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A Second-order Corrector Infeasible Interior-point Method with One-norm wide Neighborhood for Symmetric Optimization

Kheirfam Behrouz

Fundamenta Informaticae

2020

Vol. 172, nr 4

343--359

In this article, we present a second-order corrector infeasible interior-point method based on one-norm large neighborhood for symmetric optimization. We consider the classical Newton direction as the sum of two other directions associated with the negative and positive parts of the right-hand side of the centrality equation. In addition to equipping them with different step lengths, we add a corrector step that is multiplied by the square of the step length in the expression of the new iterate. The convergence analysis of the algorithm is discussed and it is proved that the new algorithm has the same complexity as small neighborhood infeasible interior-point algorithms for the Nesterov-Todd (NT) direction, and the xs and sx directions.

A New Predictor-corrector Infeasible Interior-point Algorithm for Linear Optimization in a Wide Neighborhood

Kheirfam Behrouz

Fundamenta Informaticae

2020

Vol. 177, nr 2

141--156

In this paper, we propose a Mizuno-Todd-Ye type predictor-corrector infeasible interior-point method for linear optimization based on a wide neighborhood of the central path. According to Ai-Zhang’s original idea, we use two directions of distinct and orthogonal corresponding to the negative and positive parts of the right side vector of the centering equation of the central path. In the predictor stage, the step size along the corresponded infeasible directions to the negative part is chosen. In the corrector stage by modifying the positive directions system a full-Newton step is removed. We show that, in addition to the predictor step, our method reduces the duality gap in the corrector step and this can be a prominent feature of our method. We prove that the iteration complexity of the new algorithm is 𝒪 (n log ɛ −1 ), which coincides with the best known complexity result for infeasible interior-point methods, where ɛ > 0 is the required precision. Due to the positive direction new system, we improve the theoretical complexity bound for this kind of infeasible interior-point method [1] by a factor of √n . Numerical results are also provided to demonstrate the performance of the proposed algorithm.