Infeasible Interior-Point Methods for Linear Optimization Based on Large Neighborhood

In this paper, we design a class of infeasible interior-point methods for linear optimization based on large neighborhood. The algorithm is inspired by a full-Newton step infeasible algorithm with a linear convergence rate in problem dimension that was recently proposed by the second author. Unfortunately, despite its good numerical behavior,...

An Improved and Simplified Full-Newton Step O(n) Infeasible Interior-Point Method for Linear Optimization

We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called feasibility step and a few---at most three---centering steps. In this paper each iteration consists of only a feasibility step, whereas the iteration bound improves the...

Counterexample to a Conjecture on an Infeasible Interior-Point Method

In [SIAM J. Optim., 16 (2006), pp. 1110–1136], Roos proved that the devised full-step infeasible algorithm has $O(n)$ worst-case iteration complexity. This complexity bound depends linearly on a parameter $\bar{\kappa}(\zeta)$, which is proved to be less than $\sqrt{2n}$. Based on extensive computational evidence (hundreds of thousands of...

Generic primal-dual interior point methods based on a new kernel function

In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in

New Primal-dual Interior-point Methods Based on Kernel Functions

Two important classes of polynomial-time interior-point method (IPMs) are small- and large-update methods, respectively. The theoretical complexity bound for large-update methods is a factor $\sqrt{n}$ worse than the bound for small-update methods, where $n$ denotes the number of (linear) inequalities in the problem. In practice the situation is...