The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrix A . The main drawback of the algorithm is the use of an unknown big constant x, in computing the search direction and to initiate the algorithm. We propose a modified layered-step in...

We introduce a new barrier function which is not a barrier function in the usual sense: it has finite value at the boundary of the feasible region. Despite this, its iteration bound, O (√ n logn log n ε ) , is as good as it can be: it is the best known bound for large-update methods. The recently introduced notions of superconvexity and exponential convexity are crucial in the analysis.

Re-optimization techniques for an interior point method applied to solve a sequence of linear programming problems are discussed. Conditions are given for problem perturbations that can be absorbed in merely one Newton step. The analysis is performed for both short-step and long-step feasible path-following method. A practical procedure is then derived for an infeasible path-following method. I...

This paper describes interior point methods for nonlinear programming endowed with infeasibility detection capabilities. The methods are composed of two phases, a main phase whose goal is to seek optimality, and a feasibility phase that aims exclusively at improving feasibility. A common characteristic of the algorithms is the use of a step-decomposition interior-point method in which the step ...

The interior trust region algorithm for convex quadratic programming is further developed. This development is motivated by the barrier function and the \center" path-following methods, which create a sequence of primal and dual interior feasible points converging to the optimal solution. At each iteration, the gap between the primal and dual objective values (or the complementary slackness val...

An interior point method is proposed to solve variational inequality problems for monotone functions and polyhedral sets. The method has the following advantages. 1. Given an initial interior feasible solution with duality gap 0 , the algorithm requires at most On log(0 ==)] iterations to obtain an-optimal solution. 2. The rate of convergence of the duality gap is q-quadratic. 3. At each iterat...

We present an improved version of a full Nesterov-Todd step infeasible interior-point method for linear complementarityproblem over symmetric cone (Bull. Iranian Math. Soc., 40(3), 541-564, (2014)). In the earlier version, each iteration consisted of one so-called feasibility step and a few -at most three - centering steps. Here, each iteration consists of only a feasibility step. Thus, the new...

In the adaptive step primal dual interior point method for linear programming polynomial algorithms are obtained by computing Newton directions towards targets on the central path and restricting the iterates to a neighborhood of this central path In this paper the adaptive step methodology is extended by considering targets in a certain central region which contains the usual central path and ...

A modern mathematical proof is not very diierent from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. Abstract In this paper the duality theory of Linear Optimization (LO) is built up based on ideas emerged from interior point methods. All we need is elementary calculus. We will embed the LO problem a...