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

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 New Infeasible Interior-Point Method Based on a Non-Coercive Kernel Function with Improved Centering Steps for Second-Order Cone Optimization

Numerical Functional Analysis and Optimization ◽

10.1080/01630563.2015.1065886 ◽

2015 ◽

Vol 36 (10) ◽

pp. 1322-1338

Author(s):

B. Kheirfam

Keyword(s):

Kernel Function ◽

Interior Point ◽

Interior Point Method ◽

Second Order ◽

Point Method ◽

Second Order Cone ◽

Infeasible Interior Point Method

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Simplified analysis of a full Nesterov–Todd step infeasible interior-point method for symmetric optimization

Asian-European Journal of Mathematics ◽

10.1142/s1793557115500710 ◽

2015 ◽

Vol 08 (04) ◽

pp. 1550071 ◽

Cited By ~ 2

Author(s):

Behrouz Kheirfam

Keyword(s):

Interior Point ◽

Interior Point Method ◽

Interior Point Methods ◽

Point Method ◽

Central Path ◽

Complexity Bound ◽

Symmetric Optimization ◽

Iteration Bound ◽

Infeasible Interior Point Method ◽

Simplified Analysis

We give a simplified analysis and an improved iteration bound of a full Nesterov–Todd (NT) step infeasible interior-point method for solving symmetric optimization. This method shares the features as, it (i) requires strictly feasible iterates on the central path of a perturbed problem, (ii) uses the feasibility steps to find strictly feasible iterates for the next perturbed problem, (iii) uses the centering steps to obtain a strictly feasible iterate close enough to the central path of the new perturbed problem, and (iv) reduces the size of the residual vectors with the same speed as the duality gap. Furthermore, the complexity bound coincides with the currently best-known iteration bound for full NT step infeasible interior-point methods.

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