Scalability Evaluation of Cimmino Algorithm for Solving Linear Inequality Systems on Multiprocessors with Distributed Memory

Sokolinsky, Leonid B.; Sokolinskaya, Irina M.

doi:10.14529/jsfi180202

Cited by 8 publications

(4 citation statements)

References 19 publications

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“…The article [63] investigates the efficiency of parallelization of the Cimmino-like method on Xeon Phi manycore processors. In [64], the scalability of the Cimmino method for multiprocessor systems with distributed memory is evaluated. The applicability of the Cimmino-like method for solving non-stationary systems of linear inequalities on computing clusters is considered in [65].…”

Section: Related Workmentioning

confidence: 99%

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: Related Workmentioning

confidence: 99%

Apex Method: A New Scalable Iterative Method for Linear Programming

Sokolinsky¹,

Sokolinskaya²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The article [39] investigates the efficiency of parallelization of the Cimmino-like method on Xeon Phi manycore processors. In [40], the scalability of the Cimmino method for multiprocessor systems with distributed memory is evaluated. The applicability of the Cimmino-like method for solving non-stationary systems of linear inequalities on computing clusters is considered in [41].…”

Section: Related Workmentioning

confidence: 99%

Apex Method: A New Scalable Iterative Method for Linear Programming

Sokolinsky

Sokolinskaya

2023

Mathematics

View full text Add to dashboard Cite

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 a linear programming problem. The optimal path refers to a path of the 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 the 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. AlsoAdditionally, 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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“…Finding such an interior point can be reduced to solving an additional LP problem. An alternative is the iterative projection-type methods [23,26,31], which are also self-correcting. Third, like the simplex method, the inner point method does not scale well on multiprocessor systems with distributed memory.…”

Section: Introductionmentioning

confidence: 99%

Visualizing Multidimensional Linear Programming Problems

Olkhovsky,

Sokolinsky

2022

Preprint

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The article proposes an n-dimensional mathematical model of the visual representation of a linear programming problem. This model makes it possible to use artificial neural networks to solve multidimensional linear optimization problems, the feasible region of which is a bounded non-empty set. To visualize the linear programming problem, an objective hyperplane is introduced, the orientation of which is determined by the gradient of the linear objective function: the gradient is the normal to the objective hyperplane. In the case of searching a maximum, the objective hyperplane is positioned in such a way that the value of the objective function at all its points exceeds the value of the objective function at all points of the feasible region, which is a bounded convex polytope. For an arbitrary point of the objective hyperplane, the objective projection onto the polytope is determined: the closer the objective projection point is to the objective hyperplane, the greater the value of the objective function at this point. Based on the objective hyperplane, a finite regular set of points is constructed, called the receptive field. Using objective projections, an image of a polytope is constructed. This image includes the distances from the receptive points to the corresponding points of the polytope surface. Based on the proposed model, parallel algorithms for visualizing a linear programming problem are constructed. An analytical estimation of its scalability is performed. Information about the software implementation and the results of large-scale computational experiments confirming the efficiency of the proposed approaches are presented.

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Scalability Evaluation of Cimmino Algorithm for Solving Linear Inequality Systems on Multiprocessors with Distributed Memory

Cited by 8 publications

References 19 publications

Apex Method: A New Scalable Iterative Method for Linear Programming

Apex Method: A New Scalable Iterative Method for Linear Programming

Apex Method: A New Scalable Iterative Method for Linear Programming

Visualizing Multidimensional Linear Programming Problems

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