. In this paper we present a convergence analysis for some inexact (polynomial) variants of the infeasible-interior-point-algorithm of Kojima, Megiddo and Mizuno. For this analysis we assume that the iterates are bounded. The new variants allow the use of search directions that are calculated from the defining linear system with only moderate accuracy, e.g. via the use of Krylov subspace methods like CG or QMR. Furthermore, some numerical results for the proposed methods are given. Key words. Linear programming, infeasible-interior-point method, inexact search direction AMS subject classifications. 90C05, 65K05, 90C06 1. Introduction. Considering the fact that the primal-dual algorithm of Kojima, Megiddo and Mizuno [3] (henceforth called the KMM-Algorithm) does not use a predictor-corrector approach, it is surprising that it is an efficient algorithm for solving linear programming problems in practice. Of course, this efficiency does not only follow from the use of the Newton search...

An important advantage of infeasible interior-point methods compared to feasible interior-point methods is their ability to be warm-started from approximate solutions. It is therefore important for the convergence theory of these algorithms not to depend on being able to alter the starting point. In two recent papers, Yin Zhang and Stephen Wright prove convergence results for some infeasible interior-point methods. Unfortunately, their analysis places a restriction on the starting point. It is easy to meet the restriction by altering the starting point, but this may take the point farther away from the solution, removing the advantage of warm-starting the algorithms. In this paper we extend Zhang and Wright's results to apply to arbitrary strictly positive starting points. We then present an algorithm for solving the Box-Constrained Linear Complementarity problem and prove its convergence. 1 Introduction Quite often, in using an iterative method to solve a problem, it is possible to u...

In the recent past a number of papers were written that present low complexity interior-point methods for different classes of convex programs. Goal of this article is to show that the logarithmic barrier function associated with these programs is self-concordant, and that the analyses of interiorpoint methods for these programs can thus be reduced to the analysis of interior-point methods with self-concordant barrier functions. Key words: interior-point method, barrier function, dual geometric programming, (extended) entropy programming, primal and dual l p -programming, relative Lipschitz condition, scaled Lipschitz condition, self-concordance. 1 Introduction The efficiency of a barrier method for solving convex programs strongly depends on the properties of the barrier function used. A key property that is sufficient to prove fast convergence for barrier methods is the property of self-concordance introduced in [17]. This condition not only allows a proof of polynomial convergen...

In this work we define a trust-region feasible-point algorithm for approximating solutions of the nonlinear system of equalities and inequalities F(x, y) = 0, y ≥ 0, where F: R^n × R^m → R^p is continuously differentiable. This formulation is quite general; the Karush-Kuhn-Tucker conditions of a general nonlinear programming problem are an obvious example, and a set of equalities and inequalities can be transformed, using slack variables, into such form. We will be concerned with the possibility that n, m, and p may be large and that the Jacobian matrix may be sparse and rank deficient. Exploiting the convex structure of the local model trust-region subproblem, we propose a globally convergent inexact trust-region feasible-point algorithm to minimize an arbitrary norm of the residual, say ||F(x, y)||a, subject to the nonnegativity constraints. This algorithm uses a trust-region globalization strategy to determine a descent direction as an inexact solution of the local model trust-region subproblem and then, it uses linesearch techniques to obtain an acceptable steplength. We demonstrate that, under rather weak hypotheses, any accumulation point of the iteration sequence is a constrained stationary point for f = ||F||a, and that the sequence of constrained residuals converges to zero.