A Modified Full-Newton Step Infeasible Interior-Point Algorithm for Linear Optimization

Abstract

We present a full-Newton step infeasible interior-point algorithm based on a new search direction. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. We show that the algorithm converges and finds an approximate solution in a polynomial time complexity. A numerical study is done for its numerical performance.

CurveLP-A MATLAB implementation of an infeasible interior-point algorithm for linear programming

“…We emphasize that the second derivatives are functions of σ k which is selected by using a heuristic of [5] to speed up the convergence of the proposed algorithm. Several relations follow immediately from (12) and (14).…”

“…) be an arc defined by (11) passing through a point (x, λ, s) ∈ E ∩ H, and its first and second derivatives at (x, λ, s) be (ẋ,λ,ṡ) and (ẍ,λ,s) which are defined by (12) and (14). Then the ellipse approximation of H(t) is given by…”

“…Although many more algorithms have been proposed since then (see, for example, [10,11,12,13,14]), there is no significant improvement in the best polynomial bound for interior-point algorithms, and there is no report of a better algorithm than MPC for general linear programming problems 1 . In fact, the most popular interior-point method software packages implemented MPC, for example, LOQO [16], PCx [17] and LIPSOL [18].…”

A wide neighborhood infeasible-interior-point method with arc-search for linear programming

“…We emphasize that the second derivatives are functions of σ k which is selected by using a heuristic of [5] to speed up the convergence of the proposed algorithm. Several relations follow immediately from (12) and (14).…”

“…Equations (12) and (14) can be solved in either unreduced form, or augmented system form, or normal equation form as suggested in [1]. We solve the normal equations for (ẋ,λ,ṡ) and (ẍ,λ,s) as follows:…”