A Modified Infeasible-Interior-Point Algorithm for Linear Optimization Problems

In this paper, a primal-dual interior point algorithm for solving linear optimization problems based on a new kernel function with a trigonometric barrier term which is not only used for determining the search directions but also for measuring the distance between the given iterate and the ?-center for the algorithm is proposed. Using some simple analysis tools and prove that our algorithm based on the new proposed trigonometric kernel function meets O (?n log n log n/?) and O (?n log n/?) as the worst case complexity bounds for large and small-update methods. Finally, some numerical results of performing our algorithm are presented.

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A PRIMAL-DUAL INTERIOR-POINT ALGORITHM BASED ON A NEW KERNEL FUNCTION

The ANZIAM Journal ◽

10.1017/s1446181110000908 ◽

2010 ◽

Vol 51 (4) ◽

pp. 476-491 ◽

Cited By ~ 3

Author(s):

G. M. CHO ◽

Y. Y. CHO ◽

Y. H. LEE

Keyword(s):

Kernel Function ◽

Interior Point ◽

Optimization Problems ◽

Linear Optimization ◽

Interior Point Algorithm ◽

Linear Optimization Problems ◽

Primal Dual ◽

Search Directions

AbstractWe propose a new primal-dual interior-point algorithm based on a new kernel function for linear optimization problems. New search directions and proximity functions are proposed based on the kernel function. We show that the new algorithm has $\mathcal {O}(\sqrt {n} \log n \log ({n}/{\epsilon }))$ and $\mathcal {O}(\sqrt {n}\log ({n}/{\epsilon }))$ iteration bounds for large-update and small-update methods, respectively, which are currently the best known bounds for such methods.

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A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods

The ANZIAM Journal ◽

10.1017/s1446181100012827 ◽

2007 ◽

Vol 49 (2) ◽

pp. 259-270 ◽

Cited By ~ 15

Author(s):

Keyvan Aminis ◽

Arash Haseli

Keyword(s):

Interior Point ◽

Interior Point Methods ◽

Optimization Problems ◽

Linear Optimization ◽

Point Of View ◽

Theory And Practice ◽

Theoretical Point ◽

Barrier Functions ◽

Iteration Bound ◽

Linear Optimization Problems

AbstractInterior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.

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

Statistics Optimization & Information Computing ◽

10.19139/soic-2310-5070-1175 ◽

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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An interactive satisficing approach for solving fuzzy multiobjective Linear optimization problems based on the Attainable Reference Point Method

The International Conference on Mathematics and Engineering Physics ◽

10.21608/icmep.2010.29804 ◽

2010 ◽

Vol 5 (5) ◽

pp. 1-15

Author(s):

Taghreed Hassan ◽

A. Mousa

Keyword(s):

Reference Point ◽

Optimization Problems ◽

Linear Optimization ◽

Point Method ◽

Reference Point Method ◽

Linear Optimization Problems

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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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