A full-Newton step infeasible interior-point method for linear optimization based on a trigonometric kernel function

Kheirfam, Behrouz; Haghighi, Masoumeh

doi:10.1080/02331934.2015.1080255

Cited by 9 publications

(14 citation statements)

References 12 publications

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“…In this section, we describe a new scalable iterative method for solving LP problem (38), called the "apex method". The apex method is based on the predictor-corrector framework and proceeds in two stages: quest (predictor) and target (corrector).…”

Section: Description Of Apex Methodsmentioning

confidence: 99%

“…Proposition 5. Let M ̸ = ∅ be the convex closed set representing the feasible region of LP problem (38):…”

Section: ∀I ∈ Imentioning

confidence: 99%

“…Let M ̸ = ∅ be the feasible region of LP problem (38), ψ(•) be the mapping defined by equation (95). The pseudoprojection ρ M (x) of the point x onto the feasible polytope M is the limit point of the sequence [x, ψ(x), ψ 2 (x), .…”

Section: Definitionmentioning

confidence: 99%

“…Later, Karmarkar [33] proposed a polynomial-time interior-point algorithm, which can be used in practice. This algorithm generated the whole field of modern interior-point methods [34], which are able to solve large-scale LP problems with millions of variables and millions of equations [35][36][37][38][39]. Moreover, these methods are self-correcting, and therefore provide high precision of calculations.…”

Section: Introductionmentioning

confidence: 99%

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Apex Method: A New Scalable Iterative Method for Linear Programming

Sokolinsky¹,

Sokolinskaya²

2023

Preprint

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The article presents a new scalable iterative method for linear programming called the “apex method”. The key feature of this method is constructing a path close to optimal on the surface of the feasible region from a certain starting point to the exact solution of linear programming problem. The optimal path refers to a path of minimum length according to the Euclidean metric. The apex method is based on the predictor-corrector framework and proceeds in two stages: quest (predictor) and target (corrector). The quest stage calculates a rough initial approximation of linear programming problem. The target stage refines the initial approximation with a given precision. The main operation used in the apex method is an operation that calculates the pseudoprojection, which is a generalization of the metric projection to a convex closed set. This operation is used both in the quest stage and in the target stage. A parallel algorithm using a Fejér mapping to compute the pseudoprojection is presented. An analytical estimation of the parallelism degree of this algorithm is obtained. Also, an algorithm implementing the target stage is given. The convergence of this algorithm is proven. An experimental study of the scalability of the apex method on a cluster computing system is described. The results of applying the apex method to solve problems from the Netlib-LP repository are presented.

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Section: Description Of Apex Methodsmentioning

confidence: 99%

“…Proposition 5. Let M ̸ = ∅ be the convex closed set representing the feasible region of LP problem (38):…”

Section: ∀I ∈ Imentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Apex Method: A New Scalable Iterative Method for Linear Programming

Sokolinsky¹,

Sokolinskaya²

2023

Preprint

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“…Основываясь на работе Хачияна, Кармаркар в работе [21] предложил метод внутренних точек, который демонстрирует полиномиальное время решения задачи ЛП и применим на практике. Итерационные алгоритмы, основанные на методе Кармаркара, способны решать сверхбольшие задачи ЛП с миллионами переменных и миллионами уравнений [22][23][24][25][26]. Более того, эти алгоритмы являются самокорректирующимися, и поэтому обеспечивают высокую точность вычислений.…”

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On an iterative method for solving linear programming problems on cluster computing systems

Sokolinsky

Sokolinskaya

2020

Numerical Methods and Programming (Vychislitel'nye Metody I Programmirovanie)

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Статья посвящена исследованию нового метода решения сверхбольших задач линейного программирования. Указанный метод получил название "апекс-метод". Апекс-метод работает по схеме предиктор-корректор. На фазе предиктор находится точка, лежащая на границе <em>n</em>-мерного многогранника, задающего допустимую область задачи линейного программирования. На фазе корректор организуется итерационный процесс, в результате которого строится последовательность точек, сходящаяся к точному решению задачи линейного программирования. В статье дается формальное описание апекс-метода и приводятся сведения о его параллельной реализации на языке C++ с использованием библиотеки MPI. Приводятся результаты масштабных вычислительных экспериментов на кластерной вычислительной системе по исследованию масштабируемости апекс-метода. The paper is devoted to a new method for solving large-scale linear programming (LP) problems. This method is called the apex-method. The apex-method uses the predictor–corrector framework. Thepredictor step calculates a point belonging to the feasible region of the LP problem. The corrector step calculates a sequence of points converging to the exact solution of the LP problem. The paper gives a formal description of the apex-method and provides information about its parallel implementation in C++ language using the MPI library. The results of large-scale computational experiments on a cluster computing system to study the scalability of the apex method are discussed.

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A full-Newton step infeasible interior-point method based on a trigonometric kernel function without centering steps

Kheirfam

Haghighi

2019

Numer Algor

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A full-Newton step infeasible interior-point method for linear optimization based on a trigonometric kernel function

Cited by 9 publications

References 12 publications

Apex Method: A New Scalable Iterative Method for Linear Programming

Apex Method: A New Scalable Iterative Method for Linear Programming

On an iterative method for solving linear programming problems on cluster computing systems

A full-Newton step infeasible interior-point method based on a trigonometric kernel function without centering steps

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