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Apex Method: A New Scalable Iterative Method for Linear Programming
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These authors contributed equally to this work.
Version 1
: Received: 13 March 2023 / Approved: 16 March 2023 / Online: 16 March 2023 (03:26:48 CET)
Version 2 : Received: 23 March 2023 / Approved: 24 March 2023 / Online: 24 March 2023 (03:48:13 CET)
Version 2 : Received: 23 March 2023 / Approved: 24 March 2023 / Online: 24 March 2023 (03:48:13 CET)
A peer-reviewed article of this Preprint also exists.
Sokolinsky, L.B.; Sokolinskaya, I.M. Apex Method: A New Scalable Iterative Method for Linear Programming. Mathematics 2023, 11, 1654. Sokolinsky, L.B.; Sokolinskaya, I.M. Apex Method: A New Scalable Iterative Method for Linear Programming. Mathematics 2023, 11, 1654.
Abstract
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.
Keywords
linear programming; apex method; iterative method; projection-type method; Fejér mapping; parallel algorithm; cluster computing system; scalability evaluation; Netlib-LP repository
Subject
Computer Science and Mathematics, Computational Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Commenter: Leonid Sokolinsky
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