Unified Analysis of Kernel-Based Interior-Point Methods for \u3cem\u3eP\u3c/em\u3e *(κ)-LCP

We present an interior-point method for the P∗(κ)-linear complementarity problem (LCP) that is based on barrier functions which are defined by a large class of univariate functions called eligible kernel functions. This class is fairly general and includes the classical logarithmic function and the self-regular functions, as well as many non-self-regular functions as special cases. We provide a unified analysis of the method and give a general scheme on how to calculate the iteration bounds for the entire class. We also calculate the iteration bounds of both long-step and short-step versions of the method for several specific eligible kernel functions. For some of them we match the best known iteration bounds for the long-step method, while for the short-step method the iteration bounds are of the same order of magnitude. As far as we know, this is the first paper that provides a unified approach and comprehensive treatment of interior-point methods for P∗(κ)-LCPs based on the entire class of eligible kernel functions

A Superlinear Infeasible-Interior-Point Affine Scaling Algorithm For LCP

. We present an infeasible-interior-point algorithm for monotone linear complementarity problems in which the search directions are affine scaling directions and the step lengths are obtained from simple formulae that ensure both global and superlinear convergence. By choosing the value of a parameter in appropriate ways, polynomial complexity and convergence with Q-order up to (but not including) two can be achieved. The only assumption made to obtain the superlinear convergence is the existence of a solution satisfying strict complementarity. Key words. infeasible-interior-point methods, monotone linear complementarity problems, superlinear convergence 1. Introduction. The monotone linear complementarity problem (LCP) is to find a vector pair (x; y) 2 IR n 2 IR n that satisfies the following conditions: y = Mx+ q; (1.1a) x 0; y 0; (1.1b) x T y = 0; (1.1c) where M is a positive semidefinite matrix. We use S to denote the solution set of (1.1) and S c to denote the set of..