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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.

A Wide Neighborhood Second-order Predictor-corrector Interior-point Algorithm for Semidefinite Optimization with Modified Corrector Directions

Kheirfam B., Mohamadi-Sangachin M.

Fundamenta Informaticae

2017

Vol. 153, nr 4

327--346

In this paper, we present a new second-order predictor-corrector interior-point method for semidefinite optimization. The algorithm is based on the wide neighborhood of the central path and modified corrector directions. In the corrector step, we derive the step size and corrector directions which guarantee that new iterate lies in the wide neighborhood. The iteration complexity bound is O(√nlog X0•S0/ɛ) for the Nesterov-Todd direction, which coincides with the best known complexity results for semidefinite optimization. Some numerical results are provided as well.

A Predictor-corrector Infeasible-interior-point Algorithm for Semidefinite Optimization in a Wide Neighborhood

Kheirfam B.

Fundamenta Informaticae

2017

Vol. 152, nr 1

33--50

In this paper, we propose a predictor-corrector infeasible interior-point algorithm for semidefinite optimization based on the Nesterov-Todd scaling scheme. In each iteration, the algorithm computes the new iterate using a new combination of the predictor and corrector directions. Using the Ai-Zhang's wide neighborhood for linear complementarity problems, and extended to semidefinite optimization by Li and Terlaky, it is shown that the iteration complexity bound of the algorithm is O(n5/4 log ɛ-1 1), where n is the dimension of the problem and ɛ is the required precision.