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

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.

scholarly journals A Full-Newton step infeasible-interior-point algorithm for P*(k)-horizontal linear complementarity problems

2015 ◽  
Vol 25 (1) ◽  
pp. 57-72 ◽  
Author(s):  
S. Asadi ◽  
H. Mansouri
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Linear Optimization ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Central Path ◽  
Interior Point Algorithm ◽  
Newton Step ◽  
Infeasible Interior Point Method ◽  
Full Newton Step

In this paper we generalize an infeasible interior-point method for linear optimization to horizontal linear complementarity problem (HLCP). This algorithm starts from strictly feasible iterates on the central path of a perturbed problem that is produced by suitable perturbation in HLCP problem. Then, we use so-called feasibility steps that serves to generate strictly feasible iterates for the next perturbed problem. After accomplishing a few centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The complexity of the algorithm coincides with the best known iteration complexity for infeasible interior-point methods.

An improved and modified infeasible interior-point method for symmetric optimization

2016 ◽  
Vol 09 (03) ◽  
pp. 1650059 ◽  
Author(s):  
Behrouz Kheirfam
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Interior Point Methods ◽  
Linear Optimization ◽  
Point Method ◽  
Semidefinite Optimization ◽  
Symmetric Optimization ◽  
Second Order Cone ◽  
Iteration Bound ◽  
Infeasible Interior Point Method

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.

A FULL-NEWTON STEP INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR OPTIMIZATION BASED ON AN EXPONENTIAL KERNEL FUNCTION

2014 ◽  
Vol 07 (01) ◽  
pp. 1450018
Author(s):  
Behrouz Kheirfam ◽  
Fariba Hasani
Keyword(s):  
Kernel Function ◽  
Interior Point ◽  
Interior Point Method ◽  
Polynomial Complexity ◽  
Linear Optimization ◽  
Interior Point Algorithm ◽  
Math Model ◽  
Newton Step ◽  
Infeasible Interior Point Method ◽  
Full Newton Step

This paper deals with an infeasible interior-point algorithm with full-Newton step for linear optimization based on a kernel function, which is an extension of the work of the first author and coworkers (J. Math. Model Algorithms (2013); DOI 10.1007/s10852-013-9227-7). The main iteration of the algorithm consists of a feasibility step and several centrality steps. The centrality step is based on Darvay's direction, while we used a kernel function in the algorithm to induce the feasibility step. For the kernel function, the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods.

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

2015 ◽  
Vol 08 (04) ◽  
pp. 1550071 ◽  
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.

A wide neighborhood predictor–corrector infeasible-interior-point method for Cartesian P∗(κ)-LCP over symmetric cones

2016 ◽  
Vol 09 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Marzieh Sayadi Shahraki ◽  
Maryam Zangiabadi ◽  
Hossein Mansouri
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Jordan Algebras ◽  
Point Method ◽  
Symmetric Cones ◽  
Wide Neighborhood ◽  
Euclidean Jordan Algebras ◽  
Iteration Bound ◽  
Infeasible Interior Point Method ◽  
Predictor Corrector

In this paper, we present a predictor–corrector infeasible-interior-point method based on a new wide neighborhood of the central path for linear complementarity problem over symmetric cones (SCLCP) with the Cartesian [Formula: see text]-property. The convergence of the algorithm is proved for commutative class of search directions. Moreover, using the theory of Euclidean Jordan algebras and some elegant tools, the iteration bound improves the earlier complexity of these kind of algorithms for the Cartesian [Formula: see text]-SCLCPs.

scholarly journals A Full Nesterov-Todd Step Infeasible Interior-point Method for Symmetric Optimization in the Wider Neighborhood of the Central Path

2021 ◽  
Vol 9 (2) ◽  
pp. 250-267
Author(s):  
Lesaja Goran ◽  
G.Q. Wang ◽  
A. Oganian
Keyword(s):  
Interior Point ◽  
Interior Point Method ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Point Method ◽  
Central Path ◽  
Symmetric Optimization ◽  
Statistical Disclosure Limitation ◽  
Starting Point ◽  
Linear Optimization Problems

In this paper, an improved Interior-Point Method (IPM) for solving symmetric optimization problems is presented. Symmetric optimization (SO) problems are linear optimization problems over symmetric cones. In particular, the method can be efficiently applied to an important instance of SO, a Controlled Tabular Adjustment (CTA) problem which is a method used for Statistical Disclosure Limitation (SDL) of tabular data. The presented method is a full Nesterov-Todd step infeasible IPM for SO. The algorithm converges to ε-approximate solution from any starting point whether feasible or infeasible. Each iteration consists of the feasibility step and several centering steps, however, the iterates are obtained in the wider neighborhood of the central path in comparison to the similar algorithms of this type which is the main improvement of the method. However, the currently best known iteration bound known for infeasible short-step methods is still achieved.

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