It is well known that a large neighborhood interior point algorithm for linear optimization performs much better in implementation than its small neighborhood counterparts. One of the key elements of interior point algorithms is how to update the barrier parameter. The main goal of this paper is to introduce an "adaptive'' long step interior-point algorithm in a large neighborhood of central path using the classical logarithmic barrier function having O(nlog(x0)Ts0ϵ) iteration complexity analogous to the classical long step algorithms. Preliminary encouraging numerical results are reported.
In this paper we propose a primal-dual path-following interior-point algorithm for semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main step of the algorithm consists of a feasibility step and several centering steps. At each iteration, we use only full-Newton step. Moreover, we use a more natural feasibility step, which targets at the μ+-center. The iteration bound of the algorithm coincides with the currently best iteration bound for semidefinite optimization problems.
A full Nesterov-Todd step infeasible interior-point algorithm is proposed for solving linear programming problems over symmetric cones by using the Euclidean Jordan algebra. Using a new approach, we also provide a search direction and show that the iteration bound coincides with the best known bound for infeasible interior-point methods.