Improved Full-Newton Step O(nL) Infeasible Interior-Point Method for Linear Optimization

Abstract

We present several improvements of the full-Newton step infeasible interior-point method for linear optimization introduced by Roos (SIAM J. Optim. 16(4): 2006). Each main step of the method consists of a feasibility step and several centering steps. We use a more natural feasibility step, which targets the μ + -center of the next pair of perturbed problems. As for the centering steps, we apply a sharper quadratic convergence result, which leads to a slightly wider neighborhood for the feasibility steps. Moreover, the analysis is much simplified and the iteration bound is slightly better.

Improved Complexity Analysis of Full Nesterov–Todd Step Feasible Interior-Point Method for Symmetric Optimization

“…Subsequently, Gu et al [19] proposed an improved version of full Newton-step infeasible IPM for LO where the convergence analysis of the algorithm was simplified. However, it still maintains the currently best known iteration bound.…”

A Modified Infeasible-Interior-Point Algorithm for Linear Optimization Problems

“…2, we measure the proximity to the μ-center of the perturbed problems by the quantity σ (x, s; μ) as defined in (6). So, initially we have σ (x, s; μ) = 0, due to (11). In the sequel, we assume that at the start of each iteration, just before a μ-update, σ (x, s; μ) is smaller than or equal to a (small) threshold value τ > 0.…”

“…Mansouri et al [9,10] extended this algorithm on semidefinite optimization and linear complementarity problems (LCPs). Gu et al presented an improved IIPM for LO in [11]. In all of the mentioned works, the authors proposed algorithms with one feasibility step and several centering steps to get an optimal solution of underlying problems.…”

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

“…In order to get a tight approximation for this value, we first need to estimate d f x . This is done in the next section, where we follow a similar approach as in [15].…”