving the
quadratic programming subproblem in a sequential quadratic programming
method for nonlinear optimization.
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