Convergent Infeasible Interior-Point Trust-Region Methods for Constrained Minimization

We study an infeasible interior-point trust-region method for constrained minimization. This method uses a logarithmic-barrier function for the slack variables and updates the slack variables using second-order correction. We show that if a certain set containing the iterates is bounded and the origin is not in the convex hull of the nearly active constraint gradients everywhere on this set, then any cluster point of the iterates is a 1st-order stationary point. If the cluster point satisfies an additional assumption (which holds when the constraints are linear or when the cluster point satisfies strict complementarity and a local error bound holds), then it is a 2nd-order stationary point.