On a Class of Superlinearly Convergent Polynomial Time Interior Point Methods for Sufficient LCP

A new class of infeasible interior point methods for solving sufficient linear complementarity problems requiring one matrix factorization andm backsolves at each iteration is proposed and analyzed. The algorithms from this class use a large (N− ∞) neighborhood of an infeasible central path associated with the complementarity problem and an initial positive, but not necessarily feasible, starting point. The Q-order of convergence of the complementarity gap, the residual, and the iteration sequence is m+1 for problems that admit a strict complementarity solution and (m+1)/2 for general sufficient linear complementarity problems. The methods do not depend on the handicap κ of the sufficient LCP. If the starting point is feasible (or “almost” feasible) the proposed algorithms have O((1 + κ)(1 + log m √ 1 + κ ) √ n L) iteration complexity, while if the starting point is “large enough” the iteration complexity is O((1 + κ)2+1/m(1 + log m √ 1 + κ )n L).