An arc-search infeasible interior-point method for semidefinite optimization with the negative infinity neighborhood

In this paper an improved and modified version of full Nesterov–Todd step infeasible interior-point methods for symmetric optimization published in [A new infeasible interior-point method based on Darvay’s technique for symmetric optimization, Ann. Oper. Res. 211(1) (2013) 209–224; G. Gu, M. Zangiabadi and C. Roos, Full Nesterov–Todd step infeasible interior-point method for symmetric optimization, European J. Oper. Res. 214(3) (2011) 473–484; Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization, Asian-Eur. J. Math. 8(4) (2015) 1550071, 14 pp.] is considered. Each main iteration of our algorithm consisted of only a feasibility step, whereas in the earlier versions each iteration is composed of one feasibility step and several — at most three — centering steps. The algorithm finds an [Formula: see text]-solution of the underlying problem in polynomial-time and its iteration bound improves the earlier bounds factor from [Formula: see text] and [Formula: see text] to [Formula: see text]. Moreover, our method unifies the analysis for linear optimization, second-order cone optimization and semidefinite optimization.

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A Second-Order Corrector Infeasible Interior-Point Method for Semidefinite Optimization Based on a Wide Neighborhood

Journal of Scientific Computing ◽

10.1007/s10915-020-01384-w ◽

2021 ◽

Vol 86 (1) ◽

Author(s):

B. Kheirfam ◽

A. Nasrollahi ◽

M. Mohammadi

Keyword(s):

Interior Point ◽

Interior Point Method ◽

Second Order ◽

Point Method ◽

Semidefinite Optimization ◽

Wide Neighborhood ◽

Infeasible Interior Point Method

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

Fundamenta Informaticae ◽

10.3233/fi-2020-1983 ◽

2020 ◽

Vol 177 (2) ◽

pp. 141-156

Author(s):

Behrouz Kheirfam

Keyword(s):

Interior Point ◽

Interior Point Method ◽

Linear Optimization ◽

Point Method ◽

Central Path ◽

Interior Point Algorithm ◽

Positive Direction ◽

Complexity Result ◽

Infeasible Interior Point Method ◽

Predictor Corrector

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.

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