Infeasible - Start Primal - Dual Methodsand

In this paper we present several \infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to nd a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an-solution for an-perturbation of an initial strictly (primal and dual) feasible problem in O(p ln f) iterations, where is the parameter of a self-concordant barrier for the cone, is a relative accuracy and f is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O(p ln) iterations, where is a primal or dual infeasibility measure.